Character Sums over p-adic Fields

Wen-Ching Winnie Li*

§1 Introduction.

Character sums have played a fundamental role in number theory and arithmetic algebraic geometry. To-date, most character sums are summing over points with coordinates lying in a finite field, and their estimates rely on the Riemann hypothesis for varieties over finite fields stated in the former Weil conjectures, which was proved by Hasse [H] and Weil [W1] for curves and by Deligne [D1] for varieties of higher dimension. This method originated from Weil’s work [W2], in which he estimated character sums of a certain type by constructing idèle class characters of rational function fields. The key idea is to find an idèle class character η so that the given character sum is the sum of η evaluated at a uniformizing element at each place of degree 1 where η is unramified. If η is not principal, then its associated L-function L(u, η) is a polynomial, which divides the numerator of the zeta function of some curve over a finite field, and hence the absolute value of the reciprocal roots of L(u, η) is known by the Riemann hypothesis for curves, from which a bound for the given character sum is derived. (See [W3, Appendix], [L1], [L2, §6.1] for more details.) His method can be interpreted geometrically, as Deligne did in [D2] for instance and followed by Katz. Weil’s bounds have had wide applications to many areas in pure and applied mathematics. In recent years, there is a new surge of interest in character sums because of their appearance in the explicit constructions of Ramanujan graphs. The reader is referred to [L2, Chap 9] and [L3] for more details.

The purpose of this paper is to estimate various character sums over points lying on C_{n} defined by the equation x^{p^{n}} = xz^{p^{n}−1} over the degree n unramified extension K_{n} of the p-adic field ℚ_{p}. More precisely, we shall consider sums of either an additive character evaluated at a rational function f(x) over K_{n} or a multiplicative character evaluated at a rational function g(x) over K_{n} or the product of the two over those points x ∈ C_{n}(K_{n}) where the relevant functions are defined, the same prototype as character sums over finite fields considered by Weil. Some classical character sums, though of different appearance, are of this type. For example, the Heilbronn sum

S_{a} = ∑_{0≤x≤p−1} e^{2πiax^{p}/p^{2}}

for a not divisible by p can be rephrased as

S_{a} = ∑_{x∈C_{1}(ℚ_{p})} ψ(ax),

where ψ is an additive character of ℚ_{p} of order -2. The consideration of such character sums in general form was initiated by coding theorists, who used them to construct families of periodic sequences with low correlations. The first one was the paper by Kumar, Helleseth and Calderbank [KHC], where the authors considered the sum over points of C_{n}(K_{n}) other than infinity of an additive character evaluated at a polynomial over the ring of integers O_{n} of K_{n}, the simplest kind of character sums over K_{n}. Later more complicated forms gradually evolved, although no one has obtained the general form contained in this paper. The reader is referred to the review article [HKS] by Helleseth, Kumar and Shanbhag and the references therein for more details. The results were proved case by case following Bombieri’s geometric method [B]. More precisely, instead of constructing an idèle class character η, one first searches for a curve E over a finite field so that L(u, η) is a part of the numerator of the zeta function of E, and then one computes the degree of L(u, η). In essence, this is Weil’s method carried out with more details instead of just appealing to class field theory to avoid finding an explicit E.

*Research supported in part by a grant from the National Security Agency no. MDA904-95-H-1006 and a grant from the National Science Foundation no. DMS-9622938

The method used in this paper is along the same line as [L1], by first constructing a suitable idèle class character η of a rational function field κ_{n}(x), κ_{n} being the residue field of K_{n}, and then derive an estimate of the given sum as described above. The bound for such a character sum is a constant M times p^{n/2} and M is the degree of the conductor of η minus 2. While it is not hard to obtain an upper bound of the conductor of η, much of the effort is devoted to determining the actual conductor so that M is as small as possible. We remark that the conductor of η depends on the smallest positive integer r such that the chosen additive (resp. multiplicative) character is trivial on p^{r}O_{n} (resp. 1 + p^{r}O_{n}), and is independent of n. Hence the estimate is good when n is large comparing to p^{r}, while the estimate could be worse than the trivial bound when n is small. In fact, it is a well-known difficult problem to estimate such character sums with small n and large r. It is not until 1995 that Heath-Brown [HB] found a nontrivial estimate of S_{a} by considering the 4th moment of S_{a}. Heath-Brown’s technique was further exploited by Fisher in [F] for more sums of Heilbronn’s type. Perhaps their methods can be adapted to obtain a better estimate for the case where Weil’s method fails. When r = 1 our results reduce to the known results over the finite field κ_{n}, as proved in [L2, Chap. 6] for the existence of idèle class characters and in [W2] and [B] for character sum estimates. Thus the results in this paper may be regarded as natural generalizations of the corresponding results over finite fields. When r > 1, the conductor of η is inflated. More precisely, over finite fields the conductor of the associated idèle class character is, roughly speaking, the sum of the divisor of zeros and the divisor of poles of the rational function(s) involved, while over p-adic fields, the conductor of η is a weighted sum over places in the support of the rational function(s) involved, and the weight depends on r and base p expansion of the functions involved. An interesting phenomenon is that the weight at the places 0 and ∞ is smaller than the weight at other places, and the weight for η attached to a multiplicative character is higher than that attached to an additive character. See Theorems 1-5 for more details. In some cases, the bound can be further improved using the technique introduced by Serre [S] and adapted by Moreno and Moreno [MM], as shown in the paper [HKMS] by Helleseth, Kumar, Moreno and Shanbhag. We shall not pursue this point in the paper.

In addition to deriving estimates of p-adic character sums, the knowledge of the conductor of an idèle class character η gives the exact degree of the attached Lfunction L(u, η). In a subsequent joint work with Ching-Li Chai we shall construct global L-functions by putting together suitable L(u, η) as local factors. When the degree n of L(u, η) is 2 or 3, we obtain automorphic L-functions for GL_{n} over the function field κ_{n}(x). In particular, this will give families of automorphic functions parametrized by rational functions over κ_{n}(x) which are eigenfunctions of Hecke operators with eigenvalues given by character sums almost everywhere.

Another consequence of our construction of idèle class characters is that, up to twists by principal characters, we obtain a parametrization by certain rational functions over O_{n} (Corollaries 2 and 3) of the idèle class characters of κ_{n}(x) with order a power of p. By class field theory, they can be viewed as characters of the Galois groups of wildly ramified finite abelian extensions of κ_{n}(x) with degree a power of p. On the other hand, such extensions are known to be parametrized by Witt vectors [J, Chap III]. Theorem 8 in §8 gives a comparison of these two parametrizations.

Character sum estimates obtained in this paper can be used to construct large families of periodic sequences with low cross-correlation. Such families find application in code-division multiple-access communication systems that employ phaseshift-keying modulation. This will be explained in §2. For more applications, consult the review paper [HKS].

Another way to describe the character sums considered in this paper is that they are character sums associated with rational functions on projective lines defined over characteristic zero p-adic fields. It would be interesting to investigate similar sums for curves of higher genus, in particular, to see if results by Bombieri in [B] can be generalized. In [VW], Voloch and Walker obtained some results in this direction for elliptic curves. The estimates for character sums over p-adic fields can be used to construct algebraic geometric codes with good parameters.

The paper is organized as follows. In §2 we state main results of this paper as well as applications. Their proofs are given in §§4-7 with ground work laid in §3. The comparison with the parametrization by Witt vectors is done in §8.

Acknowledgement Special thanks are due to Hendrik W. Lenstra, Jr. for stimulating conversations on Witt vectors, and to B. Fisher who simplified the proof of Proposition 1. Part of the results were proved while the author was visiting the Mathematical Sciences Research Institute at Berkeley in the spring of 1995. She would like to thank MSRI for its hospitality.

§2. Main results and applications.

Denote by K the field of p-adic numbers with residue field κ = ℤ/pℤ and let K̄ be its maximal unramified extension contained in an algebraic closure of K. The set T consisting of zero and the roots of unity in K̄ of order prime to p represents the residue field κ̄ of K̄. Each polynomial f̄ over κ̄ has a unique lifting to a polynomial f over K̄ such that the leading coefficient and the roots of f are the liftings in T of those of f̄ in κ̄. We shall call f the lifting of f̄. Clearly if f̄ has coefficients in a degree n extension κ_{n} of κ, then f has coefficients in the unique unramified degree n extension K_{n} of K contained in K̄. Denote by ∞ the place with uniformizer \frac{1}{x} of the rational function field κ_{n}(x) . For any place v of κ_{n}(x) other than ∞, denote by π_{v} the monic irreducible polynomial in κ_{n}[x] whose roots constitute the closed point represented by v. It generates the unique maximal ideal P_{v} of the ring of integers O_{v} of the completion of κ_{n}(x) at v. The integer p is a uniformizer in K̄ (and hence in all K_{n}); denote by P the maximal ideal of the ring of integers in K, and by O_{n} the ring of integers of K_{n}.

Theorem 1. Let ψ be an additive character of K of order −r, that is, ψ is trivial on P^{r} but not on P^{r−1}. Let f be a polynomial over O_{n} with f(0) = 0 and f( mod p^{r}) not identically 0. The base p expansion of f is

f(x) = f_{0}(x) + f_{1}(x)p + ⋯ + f_{r−1}(x)p^{r−1} + ⋯,

where f_{i} are polynomials with coefficients in T ∩ K_{n} and degree d_{i}. Then there exists a unique idèle class character η = η_{ψ,f} of κ_{n}(x) with the following properties:

(1.a) η is unramified at all places v of κ_{n}(x) other than ∞ and

η_{v}(π_{v}) = ψ ∘ Tr_{K_{n}/K}(∑_{γ} f(γ)),

where γ runs through all liftings in T of roots of π_{v}; (1.b) η_{∞} is trivial at the uniformizer π_{∞} = 1/x and κ_{n}^{×}; (1.c) The conductor of η is at most (D_{r,∞} + 1)∞, where

D_{r,∞} = max(d_{0}p^{r−1}, d_{1}p^{r−2}, ⋯, d_{r−1}),

called the weighted r-degree of f at ∞;

(1.d) η is the trivial character if D_{r,∞} = 0; if D_{r,∞} > 0 and none of the nonzero d_{i} with 0 ≤ i ≤ r − 1 is a multiple of p, then the conductor of η is (D_{r,∞} + 1)∞.

By convention, the degree of the zero polynomial is −∞.

Remarks. (1) Denote by σ the Frobenius automorphism of K̄ over K. Since Tr_{K_{m}/K}(z) = Tr_{K_{m}/K}(z^{p}) = Tr_{K_{m}/K}(z^{σ}) for any z in T ∩K_{m}, we get the same idèle class character using the polynomial g obtained from f = ∑_{j} a_{j}x^{j} by writing each term of f in the form a_{p^{i}s}x^{p^{i}s} with p and s coprime and replacing it by σ^{−i}(a_{p^{i}s})x^{s} so that no power of x occurring in g is a multiple of p. Note that the degree of g_{i} in the base p expansion of g is majorized by the degree of f_{i} and (1.d) holds for g. This gives a way to determine the conductor of η in case the conditions in (1.d) are not satisfied.

(2) Since any additive character ϕ of K_{n} with order −r is ψ ∘ Tr_{K_{n}/K} composed with multipication by a unit b in K_{n}, to change additive characters amounts to multiplying f by units, thus Theorem 1 is quite general.

As an application, we have the following character sum estimate:

Corollary 1.([KHC], [Du]) Let ψ, f and D_{r,∞} be as in Theorem 1. Assume that none of the powers of x occurring in f are divisible by p. If not all coefficients of f are multiples of p^{r}, then

| ∑_{x∈T ∩K_{n}} ψ ∘ Tr_{K_{n}/K}(f(x))| ≤ (D_{r,∞} − 1)p^{n/2}.

Proof. Let η be the idèle class character of κ_{n}(x) as in Theorem 1. It has conductor (D_{r,∞} + 1)∞ because of the conditions on f. Hence its associated L-function

L(u, η) = ∏_{v≠∞}(1 − η_{v}(π_{v})u^{deg v})^{−1}

is a polynomial in u of degree D_{r,∞} − 1. The Riemann hypothesis for curves over finite field proved by Weil [W1] implies that the reciprocial roots of L(u, η) have absolute value p^{n/2}. Therefore

| ∑_{deg v=1,v≠∞} η_{v}(π_{v})| ≤ (D_{r,∞} − 1)p^{n/2}.

For more details of the derivation of this fact, the reader is referred to Chapter 6 of [L2]. As the places of κ_{n}(x) of degree one other than ∞ correspond to elements a in κ_{n} and π_{a} = x − a, we have, by (1.a),

η_{a}(x − a) = ψ ∘ Tr_{K_{n}/K}(f(a)).

Substituting this into the above inequality yields Corollary 1.

I proved Theorem 1 and Corollary 1 back in 1993 after reading the preliminary version of [KHC] in which Corollary 1 was proved only for p = 2. The paper [KHC] was rewritten in the current published version after I informed the authors that I could prove their result for all p. Duursma gave another proof of Corollary 1 in [Du] by using a little bit class field theory, which is essentially the same as our proof but seen from the standpoint of Galois groups of abelian extensions of κ_{n}(x) unramified outside ∞. This is very closely related to Corollary 2 below.

When r = 1, we may pass f to the residue field κ_{n} of K_{n}, and the result in Theorem 1 was proved by Weil (cf. [W2], [L2, Chap 6, Theorem 4]). It is worth pointing out the difference of the idèle class characters η_{ψ,f} of κ_{n}(x) obtained using polynomials f over the finite field κ_{n} and over the p-adic field K_{n}. According to Theorem 1, the characters η_{ψ,f} attached to polynomials f(x) over κ_{n}, (that is, ψ has order r = 1,) will never have conductor of the form (Mp + 1)∞, while using polynomials over O_{n} we can get all idèle class characters of κ_{n}(x) unramified outside ∞. For example, if we choose ψ of order 2, then, by Theorem 1, η_{ψ,f} has conductor (p + 1)∞ for f of the form ax + p(b_{1}x + ⋯ + b_{p−1}x^{p−1}), where all coefficients are in T ∩ K_{n} and a ≠ 0. The precise statement is

Corollary 2. Given a positive integer D, let r be the integer such that p^{r−1} ≤ D < p^{r}. Let ψ be an additive character of K_{n} of order −r. Then η_{ψ,f} parametrizes the idèle class characters of κ_{n}(x), trivial at π_{∞}, with conductor M∞, M ≤ D + 1, as f runs through the polynomials of the form

∑_{0≤t≤r−1} p^{t} ∑_{m≥1,p∤m,p^{r−1−t}m≤D} c_{t,m}x^{m},  c_{t,m} ∈ T ∩ K_{n}.

Proof. Observe that there are no idèle class characters of κ_{n}(x) of conductor ∞. In fact, the idèle class characters of κ_{n}(x), trivial at π_{∞}, with conductor M∞, M ≤ D + 1, form a group dual to the quotient of the group of units U_{∞} at ∞ divided by κ_{n}^{×}(1 + P_{∞}^{D+1}), which is isomorphic to (1 + P_{∞})/(1 + P_{∞}^{D+1}). Here P_{∞} is the maximal ideal of the ring of integers in the completion of κ_{n}(x) at ∞. In particular, there are p^{nD} such characters. On the other hand, by Theorem 1, for each f as given, η_{ψ,f} has the properties as described, and the number of f is precisely p^{nD}. Thus it suffices to check that the η_{ψ,f}’s are distinct. Indeed, if f_{1} and f_{2} are two distinct polynomials as described, then f = f_{1} − f_{2} is a polynomial with integral coefficients, not identically zero modulo p^{r}, and no powers of x occurring in f is a multiple of p. Hence by Theorem 1, η_{ψ,f} = η_{ψ,f_{1}}η_{ψ,f_{2}}^{−1} is nontrivial. This proves Corollary 2.

The technique used to prove Theorem 1 can be extended to prove a similar result where f is a rational function with integral coefficients whose denominator is the lift of a polynomial over κ̄_{n}. By Chinese Remainder Theorem, we may decompose f as a sum of a polynomial and rational functions whose denominator is a power of an irreducible polynomial. Call each summand a primary component of f. Since ψ is an additive character, it suffices to work out the case where the denominator of f is a power of a monic irreducible polynomial g(x) over K_{n} with roots lying in T and the numerator of f is a polynomial over the ring of integers O_{n} of K_{n} with degree less than the degree of the denominator. Let α_{1}, ..., α_{a} be the roots of g, where a = deg g. Then we can further write f as the sum h_{1}((x−α_{1})^{−1})+⋯+h_{a}((x−α_{a})^{−1}), where h_{1}, ..., h_{a} are polynomials over the ring of integers of K_{na} with no constant term. The base p expansion of h_{i} is

h_{i}(x) = h_{0,i}(x) + ph_{1,i}(x) + ⋯ + p^{r−1}h_{r−1,i}(x) + ⋯,

where h_{j,i}(x) are polynomials of degree d_{j,i} with coefficients in T ∩ K_{na}. Since h_{i}((x−α_{i})^{−1})’s are conjugate under Gal(K_{na}/K_{n}), the degrees d_{j,i} are independent of i; denote the common value by d_{j}. Let w be the place of κ_{n}(x) such that π_{w} = ḡ. The weighted r-degree of f at w is defined to be

D_{r,w} = max{p^{r−1−j}(d_{j} + 1) − 1 : 0 ≤ j ≤ r − 1}  if w ≠ 0,
= max{p^{r−1−j}d_{j} : 0 ≤ j ≤ r − 1} if w = 0.

Note that when r = 1, we have D_{1,w} = d_{0}, the order of pole of f̄ at w.

