FUNCTION FIELDS: ARITHMETIC AND APPLICATIONS

Ching-Li Chai and Wen-Ching Winnie Li

Abstract. Recently Chai and Li constructed automorphic L-functions for GL_{n} over a function field by taking suitable product of automorphic L-functions for GL_{1}. As a consequence of this construction, the Kloosterman conjecture over a function field is established. This paper surveys their results and discusses related topics.

1. Introduction

An idele class character η of a function field K over a finite field F with q elements, if not principal, has the associated L-function L(s, η) a polynomial in q^{−s} with coefficients being character sums. Since L(s, η) is a factor of the zeta function of a finite abelian extension H of K, it is related to the arithmetic of the underlying curve of H. On the other hand, an automorphic L-function L(s) of GL_{n} over K has an Euler product over the places of K with the factor at almost all places v being the reciprocal of a polynomial in Nv^{−s} with degree n. In this exposé we explore the possibility that the local factors of L(s) are actually L-functions of idele class characters, that is, of GL_{1} over K. We shall survey the recent results of Chai and Li [CL] which gives a systematic construction of automorphic L-functions L(s) by taking product of appropriate L-functions of GL_{1}. In fact, L(s) is associated to an additive or multiplicative character of F and a rational function f over K so that the local factors of L(s) come from the reduction of f at good places. The Hasse-Weil zeta function of an elliptic curve over K is a special case of this construction. One important consequence is that by taking K = F (t) and f(x) = x + t/x, we establish the Kloosterman conjecture over a function field.

The paper is arranged as follows. The L-functions for GL_{1} are introduced in §2. In §3 we discuss analytic properties of L-functions for GL_{n} and review the converse theorem which gives an analytic criterion for the automorphicity of a prospective L-function. A prototype of the L-functions to be considered is the Hasse-Weil zeta function of a curve defined over K. This is computed in detail in §4 for an elliptic curve. In §5 we discuss the Kloosterman conjecture and explain its connection with Ramanujan graphs. This conjecture gives another example of our general results. Finally in §6 we state the main results and exhibit more examples.

The L-functions considered in this paper are in fact L-functions attached to complex automorphic representations of GL_{n} over a function field K. According to

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 11F03, 11F30; Secondary 11L05, 11R58.

Key words and phrases. automorphic L-functions, idele class characters, Hecke operators.

The research of the first author was supported in part by a grant from the National Science Foundation no. DMS95-02186. The research of the second author was supported in part by a grant from the National Science Foundation no. DMS96-22938.

Langlands philosophy, they are expected to be part of the L-functions attached to ℓ-adic representations of Gal(K̄/K) although for the time being it is only proved for GL_{2} by Drinfeld [Dr].

This gives an interesting contrast to the L-functions considered by Daqin Wan in this volume. Almost all character sum estimates obtained from using the Riemann hypothesis