Theorem 2. Let ψ be as in Theorem 1 and f be a quotient of two relatively prime polynomials over O_{n} with denominator a power of a monic irreducible polynomial g whose roots lie in T and deg f < 0. Denote by w the place with π_{w} = ḡ and define the weighted r-degree D_{r,w} of f at w as above. Then there exists a unique idèle class character η = η_{ψ,f} of κ_{n}(x) with the following properties:

(2.a) η is unramified at all places v of κ_{n}(x) other than w;

(2.b) η_{∞} is trivial, and at a place v ≠ w, ∞ we have

η_{v}(π_{v}) = ψ ∘ Tr_{K_{n}/K}(∑_{γ} f(γ)),

where γ runs through all liftings in T of roots of π_{v}; (2.c) The conductor of η is at most (D_{r,w} + 1)w; (2.d) If D_{r,w} = 0, then η is the trivial character. Suppose D_{r,w} > 0. For w = 0,

if none of the nonzero d_{j} with 0 ≤ j ≤ r−1 are divisible by p, then the conductor of η is (D_{r,w} +1)w. For w ≠ 0, if exactly one index, say, j is such that p^{r−1−j}(d_{j} +1)−1 is equal to D_{r,w}, then the conductor of η is (D_{r,w} + 1)w = p^{r−1−j}(d_{j} + 1)w provided that d_{j} is prime to p; and if f = (g_{1}/g)^{p^{s}m} with s ≥ r − 1, m prime to p, and g_{1} a polynomial over O_{n} with deg g_{1} < deg g and ḡ_{1} ≠ 0, then the conductor of η is at most (p^{r−1}m + 1)w with equality achieved when s = r − 1.

As a consequence of Theorem 2, we get a parametrization of idèle class characters which are ramified only at one place of κ_{n}(x) and whose order is a power of p, the characteristic of κ, by an argument similar to the proof of Corollary 2.

Corollary 3. Given a positive integer D, let r be the integer such that p^{r−1} ≤ D < p^{r}. Let ψ be an additive character of K of order −r. Let w be a place of κ_{n}(x) other than ∞. Then η_{ψ,f} parametrizes the idèle class characters of κ_{n}(x), trivial at π_{∞}, with order a power of p and conductor Mw for M ≤ D + 1 as f runs through the rational functions of the following form:

(i)If w = 0, then

f = ∑_{0≤t≤r−1} p^{t} ∑_{m≥1,p∤m,p^{r−1−t}m≤D} c_{t,m}/x^{m}

where c_{t,m} are in T ∩ K_{n}; (ii)If w ≠ 0, let g be the lifting of π_{w}. Then

f = ∑_{0≤t≤r−1} p^{t} ∑_{m≥1,p∤m,p^{r−1−t}m≤D}(c_{t,m}/g)^{p^{r−1−t}m},

where c_{t,m} runs through polynomials with coefficients in T ∩K_{n} and degree less than the degree of g.

Actually the statement (ii) is also valid for w = 0, only in this case raising to the pth power of an element in T is the same as applying the Frobenius automorphism to this element, which has no effect after applying the trace function, therefore one may remove the p-power part to get the simpler expression (i). As explained in Remark (8) at the end of §5, we may replace c_{t,m}/g by Tr_{K_{na}/K_{n}}(c′_{t,m}/(x − α)), where α is a fixed root of g, a is the degree of g, and c′_{t,m} runs through elements in T ∩ K_{na}.

Combining Corollaries 2 and 3 together, we obtain a parametrization of idèle class characters of κ_{n}(x), trivial at π_{∞} and with order a power of p. By class field theory, these characters can be interpreted as characters of the Galois group of wildly ramified abelian extensions of κ_{n}(x) of degree a power of p, which are known to be parametrized by Witt vectors. We shall compare the two parametrizations in §8..

For a general rational function f, the corresponding character η_{ψ,f} will be the product of the characters constructed using each primary component of f as described in Theorems 1 and 2. Each primary component of f determines a place w of κ_{n}(x) as in Theorems 1 and 2, and there is an associated r-degree D_{r,w} of this component at w. The conductor of η_{ψ,f} is at most ∑_{w}(D_{r,w} + 1)w; its degree ∑_{w}(D_{r,w} + 1) deg w is called the weighted r-degree of f. Recall that C_{n} is defined by the equation x^{p^{n}} = xz^{p^{n}−1}.

Corollary 4. Let ψ be an additive character of K_{n} of order −r ≤ −1. Let f(x) be a rational function over O_{n} whose denominator is a monic polynomial with roots in T , and f is not a constant. Denote by S the set of elements in C_{n}(K_{n}) at which f has no pole, and by D_{r,f} the weighted r-degree of f as defined above. Suppose the associated character η_{ψ,f} is nontrivial. Then we have

| ∑_{x∈S} ψ(f(x))| ≤ (D_{r,f} − 2)p^{n/2}.

In particular, denote by ∑_{w} n_{w}w the divisor of the pole of f̄. If none of the nonzero n_{w} is a multiple of p and r = 1, then D_{1,f} = ∑_{w}(n_{w} + 1) deg w.

Proof. For brevity, write η for η_{ψ,f}. Denote by D the conductor of η and by S′ the set of places v of κ_{n}(x) of degree one at which η_{v} is unramified. Then D is a nonzero effective divisor by assumption. From Corollary 3 on p. 93 of [L2] we have

(2.1)  | ∑_{v∈S′} η_{v}(π_{v})| ≤ (deg D − 2)p^{n/2}.

The natural image S̄ of S in the projective line ℙ^{1}(κ_{n}) is contained in S′. If S̄ = S′, then the sum over S is the sum over S′ by Theorems 1 and 2. Since the degree of D is at most D_{r,f}, the desired estimate follows immediately from (2.1). If S̄ is a proper subset of S′, let m be the cardinality of S′ − S̄. Then the sum over S′ has m more terms, each of absolute value 1, in addition to the sum over S; and the degree of D is at most D_{r,f} − m. As q^{n/2} > 1, the desired estimate again follows easily from (2.1).

The last statement follows immediately from Theorems 1 and 2. We remark that this was proved in Theorem 5 of [B] by Bombieri.

Example. Let f(x) = f_{1}(x) + f_{2}(x^{−1}), where f_{1}(x), f_{2}(x) are two polynomials over O_{n} such that none of the powers of x in f_{1} and f_{2} are divisible by p and not all coefficients in f_{1}, f_{2} are multiples of p^{r}. Denote by D_{r,∞} the r-degree of f_{1} at ∞ and by D_{r,0} the r-degree of f_{2}(x^{−1}) at 0. Then the weighted r-degree of f is D_{r,f} = D_{r,∞} + D_{r,0} + 2 and the character η_{ψ,f} is ramified at ∞ and 0 by Theorems 1 and 2, and hence is nontrivial. Corollary 4 gives the estimate

| ∑_{x∈T ∩K_{n}^{×}} ψ ∘ Tr_{K_{n}/K}(f_{1}(x) + f_{2}(x^{−1}))| ≤ (D_{r,∞} + D_{r,0})p^{n/2}.

This was proved in [SKH1], while Corollary 4 was announced as Theorem 8 in [HKS].

For multiplicative characters of K we have the following two results.

Theorem 3. Let χ be a multiplicative character of K_{n} with conductor r ≥ 1, that is, r is the smallest nonnegative integer such that χ is trivial on 1 + p^{r}O_{n}. Given a nonconstant polynomial f̄ over κ_{n} with the lifting f over K_{n}, denote by S the set of places of κ_{n}(x) dividing f̄. Then there exists an idèle class character η = η_{χ,f} of κ_{n}(x) with the following properties:

(3.a) η is unramified at any place v outside ∞ and S, and

η_{v}(π_{v}) = χ(∏_{γ} f(γ)),

where γ runs through all liftings of roots of π_{v} in T ;

(3.b) If f(x) = x^{a}, the conductor of η is at most 0+∞. Further, η has conductor 0 + ∞ if χ^{a} is nontrivial on T ∩ K_{n}^{×}, and it is a trivial character otherwise.

(3.c) If f is prime to x, the conductor of η is at most ∞+p^{r−1} ∑_{v∈S} v. Furthermore, denote by s_{v} the order of f̄ at v ∈ S. Let r_{v} be the conductor of χ^{s_{v}}. If χ^{deg f} is nontrivial on T ∩ K_{n}^{×}, then η has conductor ∞ + ∑_{v∈S,r_{v}≥1} p^{r_{v}−1}v; otherwise it has conductor ∑_{v∈S,r_{v}≥1} p^{r_{v}−1}v.

Remarks. (3)If r = 1 in the above three theorems, then the character ψ (resp. χ) may be regarded as a character of the residue field of K (resp. K_{n}) and the results may be formulated in terms of characters of finite fields. These were proved in [L2, Chap 6, §2].

(4) If f(x) = c is a nonzero constant, then the associated character η is unramified everywhere and satisfies

η_{v}(π_{v}) = χ(∏_{γ} f(γ)) = χ(c)^{deg v}

at any place v.

Corollary 5. Let χ be as in Theorem 3. Let f̄ be a nonconstant rational function in κ_{n}(x) which lifts to f over K_{n}. Denote by S the set of places v of κ_{n}(x) at which ord_{v} f̄ = s_{v} ≠ 0 and let r_{v} be the conductor of χ^{s_{v}}. Suppose some r_{v} ≥ 1. Let S′ consist of the elements γ in C_{n}(K_{n}) at which f has neither a zero nor a pole. Suppose further that m = |S| − |S′|. Then

| ∑_{x∈S′} χ(f(x))| ≤ ( ∑_{v∈S,r_{v}≥1} p^{d_{v}} deg v − 2)p^{n/2} + m ≤ (∑_{v∈S} p^{d_{v}} deg v − 2)p^{n/2}.

Here d_{v} = max(r_{v} − 1, 0) for v ≠ 0, ∞, and d_{v} = 0 if v = 0 or ∞.

Proof. Let η be the character attached to χ and f via Theorem 3. Then η is ramified at the places v where r_{v} ≥ 1 with conductor ∑_{v∈S,r_{v}≥1} p^{d_{v}}v by Theorem 3. The rest of the proof follows the same line as the proof of Corollary 4, using (3.a) instead of Theorems 1 and 2.

Theorem 4. Let χ be a multiplicative character of K_{n} with conductor r ≥ 1. Let f be a rational function over K_{n} whose denominator is a monic polynomial with roots in T and whose numerator has integral coefficients. Denote by S the set of places of κ_{n}(x) other than ∞ where f̄ has a pole, and for v ∈ S denote by P_{v} the lifting of π_{v}. Assume further that when expanding f in partial fractions

(2.2)  f(x) = ∑_{0≤i≤d_{∞}} a_{∞i}x^{i} + ∑_{v∈S} ∑_{1≤i≤d_{v}} a_{vi}P_{v}^{−i},

the leading coefficients a_{vd_{v}}, v ∈ S, and a_{∞d_{∞}}, if nonzero, are units in K_{n}. Then for any positive integer e, there is an idèle class character η = η_{χ,e,f} of κ_{n}(x) with the following properties.

(4.a) η is unramified at any place v outside ∞ and S and

η_{v}(π_{v}) = χ(∏_{γ}(1 + p^{e}f(γ))),

where γ runs through the lifting of the roots of π_{v};

(4.b) η is the trivial character if e ≥ r; when e < r, the conductor of η is at most ∑_{v∈S}(1 + d_{v})p^{r−e−1}v + (1 + d_{∞}p^{r−e−1})∞.

(4.c) At each place v ∈ S, v ≠ 0, the conductor of η_{v} is (1 + d_{v})p^{r−e−1}v. η is unramified at ∞ if f̄ does not have a pole at ∞, that is, d_{∞} = 0; otherwise the conductor of η_{∞} is at most (1 + d_{∞}p^{r−e−1})∞ with the upper bound achieved if and only if p ∤ d_{∞}. If 0 ∈ S, then the conductor of η_{0} is at most (1 + d_{0}p^{r−e−1})0 with the upper bound achieved if and only if p ∤ d_{0}.

Observe that the order of η_{χ,e,f} is a power of p, hence it is equal to η_{ψ,g} for some additive character ψ and some rational function g as described before. Consequently, we have, for almost all places v of κ_{n}(x),

χ(∏_{γ}(1 + p^{e}f(γ))) = ψ(∑_{γ} g(γ)),

where γ runs through the liftings of the roots of π_{v} in T .

Remarks. (5) The assumption on a_{vd_{v}}, v ∈ S, and a_{∞d_{∞}} being units is only for the sake of determining the conductor of η. For a general f, we may write

1 + p^{e}f = (1 + p^{e}f_{0})(1 + p^{e+1}f_{1}) ⋯ (1 + p^{r−1}f_{r−e−1}) ⋯,

where f_{0}, f_{1}, ... satisfy the assumptions of Theorem 4. Then the character attached to 1+p^{e}f is the product of the characters attached to 1+p^{e+i}f_{i} for 0 ≤ i ≤ r−e−1.

(6) In the case that the order of pole of f̄ at ∞ (or 0) is a multiple of p, denote the sum of the terms x^{i} (or x^{−i}) in (2.2) where p divides i by f_{2}(x) and the remaining sum f_{1}(x). Let g(x) be a polynomial in x (or x^{−1}) over K_{n} such that f_{2}(x) = g^{σ}(x^{p}), where σ is the Frobenius automorphism. From

1 + p^{e}f(x) = (1 + p^{e}f_{1}(x))(1 + p^{e}f_{2}(x))(1 − p^{2e}f_{1}(x)f_{2}(x)) ⋯,

we get that the character η_{χ,e,f} is the product of η_{χ,e,f_{1}}, η_{χ∘σ,e,g}, η_{χ,2e,−f_{1}f_{2}}, etc, in which the conductor at ∞ (or 0) of the first and third characters is determined by Theorem 4, and that of the remaining characters can be either determined using Theorem 4 or can be determined by repeating the same procedure.

By combining Theorems 3 and 4 together, we can attach an idèle class character to a multiplicative character χ of K_{n}^{×} of conductor r and a polynomial f over K_{n} with integral coefficients which is not identically zero mod p as follows. Write

f = f_{0}(1 + pf_{1})(1 + p^{2}f_{2}) ⋯ (1 + p^{r−1}f_{r−1}) ⋯,

where f_{0} is the lifting over K_{n} of f mod p, f_{i}, i ≥ 1, are rational functions with denominator a power of f_{0} and numerator a polynomial with coefficients in T ∩ K_{n} such that

f ≡ f_{0}(1 + pf_{1})(1 + p^{2}f_{2}) ⋯ (1 + p^{i}f_{i}) (mod p^{i+1}).

Then the product of the characters attached to χ and f_{0}, 1 + pf_{1}, ..., 1 + p^{r−1}f_{r−1} by Theorems 3 and 4 is the desired character η = η_{χ,f}. It is easy to see that η is unramified outside the set S of places where f̄_{0} has a zero or pole, and for v outside S, we have

η_{v}(π_{v}) = χ(∏_{γ} f(γ)),

where γ runs through the lifting of the roots of π_{v}.

Corollary 6. Let χ be a character of K_{n}^{×} of conductor r. Let

g = x^{a}(1 + pg_{1})(1 + p^{2}g_{2}) ⋯ (1 + p^{r−1}g_{r−1}),

where a is an integer, g_{1}, ..., g_{r−1} are polynomials in x and x^{−1} with coefficients in T such that no powers of x, x^{−1} occurring in these polynomials with nonzero coefficients are multiples of p. Denote by d_{∞,i} and d_{0,i} the highest nonnegative power of x, x^{−1} occurring in g_{i} for i = 1, ..., r − 1. For v = 0, ∞, let

D_{v,g} = max{1 + d_{v,i}p^{r−i−1} : 1 ≤ i ≤ r − 1}.

Suppose at least one of D_{∞,g}, D_{0,g} is greater than 1 or χ^{a} is nontrivial on T ∩ K_{n}^{×}. Then

(2.3)  | ∑_{x∈T ∩K_{n}^{×}} χ(g(x))| ≤ (D_{∞,g} + D_{0,g} − 2)p^{n/2}.

Proof. Let η be the character attached to χ and g arising as the product of the characters attached to x^{a} and 1 + p^{i}g_{i} via Theorems 3 and 4. By assumption, none of the nonzero d_{∞,i}, d_{0,i} is a multiple of p. Hence, by Theorems 3 and 4, η_{∞} (resp. η_{0}) is ramified at ∞ (resp. 0) with conductor D_{∞,g} (resp. D_{0,g}) if it is greater than 1 or if χ^{a} is nontrivial on T ∩K_{n}^{×}. The last assumption in the corollary ensures that η is ramified at least at one place and it has conductor D at most D_{∞,g}∞ + D_{0,g}0. Denote by S the set of places v of κ_{n}(x) of degree 1 such that η is unramified at v. Then by Corollary 3 on p.93 of [L2] we have

(2.4)  | ∑_{v∈S} η_{v}(π_{v})| ≤ (deg D − 2)p^{n/2}.

If η is ramified at both 0 and ∞, then D = D_{∞,g}∞ + D_{0,g}0 and the desired inequality follows from (2.4) and (4.a). If η is unramified at 0 or ∞, then one of D_{0,g}, D_{∞,g} is 1 and the left hand side of (2.4) contains one more term, with absolute value one, than the left hand side of (2.3), while the right hand side of (2.4) is p^{n/2} less than the right hand side of (2.3). Therefore (2.3) holds.

Combining the previous theorems together, we immediately get

Theorem 5. Let ψ be an additive character of K_{n} of order −r ≤ −1 and let χ be a multiplicative character of K_{n} of conductor s ≥ 1. Let f and g be two rational functions over K_{n} whose denominators are monic polynomials with roots lying in T and whose numerators are polynomials with integral coefficients, not all divisible by p. Denote by S the set of places of κ_{n}(x) consisting of (i) ∞, (ii) poles of f̄, and (iii) zeros and poles of ḡ. Let η_{ψ,f} be the idèle class character of κ_{n}(x) attached to f and ψ via Theorems 1 and 2 and η_{χ,g} be the character attached to g and χ via Theorems 3 and 4. Then the product η_{ψ,f}η_{χ,g} =: ω is an idèle class character of κ_{n}(x) with the following properties:

(5.a) ω is unramified outside S, and at each place v of κ_{n}(x) not in S, we have

ω(π_{v}) = ψ(∑_{γ} f(γ))χ(∏_{γ} g(γ)),

where γ runs through the liftings in T of roots of π_{v};

(5.b) Denote by D_{ψ,f} and D_{χ,g} the conductor of η_{ψ,f} and η_{χ,g}, respectively. Then the conductor D of ω is at most the least effective divisor dominating D_{ψ,f} and D_{χ,g}, with the upper bound achieved if there is no place where η_{ψ,f} and η_{χ,g} are both ramified and have the same local conductor. In particular, if D_{ψ,f} and D_{χ,g} are coprime, then D is the sum of these two divisors.

Suppose ω is not principal. Let S′ consist of the elements γ in C_{n}(K_{n}) such that γ̄ does not occur in D. Then, using Corollary 3 on p.93 of [L2] and arguing as before, we have

(2.5)  | ∑_{γ∈S′} ψ(f(γ))χ(g(γ))| ≤ (deg D − 2)p^{n/2}.

This character sum estimate specializes to the following results.

Corollary 7. Suppose ψ has order −r ≤ −1 and χ has conductor s = 1. Let f(x) be a polynomial in x with coefficients in the ring of integers O_{n} of K_{n} and let g be a rational function as in Theorem 5. Suppose the weighted r-degree D_{r,∞} of f at ∞ is at least 2. Let S′ consist of elements γ in T ∩ K_{n} such that γ̄ is outside the support of ḡ. Then

| ∑_{γ∈S′} ψ(f(γ))χ(g(γ))| ≤ (D_{r,∞} − 1 + ∑_{v∈Suppḡ,v≠∞} deg v)p^{n/2}.

In particular, if g is a polynomial, then the upper bound above is ≤ (D_{r,∞} + deg ḡ − 1)p^{n/2}.

This is because the attached character ω is ramified with conductor D equal to (D_{r,∞} + 1)∞ plus thoses places v occurring in the support of ḡ so that ord_{v} ḡ is not a multiple of the order of χ by Theorems 1 and 3. Of course a similar result holds with 0 replacing ∞ if f is a polynomial in x^{−1}. Moreover, if we retain the same ψ, χ and g as in Corollary 7 but choose f(x) to be f_{1}(x) + f_{2}(x^{−1}), where f_{1} and f_{2} are polynomials in x with coefficients in O_{n} and the weighted r-degree D_{r,∞} of f_{1}(x) at ∞ and the weighted r-degree D_{r,0} of f_{2}(x^{−1}) at 0 are at least 2. Then the same argument yields the estimate

| ∑_{γ∈S′} ψ(f_{1}(γ) + f_{2}(γ^{−1}))χ(g(γ))|

≤ (D_{r,∞} + D_{r,0} + ∑_{v∈Suppḡ,v≠∞,0} deg v)p^{n/2},

where S′ consists of elements γ in T ∩ K_{n}^{×} such that γ̄ is outside the support of ḡ. This summarizes the estimates proved in [SKH2], [La], [SKH3], and [SKH4].

Corollary 8. Suppose ψ has order −r ≤ −1 and χ has conductor s > 1. Let e be an integer such that s > e ≥ 1. Let f and g be polynomials in x with coefficients in O_{n} such that none of the exponents occurring in f are multiples of p, the leading coefficient of g is a unit and the degree d_{∞} of g is not a multiple of p. Denote by D_{r,∞} the weighted r-degree of f at ∞. Then

| ∑_{γ∈T ∩K_{n}} ψ(f(γ))χ(1 + p^{e}g(γ))| ≤ (max(d_{∞}p^{s−e−1}, D_{r,∞}) − 1)p^{n/2}

provided that d_{∞}p^{s−e−1} ≠ D_{r,∞}.

This follows from the fact that the conductor of η_{ψ,f} is (D_{r,∞} + 1)∞ and the conductor of η_{χ,e,g} is (d_{∞}p^{s−e−1} + 1)∞ by Theorems 1 and 4. As these two characters have different conductor, their product ω has conductor equal to the larger one, hence the result. Likewise, we get a similar result with 0 replacing ∞. When 0 and ∞ are poles of fg, we also have an estimate as follows.

Let e_{1}, e_{2} be two positive integers less than r. Let f_{1}, f_{2}, g_{1}, g_{2} be polynomials in x with coefficients in O_{n}. Denote by D_{r,∞} the weighted r-degree of f_{1}(x) at ∞ and by D_{r,0} the weighted r-degree of f_{2}(x^{−1}) at 0. Suppose none of the exponents occurring in f_{1}, f_{2} are multiples of p, the leading coefficients of g_{1} and g_{2} are units, and the degrees d_{∞} and d_{0} of g_{1}(x) and g_{2}(x) are not divisible by p. Assume further that d_{∞}p^{s−e_{1}−1} ≠ D_{r,∞} and d_{0}p^{s−e_{2}−1} ≠ D_{r,0}. Then we have

| ∑_{γ∈T ∩K_{n}^{×}} ψ(f_{1}(γ) + f_{2}(γ^{−1}))χ(1 + p^{e_{1}}g_{1}(γ) + p^{e_{2}}g_{2}(γ^{−1}))|

≤ (max(d_{∞}p^{s−e_{1}−1}, D_{r,∞}) + max(d_{0}p^{s−e_{2}−1}, D_{r,0}))p^{n/2}.

The estimates of character sums over p-adic fields were used to construct sequences of low correlations in information theory by Kumar, Helleseth and Calderbank in [KHC]. Here we briefly explain the problem and sketch the result. In codedivision multiple-access communication systems that employ phase-shift-keying modulation, the messages to be transmitted are presented by periodic sequences. It is desirable to construct a large family of cyclically different sequences with low correlations to allow a large number of intermittent users and to minimize interference due to the simultaneous presence of other users. The correlation C_{max} of a family F of sequences s⃗ = {s_{1}, s_{2}, ...} with entries in ℂ and with period L is defined as the maximum of the absolute value of ∑_{1≤i≤L} s_{i}\overline{t_{i+j}} for all s⃗, t⃗ in F. Here j runs from 0 (resp. 1) to L − 1 if s⃗ ≠ t⃗ (resp. s⃗ = t⃗). One way to construct such sequences is to attach a sequence to each idèle class character of a rational function field κ_{n}(x) with conductor M∞, M ≤ D + 1, and trivial at π_{∞} as follows. Let β be the lifting to K_{n} of a generator of the multiplicative group of κ_{n}. Let r be the integer such that p^{r−1} ≤ D < p^{r} and let ψ be an additive character of K_{n} of order −r; the characters we shall use are η_{ψ,f}, parametrized by polynomials f as described in Corollary 2. The sequence f⃗ attached to η_{ψ,f} has its ith entry being ψ(f(β^{i})) so that f⃗ is a sequence with period L = p^{n} − 1. Note that the sequence obtained from shifting f⃗ forward by j coordinates is the sequence g⃗ where g(x) = f(β^{j}x). Let F be a collection of sequences f⃗ obtained from choosing among nonzero polynomials f described in Corollary 2 such that f(x) ≠ g(β^{i}x) for 1 ≤ i < L whenever f and g are chosen (including the case f = g). This gives a family of cyclically different sequences with values in p^{r}th roots of unity, each has period L = p^{n} −1, and the correlation C_{max} is dominated by (D − 1)p^{n/2} + 1, as a consequence of Corollary 1. The maximum size of F is 1/L times the size of S consisting of polynomials f in Corollary 2 satisfying f(x) ≠ f(β^{i}x) for 1 ≤ i < L, and the size of S is at most p^{nD} − 1, which is the number of nonzero polynomial parameters. However, if the smallest prime factor ℓ of L is ≤ D, then f(x) = p^{r−1}x^{ℓ} is one of the parameters which do not lie in S since f(x) = f(β^{L/ℓ}x), and hence the size of F is less than (p^{nD} − 1)/L. This point was overlooked in [KHC]. Those parameters with nonzero linear term are contained in S, which gives a lower bound of the size of S: |S| ≥ p^{nD} − p^{n(D−r)}, which in turn shows that the size of F is at least (p^{nD} − p^{n(D−r)})/L.

The vector obtained by taking L consecutive components of a sequence f⃗ generates a cyclic code. It was explained in [Du] that the duals of such cyclic codes are generalized BCH codes. This gives another interesting description of the family F.

§3 Congruences on symmetric polynomials

Let X = {x_{1}, …, x_{N}} be a set of N independent variables. Given m, 1 ≤ m ≤ N, functions g_{1}, …, g_{m} in one variable, denote by S_{X}(g_{1}, …, g_{m}) the formal sum

S_{X}(g_{1}, …, g_{m}) = \frac{1}{(N − m)!} ∑_{σ∈S_{N}} g_{1}(x_{σ(1)}) ⋯ g_{m}(x_{σ(m)}),

where S_{N} is the group of permutations in N letters. The set of partitions of m into nonincreasing parts is denoted by P(m). For a partition q = {q_{1}, …, q_{ℓ}} of m (that is, m = q_{1} + ⋯ + q_{ℓ},) into ℓ nonzero parts, define the sign of q to be

sgn(q) = (−1)^{m−ℓ}.

For any finite set T = {t_{1}, …, t_{ℓ}} in which elements are allowed to repeat, the multiplicity of T is defined to be

m(T) = i_{1}! ⋯ i_{s}!,

where i_{1}, …, i_{s} counts the multiplicities of different elements in T.

Theorem 6. For 1 ≤ m ≤ N we have

(3.1)  S_{X}(g_{1}, …, g_{m}) = ∑_{q=\{q_{1},...,q_{ℓ}\}∈P(m)} sgn(q)\frac{1}{m(q)q_{1} ⋯ q_{ℓ}} ∑_{τ∈S_{m}} Λ_{X}(q, τ),

where

Λ_{X}(q, τ) = Λ_{X}(q, τ, g_{1}, ..., g_{m}) = ∏_{i=1}^{ℓ} S_{X}(g_{τ(q_{1}+···+q_{i−1}+1)} ⋯ g_{τ(q_{1}+···+q_{i})}).

Proof. We shall drop the variable set X since it does not change. Observe first that S(g_{1}, …, g_{m}) satisfies a recursive formula:

S(g_{1}, …, g_{m+1}) =S(g_{1}, …, g_{m})S(g_{m+1})

− ∑_{j=1}^{m} S(g_{1}, …, g_{j−1}, g_{j}g_{m+1}, g_{j+1}, …, g_{m}).

This is because

S(g_{1}, …, g_{m})S(g_{m+1})

= \frac{1}{(N − m)!} ∑_{σ∈S_{N}} g_{1}(x_{σ(1)}) ⋯ g_{m}(x_{σ(m)}) ∑_{i=1}^{N} g_{m+1}(x_{i})

= \frac{1}{(N − m − 1)!} ∑_{σ∈S_{N}} g_{1}(x_{σ(1)}) ⋯ g_{m+1}(x_{σ(m+1)})

+ \frac{1}{(N − m)!} ∑_{j=1}^{m} ∑_{σ∈S_{N}}(g_{j}g_{m+1})(x_{σ(j)}) ∏_{1≤i≤m,i≠j} g_{i}(x_{σ(i)})

= S(g_{1}, …, g_{m+1}) + ∑_{j=1}^{m} S(g_{1}, …, g_{j−1}, g_{j}g_{m+1}, g_{j+1}, …, g_{m}).

Call the right hand side of the formula in Theorem S′(g_{1}, …, g_{m}). Clearly, S(g_{1}) = S′(g_{1}). Thus it suffices to show that S′ satisfies the same recursive formula. By definition, S′(g_{1}, …, g_{m+1}) is a summation over partitions q of m+1 and permutations τ in S_{m+1}. Given a term

S = S(g_{i_{1}} ⋯)S(g_{i_{2}} ⋯) ⋯ S(g_{i_{ℓ}} ⋯)

occurring in S′(g_{1}, ..., g_{m+1}), there exist a unique partition q = {q_{1}, …, q_{ℓ}} in P(m+ 1) of m + 1 with ℓ nonincreasing parts and q_{1}! ⋯ q_{ℓ}!m(q) permutations τ of S_{m+1} so that the given term is Λ(q, τ). Hence the coefficient of S in S′(g_{1}, …, g_{m+1}) is A := sgn(q)(q_{1} − 1)! ⋯ (q_{ℓ} − 1)!.

Next we compute the coefficient B of S in

S′(g_{1}, …, g_{m})S′(g_{m+1}) − ∑_{j=1}^{m} S′(g_{1}, …, g_{j−1}, g_{j}g_{m+1}, g_{j+1}, …, g_{m}).

We distinguish two cases.

Case I. S(g_{m+1}) occurs in S. Then at least one of q_{i}, and hence q_{ℓ}, is 1. We can write S = S′S(g_{m+1}) with S′ being the product of the remaining terms, which occurs in S′(g_{1}, …, g_{m}). The partition q′ of m corresponding to S′ is {q_{1}, …, q_{ℓ−1}} and the coefficient of S′ in S′(g_{1}, …, g_{m}) is

sgn(q′)(q_{1} − 1)! ⋯ (q_{ℓ−1} − 1)! = sgn(q)(q_{1} − 1)! ⋯ (q_{ℓ} − 1)!.

As S(g_{m+1}) does not occur in S′(g_{1}, …, g_{j−1}, g_{j}g_{m+1}, g_{j+1}, …, g_{m}), we have B = A.

Case II. S(g_{m+1}) does not occur in S. Then g_{m+1} occurs in the tth term S(g_{i_{t}} ⋯) for some 1 ≤ t ≤ ℓ. Then q_{t} ≥ 2 and we may assume that q_{t} > q_{t+1} if t < ℓ. Consequently, there are q_{t} − 1 other indices j such that g_{j} also appears in S(g_{i_{ℓ}} ⋯), and these are the indices such that S occurs in S′(g_{1}, …, g_{j−1}, g_{j}g_{m+1}, g_{j+1}, …, g_{m}). By symmetry, S has the same coefficient in each of such summation, hence it suffices to compute the coefficient of S in one of S′(g_{1}, …, g_{j−1}, g_{j}g_{m+1}, g_{j+1}, …, g_{m}) and them multiply it by −(q_{t} − 1) to get B. Fix such a j. Define h_{k} = g_{k} for 1 ≤ k ≤ m and k ≠ j, and h_{j} = g_{j}g_{m+1}. Then S can also be expressed by the same product with g_{i} replaced by h_{i} for 1 ≤ i ≤ m and g_{m+1} replaced by 1. Thus the coefficient of S in S′(g_{1}, …, g_{j−1}, g_{j}g_{m+1}, g_{j+1}, …, g_{m}) is the same as that in S′(h_{1}, …, h_{m}). When viewed this way, the partition of m corresponding to S is q′ = {q′_{1}, …, q′_{ℓ}} where q′_{i} = q_{i} except q′_{t} = q_{t} − 1, and we get

B = −(q_{t} − 1) · sgn(q′)(q′_{1} − 1)! ⋯ (q′_{ℓ} − 1)! = −sgn(q′)(q_{1} − 1)! ⋯ (q_{ℓ} − 1)!,

which is equal to A since q and q′ have the same number of parts. This completes the proof of the theorem.

The symmetric sum generated by g_{1}(x_{1}), ⋯, g_{m}(x_{m}) on the variable set X is

S_{X}(g_{1}, …, g_{m}) = \frac{1}{m(\{g_{1}, …, g_{m}\})}S_{X}(g_{1}, …, g_{m}).

In particular, S_{X}(x^{k}) is the Newton polynomial of degree k in x_{1}, …, x_{N}. Using Waring’s formula we may express it in terms of the elementary symmetric polynomials c_{1}, …, c_{N} in x_{1}, …, x_{N} as follows:

(3.2)  S_{X}(x^{k}) = ∑_{(i_{1},...,i_{k})}(−1)^{i_{2}+i_{4}+...} \frac{(i_{1} + ⋯ + i_{k} − 1)!k}{i_{1}! ⋯ i_{k}!} c_{1}^{i_{1}} ⋯ c_{k}^{i_{k}},

where (i_{1}, …, i_{k}) runs through all k-tuples of non-negative integers such that i_{1} + 2i_{2} + ⋯ + ki_{k} = k.

By choosing g_{i}(x) to be a positive power of x, Theorem 6 yields an explicit expression of any symmetric polynomial in x_{1}, …, x_{N} in terms of Newton polynomials, which in turn can be expressed explicitly in terms of elementary polynomials, as remarked above.

We’ll need an estimate of the p-adic order of the fraction occurring in the Waring’s formula. This is given by

Proposition 1. Let i_{1}, …, i_{k} be non-negative integers with i_{1}+⋯+i_{k} = m ≥ 1. Suppose ord_{p} gcd(i_{1}, …, i_{k}) = r. Then ord_{p} \frac{(m−1)!}{i_{1}!···i_{k}!} ≥ −r.

Proof. By definition,

r = ord_{p} gcd(i_{1}, …, i_{k}) = min ord_{p} i_{1}, ..., ord_{p} i_{k}).

Therefore at least one of i_{1}, …, i_{k}, say, i_{1}, has p-adic order equal to r. As the p-adic order of \frac{(m−1)!}{i_{1}!···i_{k}!} is − ord_{p} i_{1} plus the p-adic order of \frac{(m−1)!}{(i_{1}−1)!···i_{k}!} and the latter

fraction is in fact an integer, we have ord_{p} \frac{(m−1)!}{i_{1}!···i_{k}!} ≥ −r, as stated.

The image of S_{X}(g_{1}, …, g_{m}) under the specialization map x_{i} → γ_{i} is denoted by S_{Γ}(g_{1}, …, g_{m}), where Γ = {γ_{1}, …, γ_{N}} in which elements are allowed to repeat. The following congruence results will be used often.

Proposition 2. Let Γ and Δ be two N-element subsets of (not necessarily distinct) integral elements in K̄. For i ≥ 1 denote by c_{i} and d_{i} the degree i elementary symmetric polynomial in Γ and Δ, respectively. Let k be a positive integer with ℓ = ord_{p} k. If, for i = 1, …, k, we have c_{i} ≡ d_{i} (mod p^{e}) with e ≥ 1, then S_{Γ}(x^{k}) ≡ S_{Δ}(x^{k}) (mod p^{e+ℓ}).

Proof. Using Waring’s formula, we may write

S_{Γ}(x^{k}) = ∑ \frac{(i_{1} + ⋯ + i_{k} − 1)!k}{i_{1}! ⋯ i_{k}!} (−1)^{i_{2}+i_{4}+···}c_{1}^{i_{1}} ⋯ c_{k}^{i_{k}}.

Here (i_{1}, …, i_{k}) runs through all k-tuples of non-negative integers such that i_{1} + 2i_{2} + ⋯ + ki_{k} = k. Take any index (i_{1}, …, i_{k}), let b be the p-adic order of gcd(i_{1}, …, i_{k}). By Proposition 1, \frac{(i_{1}+···+i_{k}−1)!}{i_{1}!···i_{k}!} has p-adic order at least −b. Since p^{b} divides i_{1}, …, i_{k}, the product c_{1}^{i_{1}} ⋯ c_{k}^{i_{k}} is a p^{b}th power. It follows from the assumption that c_{1}^{i_{1}} ⋯ c_{k}^{i_{k}} ≡ d_{1}^{i_{1}} ⋯ d_{k}^{i_{k}} (mod p^{e+b}). Put together, we see that each summand of S_{Γ}(x^{k}) is congruent to the summand of S_{Δ}(x^{k}) with the same index modulo p^{e+ℓ}, hence the same congruence holds for S_{Γ}(x^{k}) and S_{Δ}(x^{k}), as desired.

Proposition 3. Let Γ, Δ, c_{i} and d_{i} be as in Proposition 2. Let k_{1}, …, k_{m} be positive integers with sum k. If, for 1 ≤ i ≤ k, we have c_{i} ≡ d_{i} (mod p^{e}) with e ≥ 1, then

S_{Γ}(x^{k_{1}}, …, x^{k_{m}}) ≡ S_{Δ}(x^{k_{1}}, …, x^{k_{m}}) (mod p^{e}).

Furthermore, if all k_{i} are multiples of p^{ℓ}, then

S_{Γ}(x^{k_{1}}, …, x^{k_{m}}) ≡ S_{Δ}(x^{k_{1}}, …, x^{k_{m}}) (mod p^{e+ℓ}).

Proof. Write X for the set of N independent variables x_{1}, …, x_{N}. Let q = {q_{1}, …, q_{t}} be a partition of m into t parts and let τ be a permutation in S_{m}. In view of Theorem 6, we examine the sum of the coefficient of Λ_{X}(q, τ′) as τ′ runs through all permutations in S_{m} so that Λ_{X}(q, τ′) = Λ_{X}(q, τ). Due to symmetry, we may assume τ equal to the identity permutation. Suppose there are s distinct functions, say, x^{k_{1}}, …, x^{k_{s}}, among x^{k_{1}}, …, x^{k_{m}}, each repeated b_{1}, …, b_{s} times, respectively. For 1 ≤ j ≤ t, denote by b_{j1}, …, b_{js} the multiplicities of x^{k_{1}}, …, x^{k_{s}} among x^{k_{q_{1}+···+q_{j−1}+1}}, …, x^{k_{q_{1}+···+q_{j}}} so that b_{j1} + ⋯ + b_{js} = q_{j}. Then the total coefficient of

Λ_{X}(q, τ) = ∏_{j=1}^{t} S_{X}(x^{k_{q_{1}+···+q_{j−1}+1}} ⋯ x^{k_{q_{1}+···+q_{j}}})

in S_{X}(x^{k_{1}}, …, x^{k_{m}}) is

∏_{j=1}^{t} \frac{(q_{j} − 1)!}{b_{j1}! ⋯ b_{js}!}

by taking into account of the number of permutations yielding the same term.

Fix an index j, denote by t = t(j) the p-adic order of gcd(b_{j1}, …, b_{js}). Then p^{t} divides the power M = M(j) of x in S_{X}(x^{k_{q_{1}+···+q_{j−1}+1}} ⋯ x^{k_{q_{1}+···+q_{j}}}) = S_{X}(x^{M}). Since M ≤ k, by Propositions 1 and 2 we have

\frac{(q_{j} − 1)!}{b_{j1}! ⋯ b_{js}!}S_{Γ}(x^{M(j)}) ≡ \frac{(q_{j} − 1)!}{b_{j1}! ⋯ b_{js}!}S_{Δ}(x^{M(j)}) (mod p^{e}).

Consequently, the terms in S_{Γ}(x^{k_{1}}, …, x^{k_{m}}) and S_{Δ}(x^{k_{1}}, …, x^{k_{m}}) arising from specializing Λ_{X}(q, τ) are congruent modulo p^{e} for all q and τ, hence S_{Γ}(x^{k_{1}}, …, x^{k_{m}}) and S_{Δ}(x^{k_{1}}, …, x^{k_{m}}) are congruent modulo p^{e}. Finally, if all k_{j} are divisible by p^{ℓ}, then the congruence between S_{Γ}(x^{M(j)}) and S_{Δ}(x^{M(j)}) is improved by an extra factor p^{ℓ} (in addition to the factor p^{t(j)}) by Proposition 2, therefore the exponent e above may be replaced by e + ℓ.

Remark. (7) In Proposition 3 above, if the assumption is a little weaker, namely, if c_{i} ≡ d_{i} (mod p^{e}) for 1 ≤ i < k, then the same proof gives

S_{Γ}(x^{k_{1}}, …, x^{k_{m}}) + (−1)^{k} \frac{(m − 1)!}{b_{1}! ⋯ b_{s}!}kc_{k}

≡ S_{Δ}(x^{k_{1}}, …, x^{k_{m}}) + (−1)^{k} \frac{(m − 1)!}{b_{1}! ⋯ b_{s}!}kd_{k} (mod p^{e}).

Denote by σ the Frobenius automorphism in Gal(K̄/K). It sends an element in T to its pth power.

Theorem 7. Given 2N not necessarily distinct elements γ_{1}, …, γ_{N}, δ_{1}, …, δ_{N} in T and an element α in T different from γ_{i} and δ_{i}, form the sets Γ = {(γ_{i} −α)^{−1} : 1 ≤ i ≤ N} and Δ = {(δ_{i} − α)^{−1} : 1 ≤ i ≤ N}. Let c_{0} and d_{0} be the product of the inverse of elements in Γ and Δ, respectively. For i ≥ 1, denote by c_{i} and d_{i} the degree i elementary polynomial in Γ and Δ, respectively. Suppose there are three positive integers M, a and e such that c_{i} ≡ d_{i} (mod p^{e}) for 0 ≤ i ≤ Mp^{a} − 2, and, for 1 ≤ j ≤ e − 1, c_{i} ≡ d_{i} (mod p^{e−j}) for Mp^{a+j−1} − 1 ≤ i ≤ Mp^{a+j} − 2. Then c_{i}^{σ} ≡ d_{i}^{σ} (mod p^{e+1}) for 0 ≤ i ≤ Mp^{a−1} − 2, and for 0 ≤ j ≤ e − 1, c_{i}^{σ} ≡ d_{i}^{σ} (mod p^{e−j}) for Mp^{a+j−1} − 1 ≤ i ≤ Mp^{a+j} − 2. Moreover, there exists a rational integer b = ±p, independent of Γ, Δ and α, such that c_{Mp^{a−1}−1}^{σ} + bα^{p−1}c_{Mp^{a}−1} ≡ d_{Mp^{a−1}−1}^{σ} + bα^{p−1}d_{Mp^{a}−1} (mod p^{e+1}) if Mp^{a−1} − 1 ≥ 1, and c_{0}^{σ} + c_{0}^{p}bα^{p−1}c_{p−1} ≡ d_{0}^{σ} + d_{0}^{p}bα^{p−1}d_{p−1} (mod p^{e+1}) if Mp^{a−1} − 1 = 0, that is, M = a = 1.

Proof. We only need to prove that c_{i}^{σ} ≡ d_{i}^{σ} (mod p^{e+1}) for 0 ≤ i ≤ Mp^{a−1} − 2 and examine the congruence between c_{Mp^{a−1}−1}^{σ} and d_{Mp^{a−1}−1}^{σ}. Write C_{i} for c_{i}^{σ} and D_{i} for d_{i}^{σ} for brevity. To prove what we want, start with

(3.3)  γ_{i}^{p} − α^{p} = (γ_{i} − α)^{p} + ∑_{k=1}^{p−1} b_{k}α^{k}(γ_{i} − α)^{p−k}

= (γ_{i} − α)^{p}[1 + ∑_{k=1}^{p−1} b_{k}α^{k}(γ_{i} − α)^{−k}].

Here b_{k}’s are rational integral multiples of p independent of α and γ_{i}. In particular, b_{p−1} = p. When we take the product of γ_{i}^{p} − α^{p} over 1 ≤ i ≤ N to obtain C_{0}, we get c_{0}^{p} times the product over i of 1 + ∑_{k=1}^{p−1} b_{k}α^{k}(γ_{i} − α)^{−k}. The latter product can be expanded as a sum in terms of S_{Γ}(x^{k_{1}}, …, x^{k_{m}}). More precisely, we have

C_{0} = c_{0}^{p}[1 + ∑_{1≤k≤p−1} b_{k}α^{k}S_{Γ}(x^{k}) + ∑_{1≤k_{1}≤k_{2}≤p−1} b_{k_{1}}b_{k_{2}}α^{k_{1}+k_{2}}S_{Γ}(x^{k_{1}}, x^{k_{2}})

+ ∑_{1≤k_{1}≤k_{2}≤k_{3}≤p−1} b_{k_{1}}b_{k_{2}}b_{k_{3}}α^{k_{1}+k_{2}+k_{3}}S_{Γ}(x^{k_{1}}, x^{k_{2}}, x^{k_{3}}) + ⋯].

A similar expression for D_{0} with Δ replacing Γ holds. Since c_{0} ≡ d_{0} (mod p^{e}) by assumption, we get c_{0}^{p} ≡ d_{0}^{p} (mod p^{e+1}). Since each b_{k} is a multiple of p, for the terms inside the bracket, we need only show that for 1 ≤ m ≤ e and 1 ≤ k_{i} ≤ p − 1,

(3.4)  S_{Γ}(x^{k_{1}}, …, x^{k_{m}}) ≡ S_{Δ}(x^{k_{1}}, …, x^{k_{m}}) (mod p^{e+1−m})

except when a = 1. Indeed, with 1 ≤ m ≤ e and 1 ≤ k_{j} ≤ p − 1 for 1 ≤ j ≤ m, we find k = k_{1} +⋯+k_{m} ≤ m(p−1) ≤ p^{a+m−1} −2 ≤ Mp^{a+m−1} −2 if a ≥ 2. Thus (3.4) follows from Proposition 3. This proves that C_{0} ≡ D_{0} (mod p^{e+1}) if a ≥ 2. When a = 1, the above argument holds except when m = M = 1 and k = k_{1} = p − 1, in which case we have, by Remark (7), that

S_{Γ}(x^{p−1}) + (−1)^{p}c_{p−1} ≡ S_{Δ}(x^{p−1}) + (−1)^{p}d_{p−1} (mod p^{e}),

which in turn implies that

C_{0} + c_{0}^{p}bα^{p−1}c_{p−1} ≡ D_{0} + d_{0}^{p}bα^{p−1}d_{p−1} (mod p^{e+1})

for some rational integer b = (−1)^{p}b_{p−1} = ±p.

It remains to show that for 1 ≤ m ≤ Mp^{a−1} − 2, C_{m} and D_{m} are congruent modulo p^{e+1} and to check the congruence between c_{Mp^{a−1}−1}^{σ} and d_{Mp^{a−1}−1}^{σ}. To see this, we derive from (3.3) the following expression

(γ_{i}^{p} − α^{p})^{−1} = (γ_{i} − α)^{−p}[1 + ∑_{k=1}^{p−1} b_{k}α^{k}(γ_{i} − α)^{−k}]^{−1}

= (γ_{i} − α)^{−p}[1 + ∑_{j≥1}(−1)^{j}(∑_{k=1}^{p−1} b_{k}α^{k}(γ_{i} − α)^{−k})^{j}]

= (γ_{i} − α)^{−p}[1 + ∑_{j≥1} ∑_{k=j}^{jp−j} b_{jk}α^{k}(γ_{i} − α)^{−k}].

Here b_{jk} are multiples of p^{j} and are independent of α and γ_{i}. Hence modulo p^{e+1}, C_{m} is an integral linear combination of S_{Γ}(x^{k_{1}}, …, x^{k_{m}}), where k_{j} ≥ p, with coefficient whose p-adic order is at least ℓ(k_{1}, …, k_{m}) := ⌈\frac{k_{1}}{p} ⌉ + ⋯ + ⌈\frac{k_{m}}{p} ⌉ − m. Therefore it suffices to prove that for each choice of k_{1}, …, k_{m} as above with ℓ = ℓ(k_{1}, …, k_{m}) < e + 1 we have

(3.5)  S_{Γ}(x^{k_{1}}, …, x^{k_{m}}) ≡ S_{Δ}(x^{k_{1}}, …, x^{k_{m}}) (mod p^{e+1−ℓ}).

Let k = k_{1} + ⋯ + k_{m}. We have k ≤ (m + ℓ)p and equality holds if and only if all k_{j} are multiples of p. Observe that k ≤ (m+ℓ)p ≤ (Mp^{a−1} −2+ℓ)p = Mp^{a} +(ℓ−2)p. When ℓ = 0, in which case all k_{j} are p and k = mp < Mp^{a} − 2, (3.5) holds by the last assertion of Proposition 3. When ℓ = 1 we have k ≤ Mp^{a} − 2, the desired conclusion again follows from Proposition 3. When ℓ ≥ 2, we have k ≤ Mp^{a} + (ℓ − 2)p ≤ Mp^{a+ℓ−1} − 2, hence (3.5) follows from Proposition 3. Finally, we inspect the case m = Mp^{a−1} − 1 ≥ 1. The same argument as above shows that (3.5) holds for all ℓ except when ℓ = 1 and k = Mp^{a} −1, which occurs exactly when m − 1 of the k_{j}’s are p and one of them is 2p − 1. In this case we again envoke Remark (7) to conclude

S_{Γ}(x^{k_{1}}, …, x^{k_{m}}) + (−1)^{k+1}c_{k} ≡ S_{Δ}(x^{k_{1}}, …, x^{k_{m}}) + (−1)^{k+1}d_{k} (mod p^{e+1−ℓ}),

which in turn implies

c_{Mp^{a−1}−1}^{σ} + bα^{p−1}c_{Mp^{a}−1} ≡ d_{Mp^{a−1}−1}^{σ} + bα^{p−1}d_{Mp^{a}−1} (mod p^{e+1})

for some rational integer b = ±b_{p−1} = ±p as asserted. This completes the proof of Theorem 7.

We derive two consequences below.

Corollary 9. With the same notations as in Theorem 7, suppose there are three positive integers a, M and e such that c_{i} ≡ d_{i} (mod p^{e}) for 0 ≤ i < Mp^{a}, and, for 1 ≤ j ≤ e − 1, c_{i} ≡ d_{i} (mod p^{e−j}) for Mp^{a+j−1} ≤ i < Mp^{a+j}. Then c_{i}^{σ} ≡ d_{i}^{σ} (mod p^{e+1}) for 0 ≤ i < Mp^{a−1}, and for 0 ≤ j ≤ e − 1, c_{i}^{σ} ≡ d_{i}^{σ} (mod p^{e−j}) for Mp^{a+j−1} ≤ i < Mp^{a+j}.

This is because c_{Mp^{a+i}−1} ≡ d_{Mp^{a+i}−1} (mod p^{e−i}) for 0 ≤ i ≤ e − 1 so the extra terms in the congruence between c_{Mp^{a+i}−1}^{σ} and d_{Mp^{a+i}−1}^{σ} can be dropped.

Corollary 10. Let Γ and c_{i} be as in Theorem 7. Suppose there are three positive integers M, a and e such that c_{i} ≡ 0 (mod p^{e}) for 1 ≤ i ≤ Mp^{a} − 2, and, for 1 ≤ j ≤ e − 1, c_{i} ≡ 0 (mod p^{e−j}) for Mp^{a+j−1} − 1 ≤ i ≤ Mp^{a+j} − 2. Then c_{i}^{σ} ≡ 0 (mod p^{e+1}) for 1 ≤ i ≤ Mp^{a−1} − 2, and for 0 ≤ j ≤ e − 1, c_{i}^{σ} ≡ 0 (mod p^{e−j}) for Mp^{a+j−1} − 1 ≤ i ≤ Mp^{a+j} − 2. Moreover, there exists a rational integer b = ±p, independent of Γ and α, such that c_{Mp^{a−1}−1}^{σ} + bα^{p−1}c_{Mp^{a}−1} ≡ 0 (mod p^{e+1}) if Mp^{a−1} − 1 ≥ 1. Further, if c_{i} ≡ 0 (mod p^{e}) for 1 ≤ i < Mp^{a}, and, for 1 ≤ j ≤ e − 1, c_{i} ≡ 0 (mod p^{e−j}) for Mp^{a+j−1} ≤ i < Mp^{a+j}, then c_{i}^{σ} ≡ 0 (mod p^{e+1}) for 1 ≤ i < Mp^{a−1}, and for 0 ≤ j ≤ e − 1, c_{i}^{σ} ≡ 0 (mod p^{e−j}) for Mp^{a+j−1} ≤ i < Mp^{a+j}.

Proof. We follow the proof of Theorem 7 with suitable modifications. By Proposition 3, we have

S_{Γ}(x^{k_{1}}, …, x^{k_{m}}) ≡ 0 (mod p^{e+1−ℓ})

for 1 ≤ m ≤ Mp^{a−1} − 1, k_{1}, ..., k_{m} ≥ p and ℓ = ℓ(k_{1}, ..., k_{m}) < e + 1 except when ℓ = 1 and k = k_{1} + ⋯ + k_{m} = Mp^{a} − 1, in which case

S_{Γ}(x^{k_{1}}, …, x^{k_{m}}) + (−1)^{k+1}c_{k} ≡ 0 (mod p^{e+1−ℓ})

by Remark (7). This proves that c_{i}^{σ} ≡ 0 (mod p^{e+1}) for 1 ≤ m ≤ Mp^{a−1} − 2 and

c_{Mp^{a−1}−1}^{σ} + bα^{p−1}c_{Mp^{a}−1} ≡ 0 (mod p^{e+1})

for some rational integer b = ±b_{p−1} = ±p, as asserted. The last statement follows from the same reasoning as in the proof of Corollary 9.

§4. A Proof of Theorem 1.

With n fixed, we shall write F for the function field κ_{n}(x) for brevity. We begin by defining a multiplicative character η_{∞} of the completion F_{∞} of F at ∞. Let η_{∞} be trivial at π_{∞} and κ_{n}. Next we define η_{∞} on 1 + P_{∞}. Given any polynomial h̄(x) = 1 + a_{1}x + ⋯ + a_{u}x^{u} with coefficients in κ_{n}, define

η_{∞}(h̄(π_{∞})) = ψ ∘ Tr_{K_{n}/K}(f(γ_{1}) + ⋯ + f(γ_{u}))^{−1},

where γ_{1}, …, γ_{u} runs through the reciprocal of all roots (in T ) of the lifting h of h̄ over K_{n}. Since the roots of h are invariant under the Frobenius automorphism of K̄ over K_{n}, so is the sum f(γ_{1}) + ⋯ + f(γ_{u}), and hence the above is well-defined. It follows immediately from the definition that η_{∞} is multiplicative on polynomials over κ_{n} in powers of π_{∞}.

Write

f(x) = f_{0}(x) + f_{1}(x)p + ⋯ + f_{r−1}(x)p^{r−1} + ⋯,

where f_{i} are polynomials with coefficients in T and degree d_{i} as in the statement of Theorem 1. Let

D = max(d_{0}p^{r−1}, d_{1}p^{r−2}, ⋯, d_{r−1}).

Our next goal is to show that whenever h̄ is of the form

h̄(x) = 1 + a_{D+1}x^{D+1} + ⋯ + a_{u}x^{u},

that is, the coefficients a_{1}, …, a_{D} of h̄ are zero, we have η_{∞}(h̄(π_{∞})) = 1. If so, then η_{∞} is trivial on the polynomials (in π_{∞}) in 1 + P_{∞}^{D+1}, which are dense in 1 + P_{∞}^{D+1}. Consequently, we can extend η_{∞} continuously to 1+P_{∞}, whence obtain a character η_{∞} on F_{∞}^{×}, trivial on κ_{n}(1 + P_{∞}^{D+1}). As ψ is an additive character of order −r, it suffices to prove that if f_{i} is nonzero and hence of degree at least 1 (since f(0) = 0) for some 0 ≤ i ≤ r − 1, then

Tr_{K_{n}/K}(f_{i}(γ_{1}) + ⋯ + f_{i}(γ_{u})) ≡ 0 (mod p^{r−i}).

Denote by σ the Frobenius automorphism in Gal(K̄/K). It sends z ∈ T to σ(z) = z^{p}. Write g_{i} for the polynomial σ^{r−i−1}(f_{i}). Partition the roots of h into the orbits under the action of Gal(K̄/K_{n}). Let γ_{1}, …, γ_{k} be representatives of these orbits and M_{j} be the field K_{n}(γ_{j}). Then

Tr_{K_{n}/K}(f_{i}(γ_{1}) + ⋯ + f_{i}(γ_{u})) = ∑_{j=1}^{k} Tr_{M_{j}/K}(f_{i}(γ_{j})) = ∑_{j=1}^{k} Tr_{M_{j}/K}(σ^{r−i−1}(f_{i}(γ_{j})))

= ∑_{j=1}^{k} Tr_{M_{j}/K}(g_{i}(γ_{j}^{p^{r−i−1}})) = Tr_{K_{n}/K}(g_{i}(γ_{1}^{p^{r−i−1}}) + ⋯ + g_{i}(γ_{u}^{p^{r−i−1}})).

Note that

g_{i}(x) = b_{i,1}x + ⋯ + b_{i,d_{i}}x^{d_{i}}

has coefficients in T ∩ K_{n} and degree d_{i}. The lifting of h̄ is

h(x) = (1 − γ_{1}x) ⋯ (1 − γ_{u}x) = 1 − c_{1}x + ⋯ + (−1)^{D}c_{D}x^{D} + ⋯ + (−1)^{u}c_{u}x^{u}

with integral coefficients. As h (mod p) is h̄, the coefficients c_{1}, …, c_{D} are congruent to zero mod p. Notice that c_{1}, …, c_{u} are elementary polynomials in γ_{1}, …, γ_{u}. Denote by S(k) the Newton polynomial of degree k on γ_{1}, …, γ_{u}. Then

g_{i}(γ_{1}^{p^{r−i−1}}) + ⋯ + g_{i}(γ_{u}^{p^{r−i−1}}) = b_{i,1}S(p^{r−i−1}) + ⋯ + b_{i,d_{i}}S(d_{i}p^{r−i−1}),

which is congruent to zero mod p^{r−i} by Proposition 2, and hence so is its trace to K.

Thus we have a character η_{∞} of F_{∞}^{×} with conductor at most (D+1)∞. By strong approximation theory, the group of idèles I_{F} of F is

I_{F} = F^{×} · F_{∞}^{×} ∏_{v≠∞} U_{v},

where U_{v} is the group of units at the place v of F. Since the intersection of F^{×} (imbedded diagonally in I_{F}) with F_{∞}^{×} ∏_{v≠∞} U_{v} is κ_{n}^{×}, on which η_{∞} is trivial, we can extend η_{∞} to a character η of I_{F} trivial on F^{×} and ∏_{v≠∞} U_{v}. Its conductor is the conductor of η_{∞}. At a place v of F other than ∞, our choice of π_{v} is a polynomial in F:

π_{v}(x) = ∏_{i=1}^{deg v}(x − γ̄_{i}) = x^{deg v} ∏_{i=1}^{deg v}(1 − γ̄_{i}x^{−1})

which is a unit at all places of F except v and ∞. Denote by γ_{i} the lifting of γ̄_{i} in K̄. From

1 = η(π_{v}) = η_{v}(π_{v})η_{∞}(π_{v}) = η_{v}(π_{v})ψ ∘ Tr_{K_{n}/K}(f(γ_{1}) + ⋯ + f(γ_{deg v}))^{−1}

it follows that η_{v}(π_{v}) is as described in (1.a).

Finally, assuming that none of the nonzero d_{i}, 0 ≤ i ≤ r −1 is a multiple of p, we show that η_{∞} has conductor (D + 1)∞. By assumption, D = d_{i}p^{r−i−1} for exactly one 0 ≤ i ≤ r − 1 and hence D > d_{j}p^{r−j} for j ≠ i. Consider h̄(x) = 1 + a_{D}x^{D} with a_{D} ∈ κ_{n}. Its lifting h(x) = 1 + c_{1}x + ⋯ + c_{D}x^{D} has integral coefficients in K_{n} with c_{j} ≡ 0 (mod p) for 1 ≤ j ≤ D − 1 and c_{D} (mod p) is a_{D}. Let γ_{1}, …, γ_{D} be the reciprocal roots of h. Since for 0 ≤ j ≤ r−1 and j ≠ i we have d_{j}p^{r−j} ≤ D−1, thus Tr_{K_{n}/K}(f_{j}(γ_{1}) + ⋯ + f_{j}(γ_{D})) is congruent to zero mod p^{r−j} by the same argument as before, and consequently, p^{j}f_{j} has no contribution to the value of η_{∞}(h̄(π_{∞})). On the other hand, using

g_{i}(x) = σ^{r−i−1}(f_{i})(x) = b_{i,1}x + ⋯ + b_{i,d_{i}}x^{d_{i}}

and the symmetric powers S(k) in γ_{j}’s and arguing the same way as bofore, we obtain

Tr_{K_{n}/K}(f_{i}(γ_{1}) + ⋯ + f_{i}(γ_{D})) = Tr_{K_{n}/K}(g_{i}(γ_{1}^{p^{r−i−1}}) + ⋯ + g_{i}(γ_{D}^{p^{r−i−1}}))
= Tr_{K_{n}/K}(b_{i,1}S(p^{r−i−1}) + ⋯ + b_{i,d_{i}}S(d_{i}p^{r−i−1}))
≡ Tr_{K_{n}/K}(±b_{i,d_{i}}d_{i}p^{r−i−1}c_{D}) (mod p^{r−i})

since all terms occurring in the expression (3.2) of relevant symmetric powers not involving c_{D} are multiples of p^{r−i}, and c_{D} occurs only once, in S(D), which is multiplied by ±1. This gives

η_{∞}(h̄(π_{∞})) = ψ ∘ Tr_{K_{n}/K}(±b_{i,d_{i}}d_{i}p^{r−1}c_{D}).

Since b_{i,d_{i}} and d_{i} are both units in K_{n}, as a_{D} runs through all elements in κ_{n}, Tr_{K_{n}/K}(±b_{i,d_{i}}d_{i}p^{r−1}c_{D}) runs through all elements in P^{r−1}/P^{r}, on which ψ is nontrivial. This shows that η_{∞} is nontrivial on 1+P_{∞}^{D}, hence it has conductor (D+1)∞.

§5. A Proof of Theorem 2.

Let α_{1}, ..., α_{a} be the roots of g in T . Note that these roots are conjugate under the group Gal(K_{na}/K_{n}). Define η_{w} on the group of units U_{w} at w as follows. Given a polynomial h̄(x) in κ_{n}[x] prime to ḡ with γ_{i}, 1 ≤ i ≤ u, being the lift of its roots in T , define

η_{w}(h̄(x)) = ψ ∘ Tr_{K_{n}/K}(f(γ_{1}) + ⋯ + f(γ_{u}))^{−1}.

Clearly η_{w} is multiplicative on the polynomials in U_{w} and it is trivial on κ_{n}^{×}. To continue, we distinguish three cases.

Case I. g(x) = x so that w = 0. Write

f(x) = f_{0}(x^{−1}) + pf_{1}(x^{−1}) + ⋯ + p^{r−1}f_{r−1}(x^{−1}) + ⋯,

where f_{i} are polynomials of degree d_{i} with coefficients in T ∩K_{n} and f_{i}(0) = 0. Let

D = D_{r,0} = max(p^{r−1}d_{0}, p^{r−2}d_{1}, ..., d_{r−1}).

Suppose h̄ ∈ 1 + P_{0}^{D+1}. Then with π_{0} = x we write

h̄(x) = 1 + b_{D+1}x^{D+1} + ⋯ + b_{u}x^{u} = b_{u} ∏_{1≤i≤u}(x − γ̄_{i})

= ∏_{1≤i≤u}(1 − γ̄_{i}^{−1}π_{0}),

where γ_{i}^{−1} are the roots of the lift of the inverse polynomial of h̄:

H(y) = y^{u} + a_{1}y^{u−1} + ⋯ + a_{u},

with a_{j} ≡ b_{j} (mod p). In particular, a_{j} ≡ 0 (mod p) for 1 ≤ j ≤ D. From definition,

η_{0}(h̄(x)) = ψ ∘ Tr_{K_{n}/K}( ∑_{1≤i≤u} f_{0}(γ_{i}^{−1}) + pf_{1}(γ_{i}^{−1}) + ⋯ + p^{r−1}f_{r−1}(γ_{i}^{−1})).

By the same argument as in the proof of Theorem 1, we find that η_{0}(h̄(x)) = 1 for all h̄ ∈ 1 + P_{0}^{D+1}. Consequently, we may extend η_{0} to the whole group U_{0} by continuity. Further, if none of the nonzero d_{i}’s, 0 ≤ i ≤ r − 1, are a multiple of p, then η_{0} has conductor equal to (D + 1)0.

Case II. g(x) = x − α so that w is a place of degree one other than ∞ and 0. Write

f(x) = f_{0}((x − α)^{−1}) + pf_{1}((x − α)^{−1}) + ⋯ + p^{r−1}f_{r−1}((x − α)^{−1}) + ⋯,

where f_{i} are polynomials with coefficients in T ∩ K_{n} with degree d_{i} and f_{i}(0) = 0. The weighted degree of f at w is

(5.1)  D = D_{r,w} = max(p^{r−1}(d_{0} + 1) − 1, p^{r−2}(d_{1} + 1) − 1, ..., d_{r−1}).

Suppose h ∈ 1 + P_{w}^{D+1}. Then with π_{w} = ḡ we may write

h̄(x) = b_{u} ∏_{1≤i≤u}(x − γ̄_{i}) = b_{u} ∏_{1≤i≤u}((x − ᾱ) − (γ̄_{i} − ᾱ))

= ∏_{1≤i≤u}(1 − (γ̄_{i} − ᾱ)^{−1}π_{w})

= 1 + b_{D+1}π_{w}^{D+1} + ⋯ + b_{u}π_{w}^{u}.

Let β_{i} be the lifting of (γ̄_{i} − ᾱ)^{−1} in T . Then β_{i} are the roots of the lift of the inverse polynomial of h̄:

H(y) = y^{u} + a_{1}y^{u−1} + ⋯ + a_{u},

where a_{j} ≡ b_{j} (mod p). In particular, a_{j} ≡ 0 (mod p) for 1 ≤ j ≤ D. Further, since β_{i} ≡ (γ_{i} −α)^{−1} (mod p) by construction, the ith elementary symmetric polynomial c_{i} on the set Γ = {(γ_{j} −α)^{−1} : 1 ≤ j ≤ u} is congruent to zero mod p for 1 ≤ i ≤ D. It follows from the definition of η_{w} that

η_{w}(h̄(x)) = ψ ∘ Tr_{K_{n}/K}( ∑_{1≤i≤u} ∑_{0≤j≤r−1} p^{j}f_{j}((γ_{i} − α)^{−1}))^{−1}.

Similar to the argument in the proof of Theorem 1, we show that

∑_{1≤i≤u} f_{j}((γ_{i} − α)^{−1})^{σ^{r−j−1}} ≡ 0 (mod p^{r−j})

for 0 ≤ j ≤ r − 1. Here σ denotes the Frobenius automorphism in Gal(K̄/K). Indeed, f_{j}^{σ^{r−j−1}} is a polynomial of degree d_{j} with coefficients in T ∩ K_{n} with no constant term and the sum in question is a linear combination with coefficients in T ∩ K_{n} of the symmetric polynomials S_{X}(x^{k}), where X = Γ^{σ^{r−j−1}} and 1 ≤ k ≤ d_{j}. Applying Corollary 10 (r − j − 1) times by starting with M = d_{j} + 1, a = r − j − 1, e = 1 and then successively reducing a by 1 and increasing e by 1, we arrive at c_{i}^{σ^{r−j−1}} ≡ 0 (mod p^{r−j}) for 1 ≤ i ≤ d_{j}, which then imply S_{X}(x^{k}) ≡ 0 (mod p^{r−j}) by Proposition 2. This shows that η_{w}(h̄) = 1 for polynomials h̄ ∈ 1 + P_{w}^{D+1}. Hence we may extend η_{w} to a character of U_{w} with conductor at most (D + 1)w by continuity. Moreover, if h̄ is a polynomial in 1 + P_{w}^{D}, p^{r−j−1}(d_{j} + 1) − 1 is the only term occurring in (5.1) achieving the maximum D and p does not divide d_{j}, then, following the same argument as above by applying Corollary 10 repeatedly, we get

η_{w}(h̄(x)) = ψ ∘ Tr_{K_{n}/K}(p^{j}bc_{d_{j}}^{σ^{r−j−1}}),

where b is the leading coefficient of f_{j}^{σ^{r−j−1}}, and

c_{d_{j}}^{σ^{r−j−1}} ± p^{r−j−1}α^{r−j−1}c_{D} ≡ 0 (mod p^{r−j}).

As h̄ varies in 1+P_{w}^{D}, the coefficient a_{D} of the inverse polynomial of h runs through all residues in O_{n} mod p, and so does c_{D}. This shows that η_{w} is nontrivial on 1+P_{w}^{D}. In other words, the conductor of η_{w} is (D + 1)w in this case.

Finally assume r ≥ 2 and f = c/(x − α)^{k} with c ∈ T ∩ K_{n}^{×} and k = p^{s}m, where m is prime to p and s ≥ r − 1. Let D′ = p^{r−1}m. We show that the conductor of η_{w} for this case is at most (D′ + 1)w, lower than (D + 1)w. Let h̄ be a polynomial in 1 + P_{w}^{D′+1}. Proceeding as before with the same notation, we have to compute

η_{w}(h̄(x)) = ψ ∘ Tr_{K_{n}/K}( ∑_{1≤i≤u} c/(γ_{i} − α)^{k}))^{−1} = ψ ∘ Tr_{K_{n}/K}(cS_{Γ}(x^{k})).

Express S_{Γ}(x^{k}) as a sum (3.2). We see that the terms with b := ord_{p} gcd(i_{1}, ..., i_{k}) ≤ s − r are congruent to zero mod p^{r} since each c_{i} is integral; while if b ≥ s − r + 1, we see that the largest possible j with i_{j} nonzero is j = kp^{−b} ≤ p^{r−1}m. As c_{j} ≡ 0 (mod p) for 1 ≤ j ≤ kp^{−b} ≤ D′ by assumption, for such choice of i_{1}, ..., i_{k} we have c_{1}^{i_{1}} ⋯ c_{k}^{i_{k}} ≡ 0 (mod p^{p^{b}}), which, when combined with Proposition 1, in turn implies that the corresponding full term is congruent to 0 mod p^{r}. This shows that S_{Γ}(x^{k}) ≡ 0 (mod p^{r}), and consequently, η_{w}(h̄(x)) = 1, as desired. Furthermore, if s = r − 1 and h̄(x) is a polynomial in 1 + P_{w}^{D′}, then S_{Γ}(x^{p^{r−1}m}) ≡ ±p^{r−1}mc_{D′} (mod p^{r}) by Remark 7. So as h̄(x) varies among all such polynomials, c_{D′} runs through all residues in O_{n} mod p and some h̄(x) will make η_{w}(h̄(x)) ≠ 1. This proves that the conductor of η_{w} is (D′ + 1)w when s = r − 1.

Case III. a = deg g = deg w > 1. Decompose f as a sum h_{1}((x − α_{1})^{−1}) + ⋯ + h_{a}((x − α_{a})^{−1}), where h_{1}, ..., h_{a} are polynomials with coefficients in T ∩ K_{an}. The place w of κ_{n}(x) decomposes into a places w_{1}, ..., w_{a} of κ_{na}(x) of degree one, which are not equal to 0 or ∞. For 1 ≤ i ≤ a, let η_{w_{i}} be the character of the group of units U_{w_{i}} of the completion of κ_{na}(x) at w_{i} attached to ψ and h_{i}((x − α_{i})^{−1}) as in Case II. Since U_{w} is contained in each U_{w_{i}}, it follows from the definition of the characters that for a polynomial h̄ ∈ U_{w} with roots γ̄_{1}, ..., γ̄_{u}, we have

η_{w_{i}}(h̄) = ψ ∘ Tr_{K_{na}/K}( ∑_{1≤j≤u} h_{i}((γ_{j} − α_{i})^{−1}))^{−1}.

Further, since f is over K_{n}, the functions h_{1}((x − α_{1})^{−1}), ..., h_{a}((x − α_{a})^{−1}) are conjugate under Gal(K̄/K_{n}); let τ_{i} be an element in Gal(K_{na}/K_{n}) such that τ_{i}(h_{1}((x − α_{1})^{−1})) = h_{i}((x − α_{i})^{−1}). As τ_{i} permutes γ_{1}, ..., γ_{u}, we find

η_{w_{1}}(h̄) = ψ ∘ Tr_{K_{na}/K}( ∑_{1≤j≤u} h_{1}((γ_{j} − α_{1})^{−1}))^{−1}

= ψ ∘ Tr_{K_{na}/K}( ∑_{1≤j≤u} h_{i}((γ_{j} − α_{i})^{−1}))^{−1} = η_{w_{i}}(h̄)

for 1 ≤ i ≤ a. Therefore

η_{w_{1}}(h̄) = ψ ∘ Tr_{K_{na}/K}( ∑_{1≤j≤u}(τ(h_{1}((γ_{j} − α_{1})^{−1})))^{−1}

= ψ ∘ Tr_{K_{n}/K}( ∑_{1≤j≤u} ∑_{1≤i≤a} h_{i}((γ_{j} − α_{i})^{−1}))^{−1}

= ψ ∘ Tr_{K_{n}/K}( ∑_{1≤j≤u} f(γ_{j}))^{−1} = η_{w}(h̄).

Denote by D the weighted r-degree D_{r,w} of f at w, which is also the weighted r-degree D_{r,w_{1}} of h_{1}((x − α_{1})^{−1}) at w_{1} by definition. Thus if h̄ ∈ 1 + P_{w}^{D+1}, then it lies in 1 + P_{w_{1}}^{D+1} on which η_{w_{1}} is trivial. Hence η_{w} is trivial on 1 + P_{w}^{D+1} and we may extend η_{w} by continuity to get a character on U_{w} which has conductor at most (D + 1)w. As w and w_{1} have the same residue field κ_{na} and ḡ is a uniformizer at w and at w_{1}, the conductor of η_{w} is mw if and only if the conductor of η_{w_{1}} is mw_{1}. Thus if exactly one of p^{r−j−1}(d_{j} + 1) − 1 in (5.1) is equal to D and p does not divide d_{j}, we conclude from Case II that the conductor of η_{w} is equal to (D + 1)w. Finally, we examine the conductor of η_{w} attached to f = (g_{1}/g)^{p^{s}m}, where g_{1} is a polynomial over O_{n} with deg g_{1} < deg g and ḡ_{1} ≠ 0, m is prime to p and s ≥ r − 1. Write g_{1}(x)/g(x) = ∑_{1≤i≤a} b_{i}/(x − α_{i}), where b_{i} ∈ O_{na} is a sum of c_{i} ∈ T ∩ K_{na}^{×} (since ḡ_{1} ≠ 0) and pe_{i} ∈ pO_{na}. Consequently, we may write g_{1}/g as X + pY with X = ∑_{1≤i≤a} c_{i}/(x − α_{i}). Given a polynomial h̄ in U_{w} of degree u with the roots of h being γ_{1}, ..., γ_{u}, it follows from the definition that

η_{w}(h̄) = ψ ∘ Tr_{K_{na}/K}( ∑_{1≤j≤u}(X(γ_{j}) + pY (γ_{j}))^{p^{s}m})

= ψ ∘ Tr_{K_{na}/K}( ∑_{1≤j≤u} X(γ_{j})^{p^{s}m})

since X(γ_{j}) and Y (γ_{j}) are integral and

(X(γ_{j}) + pY (γ_{j}))^{p^{s}m} ≡ X(γ_{j})^{p^{s}m} (mod p^{r}).

Therefore we may assume that b_{i} = c_{i} to begin with. As proved in Case II, the conductor of η_{w_{1}} is at most (p^{r−1}m + 1)w_{1} with equality when s = r − 1, so the conductor of η_{w} is at most (p^{r−1}m + 1)w with equality when s = r − 1.

In conclusion, we have defined a character η_{w} on U_{w} whose conductor is at most (D +1)w, with the precise conductor as described in (2.d). For a place v ≠ w, let η_{v} be trivial on U_{v} and define η_{∞}(1/x) = 1. Put together, we have defined a character η on F_{∞}^{×} ∏_{v≠∞} U_{v}, trivial on κ_{n}^{×}. Here F denotes the function field κ_{n}(x). Thus it extends to a character η of the idèles I_{F} trivial on F^{×}. It satisfies (2.a), (2.c) and (2,d). Finally, at a place v other than ∞ and w, write π_{v} = ∏_{1≤i≤deg v}(x − γ̄_{i}). Then

η_{v}(π_{v}) = η_{w}(π_{v})^{−1} = ψ ∘ Tr_{K_{n}/K}(f(γ_{1}) + ⋯ + f(γ_{deg v})),

as described in (2.b).

Remark. (8) As shown in Cases II and III, the character η attached to f = (g_{1}/g)^{p^{(r−1)}m} with g_{1} a polynomial over T ∩ K_{n} of degree less than a, the degree of g, and m prime to p is identical with the character attached to (Tr_{K_{na}/K_{n}}(c/(x − α_{1})))^{p^{(r−1)}m} for some c in T ∩ K_{na}. Here α_{1} is a fixed root of g. Since different g_{1} yields different c and different choices of c give rise to different idèle class characters, so as g_{1} varies among the polynomials as described, we obtain q^{na} −1 different idèle class characters of conductor (p^{r−1}m+1)w from nonzero g_{1} and one trivial character from g_{1} = 0.

§6 A proof of Theorem 3.

Given f as in Theorem 3 with S denoting the set of places of F := κ_{n}(x) which divides f̄ and T denoting the union of S and {∞}, define, at each place v of F outside T, a character η_{v} of F_{v}^{×} so that η_{v} is trivial on the group of units at v and at the uniformizer π_{v} it is given by

η_{v}(π_{v}) = χ(∏_{γ} f(γ)),

where γ runs through all roots (in T ) of the lifting P_{v} of π_{v}, as in (3.a). Thus the product of η_{v} over the places v not in T defines an unramified character η^{T} on the subgroup I_{F}^{T} of idèles of F whose components are trivial at places in T. Since

I_{F} = F^{×} · I_{F}^{T} ∏_{v∈T}(1 + P_{v}^{e_{v}})

for any positive integers e_{v}, if we can show that η^{T} is trivial on the intersection of F^{×} with I_{F}^{T} ∏_{v∈T} (1 + P_{v}^{e_{v}}) with e_{v} as indicated in (3.b) and (3.c), then η^{T} extends to an idèle class character η of I_{F} of conductor at most ∑_{v∈T} e_{v}v, and thus establishes Theorem 3.

We first deal with the case where f is a power of x. It suffices to prove the statement for f = x, which we shall assume. The set T consists of two places 0 and ∞. We shall prove that η^{T} is trivial on the intersection of F^{×} with I_{F}^{T} ∏_{v∈T} (1 + P_{v}). Take an element in this intersection, write it as the quotient of two coprime polynomials ḡ(x)/h̄(x) in F. Then ḡ(0) = h̄(0) is a nonzero element in κ_{n}, deg ḡ = deg h̄, and ḡ and h̄ have the same leading coefficient, which we may assume to be 1. The liftings g and h over K_{n} are monic polynomials of the same degree, prime to x, and g(0) ≡ h(0) (mod p). As g(0) and h(0) are roots of unity in T , being congruent mod p implies that they are in fact equal. It then follows from definition that

η^{T} (ḡ/h̄) = χ(g(0)/h(0)) = 1,

as desired. Therefore η^{T} extends to an idèle class character η with conductor at most 0 + ∞.

Since given any nonzero element α in κ_{n}, we can find two monic polynomials ḡ and h̄ in F of the same degree such that ḡ(0)/h̄(0) = α, from

1 = η(ḡ/h̄) = η^{T} (ḡ/h̄)η_{0}(ḡ/h̄)

it follows that

η_{0}(ḡ/h̄) = χ(g(0)/h(0))^{−1}.

This shows that η is unramified at 0 if and only if χ is trivial on T ∩ K_{n}^{×}. Further, since there is no idèle class character with conductor of degree 1, η is unramified at 0 if and only if it is unramified. Hence we conclude that for f = x, η has conductor 0 + ∞ if χ is nontrivial on T ∩ K_{n}^{×}, and trivial otherwise.

Next consider the case where f is prime to x. We may assume that f is monic. We shall show that η^{T} is trivial on the intersection of F^{×} with I_{F}^{T} · ∏_{v∈S}(1 + P_{v}^{p^{r−1}}) · (1 + P_{∞}). As before, take any element in this intersection and write it as the quotient ḡ/h̄ of two coprime polynomials in F. The condition at ∞ implies that ḡ and h̄ have the same degree M and the same leading coefficient, which we may assume to be 1. The condition at places in S implies that both ḡ and h̄ are units at the places in S, and for each root α of f, when we express the liftings g and h as polynomials in powers of x − α, the last p^{r−1} coefficients are congruent mod p. In other words, write

g(x) = ∑_{i=0}^{M} a_{i}(x − α)^{i}

and

h(x) = ∑_{i=0}^{M} b_{i}(x − α)^{i},

then

a_{i} ≡ b_{i} (mod p)  for  0 ≤ i < p^{r−1}

and a_{0} and b_{0} are nonzero mod p. Here of course the coefficients depend on α. To express the value of η^{T} at ḡ/h̄, factor

g(x) = ∏_{i=1}^{M}(x − γ_{i})  and  h(x) = ∏_{i=1}^{M}(x − δ_{i}).

Then

η^{T} (ḡ/h̄) = χ(∏_{α} ∏_{i=1}^{M}(γ_{i} − α)/(δ_{i} − α)) = χ(∏_{α} g(α)/h(α)),

where α runs through all roots of f. For our purpose it suffices to prove that, for each α,

a_{0} ≡ g(α) ≡ (−1)^{M} ∏_{i}(γ_{i} − α) ≡ (−1)^{M} ∏_{i}(δ_{i} − α) ≡ h(α) ≡ b_{0} (mod p^{r}).

Denote by σ the Frobenius automorphism in Gal(K̄/K). The above is then equivalent to

a_{0}^{σ^{r−1}} ≡ b_{0}^{σ^{r−1}} (mod p^{r}).

The i-th symmetric power c_{i} (resp. d_{i}) of (γ_{j} −α)^{−1} (resp. (δ_{j} −α)^{−1}) is (−1)^{i}a_{i}/a_{0} (resp. (−1)^{i}b_{i}/b_{0}) for 1 ≤ i ≤ M, and the product c_{0} (resp. d_{0}) of γ_{j} − α (resp. δ_{j} − α) over 1 ≤ j ≤ M is (−1)^{M}a_{0} (resp. (−1)^{M}b_{0}). We have c_{i} ≡ d_{i} (mod p) for 0 ≤ i ≤ p^{r−1}. Applying Corollary 9 to c_{i}, d_{i}’s, we get congruence information on c_{i}^{σ} and d_{i}^{σ}. Apply Corollary 9 to c_{i}^{σ} and d_{i}^{σ} to get congruence relation on c_{i}^{σ^{2}} and d_{i}^{σ^{2}}. Repeating this procedure r − 1 times, finally we arrive at the desired congruence on a_{0} and b_{0}. Hence η^{T} can be extended to an idèle class character η of F satisfying (3.a) and has conductor at most ∞ + ∑_{v∈S} p^{r−1}v.

It remains to check the conductor of η. Assume first that f = P_{v}, the lifting of π_{v} at a place v of F other than 0 and ∞. Choose polynomials ḡ and h̄ in F such that

(6.1) at ∞, ḡ ≡ h̄ (mod P_{∞});

(6.2) at v, h̄ ≡ 1 (mod P_{v}^{p^{r−1}}) and ḡ ≡ 1 (mod P_{v}^{p^{r−1}−1}).

¿From (6.1) we have

1 = η(ḡ/h̄) = η^{T} (ḡ/h̄)η_{v}(ḡ/h̄)

and hence

η_{v}(ḡ/h̄) = η^{T} (ḡ/h̄)^{−1} = χ(∏_{α} g(α)/h(α))^{−1},

where α runs through all roots of f. In view of (6.1), we know that g and h have the same degree M and the same leading coefficient which we may assume to be 1. Choose a root α of f and write g and h as polynomials in powers of x − α as before with coefficients a_{i} and b_{i}, respectively. The assumption (6.2) implies that a_{i} ≡ b_{i} (mod p) for 0 ≤ i ≤ p^{r−1} −2, b_{0} ≡ 1 (mod p) and b_{i} ≡ 0 (mod p) for 1 ≤ i < p^{r−1}. Denote by Γ (resp. Δ) the set consisting of (γ_{i} − α)^{−1} (resp. (δ_{i} − α)^{−1}), where γ_{i} (resp. δ_{i}) are roots of g (resp. h) respectively. As in Theorem 7, denote by c_{0}, d_{0} the product of the reciprocals of the elements in Γ, Δ, respectively, and for i ≥ 1, let c_{i}, d_{i} be the degree i symmetric polynomial in Γ and Δ, respectively. Then a_{0} = (−1)^{M}c_{0}, b_{0} = (−1)^{M}d_{0}, and for i ≥ 1, a_{i} = (−1)^{i}a_{0}c_{i} and b_{i} = (−1)^{i}b_{0}d_{i}. Since a_{0} ≡ b_{0} ≡ 1 (mod p), the congruence conditions on a_{i}, b_{i} pass to c_{i} and d_{i}. The dependence of a_{i}, b_{i}, c_{i}, d_{i} on α will be indicated when needed.

We discuss first the case r ≥ 2. Applying Corollary 10 to d_{i}, we get d_{i}^{σ} ≡ 0 (mod p^{2}) for 1 ≤ i < p^{r−2}. Applying Theorem 7, we get c_{i}^{σ} ≡ d_{i}^{σ} (mod p^{2}) for 0 ≤ i ≤ p^{r−2} − 2 and, if r ≥ 3,

c_{p^{r−2}−1}^{σ} ± pα^{p−1}c_{p^{r−1}−1} ≡ d_{p^{r−2}−1}^{σ} ± pα^{p−1}d_{p^{r−1}−1} ≡ 0 (mod p^{2}).

Applying Corollary 10 to d_{i}^{σ^{j}} and Theorem 7 to Γ^{σ^{j}}, Δ^{σ^{j}} and α^{σ^{j}} successively for j = 1, …, r − 2 and using results obtained at the previous stage, we arrive at d_{i}^{σ^{r−2}} ≡ 0 (mod p^{r−1}) for 1 ≤ i < p and c_{i}^{σ^{r−2}} ≡ d_{i}^{σ^{r−2}} (mod p^{r−1}) for 0 ≤ i ≤ p−2 and

(6.3)  c_{p−1}^{σ^{r−2}} ≡ ±p^{r−2}α^{p^{r−2}−1}c_{p^{r−1}−1} (mod p^{r−1}).

Applying Theorem 7 once more, we get

c_{0}^{σ^{r−1}} ± p(c_{0}^{σ^{r−2}})^{p}α^{p^{r−2}(p−1)}c_{p−1}^{σ^{r−2}} ≡ d_{0}^{σ^{r−1}} (mod p^{r}),

which, combined with (6.3) and the congruences obtained at the previous stage, yields

c_{0}^{σ^{r−1}} ≡ d_{0}^{σ^{r−1}} ± p^{r−1}α^{p^{r−1}−1}c_{p^{r−1}−1} ≡ d_{0}^{σ^{r−1}}[1 + ±p^{r−1}α^{p^{r−1}−1}c_{p^{r−1}−1}] (mod p^{r})

since d_{0} ≡ (−1)^{M} (mod p). Notice that this congruence also holds when r = 2. Hence for r ≥ 2, we obtain

χ((−1)^{M}g(α)) = χ(c_{0})
= χ((−1)^{M}h(α))χ(σ^{−(r−1)}(1 ± p^{r−1}α^{p^{r−1}−1}c_{p^{r−1}−1})).

Let m be the degree of f, then the splitting field of f over K_{n} is the unramified extension K_{mn} of K. The roots α of f are conjugate under Gal(K_{mn}/K_{n}), so are the corresponding values c_{p^{r−1}−1}(α). Therefore we have

η_{v}(ḡ/h̄) = ∏_{f(α)=0} χ(σ^{−(r−1)}(1 ± p^{r−1}α^{p^{r−1}−1}c_{p^{r−1}−1}(α)))^{−1}
= χ(1 ± p^{r−1}σ^{−(r−1)}(Tr_{K_{mn}/K_{n}}(α^{p^{r−1}−1}c_{p^{r−1}−1}(α))))^{−1}.

Now we take a closer look of the coefficient c_{p^{r−1}−1}(α) modulo p. From (6.2), ḡ(x) = q̄(x)f̄(x)^{p^{r−1}−1} + 1 for some polynomial q̄(x) in F. When we expand f̄(x) as a polynomial in powers of x − ᾱ, the lowest degree term is x − ᾱ with coefficient f̄′(ᾱ), where f′(x) denotes the derivative of f(x). Therefore

ḡ(x) = 1 + q̄(ᾱ)(f̄′(ᾱ))^{p^{r−1}−1}(x − ᾱ)^{p^{r−1}−1} + ⋯,

which implies

α^{p^{r−1}−1}c_{p^{r−1}−1}(α) ≡ q(α)(αf′(α))^{p^{r−1}−1} (mod p).

Since any element z in κ_{mn} can be written as a linear combination ∑_{0≤i≤m−1} z_{i}ᾱ^{i} with coefficients z_{i} in κ_{n}, the polynomial q̄(x) = ∑_{0≤i≤m−1} z_{i}x^{i} in F has the property q̄(ᾱ) = z. In other words, as polynomials q̄ in F vary, the values q(α) (mod p) run through all elements in κ_{mn}. Since α and f′(α) are units in K_{mn}, Tr_{K_{mn}/K_{n}} on O_{mn} modulo p reduces to Tr_{κ_{mn}/κ_{n}}, which has image κ_{n}, and σ induces the Frobenius automorphism on κ_{n}, we conclude that there is a choice of ḡ and h̄ satisfying (6.1) and (6.2) such that η_{v}(ḡ/h̄) ≠ 1. This shows that η_{v} has conductor equal to p^{r−1}v when r ≥ 2. When r = 1, we have h(α) = b_{0} ≡ 1 (mod p) and g(α) = a_{0}, which can be any nonzero element in κ_{mn} when reduced modulo p. In this case we see from

η_{v}(ḡ/h̄) = ∏_{f(α)=0} χ(g(α)/h(α))^{−1} = χ ∘ N_{K_{mn}/K_{n}}(g(α))^{−1}

that η_{v} has conductor v. In conclusion, for f = P_{v}, the conductor of the associated η at v is p^{r−1}v. Since η_{χ,f} is multiplicative in f, for a general f such that the order of f̄ at v ∈ S is s_{v} and the conductor of χ^{s_{v}} is r_{v}, the conductor of the associated η_{χ,f} at v is p^{r_{v}−1}v, as asserted in (3.c).

Finally we examine the conductor of η at ∞. Since the multiplicative group of the residue field at ∞ has order prime to p, η is ramified at ∞ if and only if η^{p^{r−1}} is. But η^{p^{r−1}} is constructed from f and χ^{p^{r−1}}, which has conductor at most 1, and which is nontrivial if and only if χ is nontrivial on T ∩ K_{n}^{×}. Replacing χ by χ^{p^{r−1}} if necessary, we may assume that χ has conductor 1 to begin with. Again, we study first the case f = P_{v}. Write m for the degree of v and denote by T the set consisting of the places v and ∞ of F. Given a root of unity a in K_{n}, consider the polynomials

ḡ = āf̄+ 1  and  h̄ = f̄+ 1

in F. From

1 = η(ḡ/h̄) = η^{T} (ḡ/h̄)η_{∞}(ā)

we get

η_{∞}(ā) = η^{T} (ḡ/h̄)^{−1} = χ(∏_{i=1}^{m} f(γ_{i})/f(δ_{i}))^{−1},

where γ_{i} and δ_{i} are the roots of the lifting g, h of ḡ and h̄, respectively. Denote by α_{i}, i = 1, …, m, the roots of f. Then

∏_{i=1}^{m} f(γ_{i}) = ∏_{i,j=1}^{m}(γ_{i} − α_{j}) = (−1)^{m} ∏_{j=1}^{m} a^{−1}g(α_{j}) ≡ (−1)^{m}a^{−m} (mod p),

and similarly

∏_{i=1}^{m} f(δ_{i}) ≡ (−1)^{m} (mod p).

This implies that

η_{∞}(ā) = χ(a)^{m}

since χ has conductor 1. Hence for a general f, the associated η = η_{χ,f} at ∞ satisfies

η_{∞}(ā) = χ(a)^{deg f}  for  a ∈ T ∩ K_{n}^{×}.

This shows that η_{∞} ramifies if and only if χ^{deg f} is nontrivial on T ∩ K_{n}^{×}. This completes the proof of Theorem 3.

§7 A proof of Theorem 4.

Let f, χ and S be as in Theorem 4. For each place v of F, let d_{v} denote the order of pole of f at v. Denote by T the union of S and {∞}. Given an integer e ≥ 1, at each place v of F = κ_{n}(x) outside T, define an unramified character η_{v} on F_{v}^{×} whose value at the uniformizer π_{v} is given by

η_{v}(π_{v}) = χ(∏_{γ}(1 + p^{e}f(γ)),

where γ runs through the lifting over K_{n} of the roots of π_{v} . As in the proof of Theorem 3, the product of η_{v} over the places v of F not in T, denoted by η^{T}, is an unramified character of the subgroup I_{F}^{T} of the group of idèles I_{F}, and it extends to an idèle class character η trivial on ∏_{v∈T} (1 + P_{v}^{e_{v}}) if and only if η^{T} is trivial on the elements of F^{×} which are in 1 + P_{v}^{e_{v}} for all v ∈ T. We shall show that this can be achieved with the choice e_{v} = (d_{v} + 1)p^{r−e−1} for v ∈ S, and e_{∞} = 1 + d_{∞}p^{r−e−1}.

Let ḡ and h̄ be two polynomials in F prime to π_{v} for all v ∈ S and such that ḡ ≡ h̄ (mod P_{v}^{e_{v}}) for each v in T. Let g and h be their lifting over K_{n}, thus they have the same degree N, say, and the same leading coefficient which we may assume to be 1. Denote by Γ and Δ the set of roots (with multiplicity) of g and h in T , respectively. Our goal is to show

χ(∏_{γ∈Γ}(1 + p^{e}f(γ))) = χ( ∏_{δ∈Δ}(1 + p^{e}f(δ))).

To prove this, observe that

∏_{γ∈Γ}(1 + p^{e}f(γ))

= 1 + p^{e}S_{Γ}(f) + p^{2e}S_{Γ}(f, f) + p^{3e}S_{Γ}(f, f, f) + ⋯

= (1 + p^{e}S_{Γ}(f))(1 + p^{2e}S_{Γ}(f, f))(1 + p^{3e}(S_{Γ}(f, f, f) − S_{Γ}(f)S_{Γ}(f, f))) · · · .

Since χ has conductor r, when we apply χ to the above product, the value depends only on the product of the first ⌊r/e⌋ terms, hence it suffices to show that, for each 1 ≤ m < r/e and f_{1} = ⋯ = f_{m} = f, we have

S_{Γ}(f_{1}, …, f_{m}) ≡ S_{Δ}(f_{1}, …, f_{m}) (mod p^{r−ie})

for all m ≤ i < r/e. As the Frobenius automorphism σ preserves the p-adic order of elements in K̄, the above is equivalent to

(7.1)  S_{Γ}(f_{1}, …, f_{m})^{σ^{r−ie−1}} ≡ S_{Δ}(f_{1}, …, f_{m})^{σ^{r−ie−1}} (mod p^{r−ie})

Observe that (7.1) will follow from

(7.2)  S_{Γ}(f^{m})^{σ^{r−ie−1}} ≡ S_{Δ}(f^{m})^{σ^{r−ie−1}} (mod p^{r−ie})

for all m, i with 1 ≤ m ≤ i < r/e. Indeed, applying Theorem 7 to f_{1} = ⋯ = f_{m} = f, we get

S_{Γ}(f_{1}, …, f_{m}) = ∑_{q=\{q_{1},...,q_{ℓ}\}∈P(m)} sgn(q)\frac{m!}{m(q)q_{1} ⋯ q_{ℓ}} S_{Γ}(f^{q_{1}}) ⋯ S_{Γ}(f^{q_{ℓ}}).

Recall that the number m(q) is i_{1}! ⋯ i_{s}!, where s is the number of distinct parts in the partition q = {q_{1}, …, q_{ℓ}}, and i_{1}, …, i_{s} counts the number of times the distinct parts are repeated. Thus the coefficient \frac{m!}{m(q)q_{1}···q_{ℓ}} is an integer. Hence (7.1) follows from (7.2)

Now we proceed to prove (7.2). By assumption, all poles of f are in T . Express f in its partial fractions:

f = ∑_{j=0}^{d_{∞}} a_{∞j}x^{j} + ∑_{α} ∑_{j=1}^{d_{α}} a_{αj}(x − α)^{−j},

where a_{vj} are integral elements in K̄, α runs through the set A of distinct poles of f, and d_{α} denotes the order of α as a pole of f. Likewise, f^{m} has a similar expression as a linear combination of powers of x and (x − α)^{−1} with coefficients integral in K̄. Since the symmetric function S_{X}(z) is linear in z, it suffices to prove

(7.3) S_{Γ}(x^{j})^{σ^{r−ie−1}} ≡ S_{Δ}(x^{j})^{σ^{r−ie−1}} (mod p^{r−ie})  for  0 ≤ j ≤ md_{∞}

and, for each α ∈ A,

(7.4) S_{Γ}((x−α)^{−j})^{σ^{r−ie−1}} ≡ S_{Δ}((x−α)^{−j})^{σ^{r−ie−1}} (mod p^{r−ie}) for0 ≤ j ≤ md_{α}.

For (7.3), note that S_{Γ}(x^{j})^{σ^{r−ie−1}} = S_{Γ}(x^{jp^{r−ie−1}}) and the two sides of (7.3) are in fact equal when j = 0. So assume j ≥ 1 and d_{∞} ≥ 1. The congruence condition on ḡ and h̄ at ∞ implies that g and h have the same leading term, and the degree k elementary polynomials in Γ and Δ are congruent mod p for 1 ≤ k ≤ d_{∞}p^{r−e−1}. With 1 ≤ j ≤ md_{∞} and m ≤ i ≤ p^{(i−1)e}, we have md_{∞}p^{r−ie−1} ≤ d_{∞}p^{r−e−1}. Hence (7.3) follows from Proposition 2.

To prove (7.4), denote by Γ(α) (resp. Δ(α)) the set consisting of (γ − α)^{−1}, one term for each γ ∈ Γ (resp. (δ − α)^{−1}, δ ∈ Δ). Then we can rewrite S_{Γ}((x − α)^{−j})^{σ^{r−ie−1}} as S_{Γ(α)^{σ^{r−ie−1}}} (x^{j}). Let v be the place corresponding to the Galois orbit of α. The congruence condition on ḡ and h̄ at the place v implies that g(α) and h(α) are units and they are congruent mod p, and, furthermore, the degree k elementary polynomials c_{k}, d_{k} in Γ(α) and Δ(α) are congruent modulo p for 1 ≤ k < (d_{α} + 1)p^{r−e−1} since d_{v} = d_{α}. Applying Corollary 9 (r − ie − 1) times, we arrive at

(7.5) c_{k}^{σ^{r−ie−1}} ≡ d_{k}^{σ^{r−ie−1}} (mod p^{r−ie})  for  1 ≤ k < (d_{α} + 1)p^{(i−1)e}.

The assumption 1 ≤ m ≤ i implies md_{α} < (d_{α} + 1)p^{(i−1)e}, hence (7.4) follows from (7.5) and Proposition 2. Therefore η^{T} extends to an idèle class character η of F with conductor at most (1 + d_{∞}p^{r−e−1})∞ + ∑_{v∈S}(d_{v} + 1)p^{r−e−1}v.

It remains to check the conductor of η. Choose a monic polynomial h̄ in F of degree N, say, such that x^{−N} h̄ ≡ 1 (mod P_{∞}^{(d_{∞}+1)p^{r−e−1}}) and at each pole v of f other than ∞, h̄ ≡ 1 (mod P_{v}^{(d_{v}+1)p^{r−e−1}}). We start with the place at ∞. Choose a polynomial ḡ in F of the same degree N such that at each place v ≠ ∞ where f has a pole, ḡ ≡ h̄ (mod P_{v}^{(d_{v}+1)p^{r−e−1}}), and at the place ∞, ḡ ≡ h̄ (mod P_{∞}^{d_{∞}p^{r−e−1}}). Denote by c_{i}, b_{i} the coefficient of x^{N−i} of the lifting g and h, respectively. Then c_{i} ≡ b_{i} (mod p) for 0 ≤ i < d_{∞}p^{r−e−1}, and b_{0} = 1, b_{i} ≡ 0 (mod p) for 1 ≤ i ≤ d_{∞}p^{r−e−1}. Let Γ and Δ be the set of roots of g and h respectively. Proceeding as before, we see that (7.4) holds for all α ∈ A and all 1 ≤ m ≤ i < r/e , while (7.3) holds for all 1 ≤ m ≤ i < r/e except when m = i = 1 and j = d_{∞}. Therefore we may assume that f is a polynomial of degree d_{∞} with the leading coefficient a_{∞d_{∞}} a unit in K_{n}. We examine the exceptional case.

The only obstruction for (7.1) to hold in the exceptional case comes from the failure of (7.3) with i = 1 and j = d_{∞}. Noting S_{Γ}(x^{d_{∞}})^{σ^{r−e−1}} = S_{Γ}(x^{d_{∞}p^{r−e−1}}) and using Remark (7), we have

S_{Γ}(f)^{σ^{r−e−1}} ± a_{∞d_{∞}}^{σ^{r−e−1}}d_{∞}p^{r−e−1}c_{d_{∞}p^{r−e−1}}
≡ S_{Δ}(f)^{σ^{r−e−1}} ± a_{∞d_{∞}}^{σ^{r−e−1}}d_{∞}p^{r−e−1}b_{d_{∞}p^{r−e−1}} ≡ S_{Δ}(f)^{σ^{r−e−1}} (mod p^{r−e}).

Consequently,

η_{∞}(ḡ/h̄) = χ(1 ± p^{r−1}a_{∞d_{∞}}d_{∞}σ^{−(r−e−1)}(c_{d_{∞}p^{r−e−1}}))^{−1}.

As ḡ varies among the polynomials in F satisfying the congruence conditions, the corresponding coefficient c_{d_{∞}p^{r−e−1}} and hence σ^{−(r−e−1)}(c_{d_{∞}p^{r−e−1}}) modulo p runs through all elements in κ_{n}. Since a_{∞d_{∞}} is a unit, so if d_{∞} is not a multiple of p, then there is a polynomial ḡ such that η(ḡ/h̄) ≠ 1. This proves that the conductor of η at ∞ is (1 + d_{∞}p^{r−e−1})∞ if and only if p ∤ d_{∞}.

If 0 ∈ S, we get the same conclusion (with 0 replacing ∞) for the conductor of η_{0} by the same argument as above, with Γ, Δ consisting of the reciprocal roots of g and h, respectively.

Finally we check the conductor of η at v ∈ S and v ≠ 0. For this purpose, choose a monic polynomial ḡ in F of degree N such that ḡ ≡ h̄ (mod P_{w}^{(d_{w}+1)p^{r−e−1}}) at all places w in T other than v, and ḡ ≡ h̄ (mod P_{v}^{(d_{v}+1)p^{r−e−1}−1}). We may assume that f is a polynomial in powers of P_{v}(x) with leading coefficient a_{d_{v}} a unit in K_{n}. Let α be a root of P_{v}(x). Then d_{v} = d_{α}. As in the proof above, denote by Γ and Δ the set of roots of g and h, and set Γ(α) = {(γ − α)^{−1} : γ ∈ Γ} and Δ(α) = {(δ − α)^{−1} : δ ∈ Δ}. Denote by c_{i}(α), b_{i}(α) the degree i symmetric polynomial in Γ(α) and Δ(α), respectively. The congruence conditions on ḡ and h̄ imply that c_{i}(α) ≡ b_{i}(α) (mod p) for 0 ≤ i ≤ (d_{v} + 1)p^{r−e−1} − 2, and b_{i}(α) ≡ 0 (mod p) for 1 ≤ i ≤ (d_{v} + 1)p^{r−e−1} − 1. Then (7.3) and (7.4) hold except for the congruence between S_{Γ}((x − α)^{−d_{α}})^{σ^{r−e−1}} = S_{Γ(α)}(x^{d_{α}})^{σ^{r−e−1}} and S_{Δ(α)}(x^{d_{α}})^{σ^{r−e−1}}, for which we apply Theorem 7 and Corollary 10 and argue as in the proof of Theorem 3 to arrive at

(∏_{γ∈Γ}(1 + p^{e}f(γ)))( ∏_{δ∈Δ}(1 + p^{e}f(δ)))^{−1}

≡ ∏_{P_{v}(α)=0}(1 ± p^{r−1}a_{αd_{v}}α^{1−σ^{−(r−e−1)}}σ^{−(r−e−1)}(c_{(d_{v}+1)p^{r−e−1}−1}(α))) (mod p^{r}).

Therefore

η_{v}(ḡ/h̄) = χ( ∏_{P_{v}(α)=0}(1 ± p^{r−1}a_{αd_{v}}α^{1−σ^{−(r−e−1)}}σ^{−(r−e−1)}(c_{(d_{v}+1)p^{r−e−1}−1}(α)))^{−1}

= χ(1 ± p^{r−1} Tr_{K_{mn}/K_{n}}(a_{αd_{v}}α^{1−σ^{−(r−e−1)}}σ^{−(r−e−1)}(c_{(d_{v}+1)p^{r−e−1}−1}(α))))^{−1}.

Here m is the degree of the place v. Since Tr_{K_{mn}/K_{n}}(a_{αd_{v}}) = a_{d_{v}} is a unit, we conclude that a_{αd_{v}} itself is a unit, and so is α. As explained in the proof of Theorem 3, the above trace modulo p runs through all elements in κ_{n} when ḡ varies among polynomials in F satisfying the prescribed congruence conditions. Hence there is a suitable choice of ḡ such that η_{v}(ḡ/h̄) ≠ 1. This proves that the conductor of η at each place v ∈ S is (d_{v} + 1)p^{r−e−1}v.

§8 Comparison with the Parametrization by Witt Vectors.

Fix an integer r ≥ 1. Let ψ be an additive character of K of order −r. For f, g ∈ O_{n}[x] with g a nonzero lifting from κ_{n}[x], we have attached an idèle class character η_{f/g} := η_{ψ,f/g} of κ_{n}(x). By class field theory, the cyclic group generated by η_{f/g} may be viewed as the group of characters of the Galois group over κ_{n}(x) of a unique cyclic extension κ_{n}(x)_{f/g} of κ_{n}(x) such that the kernel of η_{f/g} is κ_{n}(x) times the image of the idèles of κ_{n}(x)_{f/g} under the norm map from κ_{n}(x)_{f/g} to κ_{n}(x). The field κ_{n}(x)_{f/g} is determined by the set of places in κ_{n}(x) which split completely in κ_{n}(x)_{f/g} up to finitely many exceptions. Such places can be described in terms of trace as follows.

Lemma 1. Let v be a place of degree m of κ_{n}(x) which is not ∞ nor a pole of ḡ. Let α be a root of the lifting of the uniformizer π_{v}. The field K_{n}(α) is a finite unramified extension K_{mn} of K_{n} of degree m, independent of the choice of the root α. Then v splits completely in κ_{n}(x)_{f/g} if and only if Tr_{K_{mn}/K}(f/g(α)) ≡ 0 (mod p^{r}). Proof. v splits in κ_{n}(x)_{f/g} if and only if it is unramified in κ_{n}(x)_{f/g} and π_{v} lies in the kernel of η_{f/g}. This is equivalent to η_{f/g} being unramified at v and

η_{f/g}(π_{v}) = ψ ∘ Tr_{K_{mn}/K}(f/g(α)) = 1.

Since T r_{K_{mn}/K}(f/g(α)) lies in O_{1} and ψ restricted to O_{1}/p^{r}O_{1} ≅ ℤ/p^{r}ℤ has order p^{r}, the latter condition is equivalent to Tr_{K_{mn}/K}(f/g(α)) ≡ 0 (mod p^{r}).

Consequently, the field κ_{n}(x)_{f/g} is independent of the choice of ψ. The trace condition in Lemma 1 has another interpretation:

Lemma 2. Let β be an element in O_{m}. Then Tr_{K_{m}/K}(β) ≡ 0 (mod p^{r}) if and only if β ≡ ξ^{σ} − ξ (mod p^{r}) for some element ξ ∈ O_{m}. Here σ denotes the Frobenius automorphism of K_{m} over K.

Proof. Since Tr_{K_{m}/K}(ξ^{σ} − ξ) = 0 for ξ ∈ O_{m}, the sufficiency is clear. We prove the necessity. Suppose β ∈ O_{m} satisfies Tr_{K_{m}/K}(β) ≡ 0 (mod p^{r}). Then Tr_{K_{m}/K}(β) ≡ 0 (mod p). By Hilbert’s Theorem 90, there exists ξ_{0} ∈ O_{m}∩T such that β ≡ ξ_{0}^{p}−ξ_{0} (mod p). Write β = (ξ_{0}^{p} − ξ_{0}) + pβ_{1} for some β_{1} ∈ O_{m}. Thus Tr_{K_{m}/K}(β_{1}) ≡ 0 (mod p^{r−1}). By induction on r, we may assume β_{1} ≡ ξ_{1}^{σ} − ξ_{1} (mod p^{r−1}) for some ξ_{1} ∈ O_{m}. Then ξ = ξ_{0} + pξ_{1} has the required property.

On the other hand, a constructive description of all abelian p-extensions of κ_{n}(x) was given by Witt using Witt vectors, which we briefly review below. For more details, refer to [J, Chap III, sec. 4] and [Lg, pp. 233-235].

Given a ring R of characteristic p, the Witt ring W_{r}(R) over R with length r is a ring on the set R^{r} under special addition and multiplication rules. More precisely, for 0 ≤ i ≤ r − 1, there are polynomials s_{i}(x_{0}, ..., x_{i}, y_{0}, ..., y_{i}) = x_{i} + y_{i} + t_{i}(x_{0}, ..., x_{i−1}, y_{0}, ..., y_{i−1}) and p_{i}(x_{0}, ..., x_{i}, y_{0}, ..., y_{i}) over ℤ/pℤ with indeterminates x_{0}, ..., x_{i}, y_{0}, ..., y_{i} such that given two elements w = (w_{0}, ..., w_{r−1}) and w′ = (w′_{0}, ..., w′_{r−1}) in W_{r}(R), the i-th component of their sum (resp. product) is s_{i}(w_{0}, ..., w_{i}, w′_{0}, ..., w′_{i}) (resp. p_{i}(w_{0}, ..., w_{i}, w′_{0}, ..., w′_{i})) for 0 ≤ i ≤ r − 1. Hence W_{r}(R′) is a subring of W_{r}(R) if R′ is a subring of R.

The map F sending w = (w_{0}, ..., w_{r−1}) in W_{r}(R) to (w_{0}^{p}, ..., w_{r−1}^{p}) is a ring homomorphism. Given w = (w_{0}, ..., w_{r−1}) ∈ W_{r}(κ_{n}(x)), there exists an element y = (y_{0}, ..., y_{r−1}) ∈ W_{r}(\overline{κ_{n}(x)}) such that F (y) − y = w. Two different choices of y have their components differ in ℤ/pℤ. Hence the extension of κ_{n}(x) obtained by joining the components y_{0}, ..., y_{r−1} of y is independent of the choice of y, which we denote by κ_{n}(x)_{w}. It can be shown that κ_{n}(x)_{w} is a Galois extension of κ_{n}(x), and the cyclic extensions of κ_{n}(x) of degree p^{r} are precisely those κ_{n}(x)_{w} with w_{0} ≠ y_{0}^{p} − y_{0} for any y_{0} in κ_{n}(x).

Lemma 3. Let w and y be as above. Let v ≠ ∞ be a place of κ_{n}(x) such that none of y_{i}, w_{i} have poles at places above v and that the extension κ_{n}(x)_{w} over κ_{n}(x) is unramified at v. Then v splits completely in κ_{n}(x)_{w} if and only if for any root z of π_{v} in an algebraic closure of κ_{n}, the components y_{0}(z), ..., y_{r−1}(z) of y(z), the image of y under the specialization map x ↦ z, all lie in κ_{n}(z).

Proof. Write M_{0} for the field κ_{n}(x) and, for 0 ≤ i ≤ r − 1, denote by M_{i+1} the field M_{0}(y_{0}, ..., y_{i}) so that κ_{n}(x)_{w} is M_{r}. With v as given, it splits completely in M_{r} if and only if it splits completely in each intermediate field M_{i+1} for 0 ≤ i ≤ r − 1. Observe that M_{i+1} = M_{i}(y_{i}) with y_{i} being a root of the polynomial g_{i}(X) = X^{p} − X − w_{i} + t_{i}(y_{0}^{p}, ..., y_{i−1}^{p}, −y_{0}, ..., −y_{i−1}) over M_{i}. Assume inductively that v splits completely in M_{i}, i ≥ 0. This is obvious when i = 0. Let v′ be a place of M_{i} dividing v. Then the completion of M_{i} at v′ is equal to the completion of M_{0} at v with π_{v} being its uniformizer. Thus v′ splits completely in M_{i+1} if and only if g_{i} splits completely over the completion of M_{i} at v′, which holds if and only if over the residue field at v′, ḡ_{i} splits completely. Given any root z of π_{v}, the specialization map x ↦ z induces an isomorphism from the residue field at v′ to the field κ_{n}(z). Under this isomorphism ḡ_{i} is mapped to

h_{i}(X) = X^{p} − X − w_{i}(z) + t_{i}(y_{0}(z)^{p}, ..., y_{i−1}(z)^{p}, −y_{0}(z), ..., −y_{i−1}(z)),

of which y_{i}(z) is a root. Thus ḡ_{i} splits completely over the residue field at v′ if and only if h_{i} splits completely over κ_{n}(z), which happens if and only of y_{i}(z) lies in κ_{n}(z) as all roots of h_{i} are y_{i}(z) + j with j ∈ ℤ/pℤ. This proves Lemma 3.

The ring structure of W_{r}(κ_{n}[x]) can also be seen from the ”ghost” coordinates (in a ring of characteristic zero) of the elements in the ring. For this purpose, we choose the ring of characteristic zero to be O_{n}[x] and regard κ_{n}[x] as O_{n}[x]/pO_{n}[x]. The Teichmuller lifting from κ_{n} to O_{n} ∩ T extends to a lifting, called the Teichmuller lifting, from κ_{n}[x] to O_{n}[x] by lifting the coefficients of polynomials (which is different from the lifting given in §2). Thus the ghost coordinate of

w = (w_{0}, w_{1}, ..., w_{r−1}) ∈ W_{r}(κ_{n}[x])

is

W = (ω_{0}, ω_{0}^{p} + pω_{1}, ..., ω_{0}^{p^{r−1}} + pω_{1}^{p^{r−2}} + ⋯ + p^{r−1}ω_{r−1}) ∈ O_{n}[x]^{r},

where ω_{i}(x) is the Teichmuller lifting of w_{i}(x). If w, w′ are two elements in W_{r}(κ_{n}[x]) with ghost coordinates W, W′, respectively, then w + w′ and ww′ are the elements in W_{r}(κ_{n}[x]) whose ghost coordinates are given by the coordinatewise addition and multiplication of W and W′, respectively. A priori, the ghost coordinate W of w depends on the choice of lifting, nonetheless the ith coordinate of W mod p^{i+1} is independent of the choice of lifting for 0 ≤ i ≤ r − 1, and this is all we need to recover the original Witt vector w. Note that if w lies in W_{r}(κ_{n}), then W is the canonical ghost coordinate of w and in this case W_{r}(κ_{n}) is canonically isomorphic to O_{n}/p^{r}O_{n} via the map sending w to ω_{0} + ω_{1}^{1/p}p + ⋯ + ω_{r−1}^{1/p^{r−1}}p^{r−1}.

Given f ∈ O_{n}[x], write f in its base p expansion

f(x) = f_{0}(x) + pf_{1}(x) + p^{2}f_{2}(x) + ⋯,

where f_{i}(x) = ∑_{j≥0} a_{ij}x^{j} is a polynomials with coefficients a_{ij} in O_{n} ∩ T . In other words, we may write

f(x) ≡ ∑_{j≥0}( ∑_{0≤i≤r−1} a_{ij}p^{i})x^{j} (mod p^{r}).

Denote by ā_{ij} the image of a_{ij} in κ_{n}. Let

w_{f} = ∑_{j≥0}(ā_{0j}x^{j}, ā_{1j}^{p}x^{pj}, ..., ā_{r−1j}^{p^{r−1}j}x^{p^{r−1}j}) ∈ W_{r}(κ_{n}[x]).

Therefore the ghost coordinate of w_{f} is

W_{f} = ∑_{j≥0}(a_{0j}x^{j}, (a_{0j}^{p} + pa_{1j}^{p})x^{pj}, ..., (a_{0j}^{p^{r−1}} + pa_{1j}^{p^{r−1}} + ⋯ + p^{r−1}a_{r−1j}^{p^{r−1}})x^{p^{r−1}j})

= (∑_{j≥0} a_{0j}x^{j}, ∑_{j≥0}(a_{0j}^{p} + pa_{1j}^{p})x^{pj}, ..., ∑_{j≥0}(a_{0j}^{p^{r−1}} + pa_{1j}^{p^{r−1}} + ⋯ + p^{r−1}a_{r−1j}^{p^{r−1}})x^{p^{r−1}j})

= (f_{0}(x), (f_{0}^{σ} + pf_{1}^{σ})(x^{p}), ..., (f_{0}^{σ^{r−1}} + pf_{1}^{σ^{r−1}} + ⋯ + p^{r−1}f_{r−1}^{σ^{r−1}})(x^{p^{r−1}})).

Here σ denotes the Frobenius automorphism of K_{m} over K. It sends an element in O_{n} ∩ T to its p-th power.

Proposition 4. The map ϕ : f ↦ w_{f} yields an imbedding from the ring O_{n}[x]/p^{r}O_{n}[x] to the Witt ring W_{r}(κ_{n}[x]). Moreover, if g ∈ O_{n}[x] is a lifting from a nonzero element in κ_{n}[x], then w_{g} is invertible in W_{r}(κ_{n}(x)).

Proof. To see that ϕ is a homomorphism, it suffices to show that the map sending f(x) = f_{0}(x) + pf_{1}(x) + p^{2}f_{2}(x) + ⋯ to

W′_{f} = (f(x) (mod p), f^{σ}(x^{p}) (mod p^{2}), ..., f^{σ^{r−1}}(x^{p^{r−1}}) (mod p^{r}))

is a ring homomorphism from O_{n}[x]/p^{r}O_{n}[x] to (O_{n}[x]/pO_{n}[x])×(O_{n}[x]/p^{2}O_{n}[x])× ⋯ × (O_{n}[x]/p^{r}O_{n}[x]). This is so since the action of σ commutes with polynomial addition and polynomial multiplication. Further, the map f ↦ W_{f} is 1-1, as seen from the last coordinate of W_{f}. Hence ϕ is an imbedding. Finally, if g is a lifting from a nonzero element in κ_{n}[x], then g_{0} ≡ g ≢ 0 (mod p), so W_{g} is invertible in O_{n}(x)^{r}, and hence w_{g} is invertible in W_{r}(κ_{n}(x)).

The map ϕ then induces an imbedding from the ring

R_{r} = {f/g : f, g ∈ O_{n}[x]/p^{r}O_{n}[x], g is a lifting}

into the Witt ring W_{r}(κ_{n}(x)) by sending f/g to w_{f}/w_{g}.

Theorem 8. it. Given f/g in R_{r}, we have κ_{n}(x)_{f/g} = κ_{n}(x)_{w_{f} /w_{g}}.

Proof. Denote by M the field κ_{n}(x)_{f/g} and by M′ the field κ_{n}(x)_{w_{f} /w_{g}}. They are both finite abelian extensions of κ_{n}(x). To show that they are equal, it suffices to show that the places of κ_{n}(x) splitting completely in M agree with those in M′ almost everywhere. As explained above, M′ = κ_{n}(x)(y_{0}, ..., y_{r−1}) for some element y = (y_{0}(x), y_{1}(x), ..., y_{r−1}(x)) in W_{r}(\overline{κ_{n}(x)}) such that F (y) − y = w_{f}/w_{g}. Let v be a place of degree m of κ_{n}(x) away from ∞ and the places ramified in M or M′ such that ḡ, all y_{i} and all w_{i} do not have poles over places above v. Let α ∈ T be a root of the lifting of π_{v}. Then we have

w_{f}/w_{g}(ᾱ) = (y_{0}(ᾱ)^{p}, ..., y_{r−1}(ᾱ)^{p}) − (y_{0}(ᾱ), ..., y_{r−1}(ᾱ)).

On the other hand, it follows from the definition of ϕ that f(α) (mod p^{r}) and g(α) (mod p^{r}) are the elements in O_{mn}/p^{r}O_{mn} corresponding to w_{f}(ᾱ) and w_{g}(ᾱ) under the isomorphism, and since g(α) is invertible, we get that f/g(α) (mod p^{r}) is the element in O_{mn}/p^{r}O_{mn} corresponding to w_{f}/w_{g}(ᾱ).

Suppose v splits completely in M′. Then all y_{i}(ᾱ) lie in κ_{n}(ᾱ) = κ_{mn} by Lemma 3. Let b = ∑_{0≤i≤r−1} b_{i}p^{i} with b_{i} ∈ O_{mn} ∩ T be the element in O_{mn}/p^{r}O_{mn} corresponding to y(ᾱ) under the canonical isomorphism. Thus b^{σ} − b ≡ f/g(α) (mod p^{r}), which implies that v splits completely in M by Lemmas 1 and 2. Conversely, suppose v splits completely in M. By Lemmas 1 and 2, there is an element ξ ∈ O_{mn}/p^{r}O_{mn} such that ξ^{σ} − ξ ≡ f/g(α) (mod p^{r}). Write ξ = ∑_{0≤i≤r−1} ξ_{i}p^{i}

with ξ_{0}, ..., ξ_{r−1} ∈ O_{mn} ∩ T . Let u = (ξ̄_{0}, ξ̄_{1}^{p}, ..., ξ̄_{r−1}^{p^{r−1}}) be the element in W_{r}(κ_{mn}) corresponding to ξ under the canonical isomorphism. From

w_{f}/w_{g}(ᾱ) = F (u) − u = F (y(ᾱ)) − y(ᾱ)

we conclude that the components of y(ᾱ) and u lie in the same field κ_{mn}. Hence v splits completely in M′ by Lemma 3. Therefore M = M′.

Since a finite abelian extension of κ_{n}(x) is a composite of finitely many cyclic extensions, Theorem 8 extends easily to two parametrizations of finite abelian extensions of κ_{n}(x).

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Wen-Ching Winnie Li Department of Mathematics, Pennsylvania State University, University Park, PA 16802 U. S. A. e-mail address: wli@math.psu.edu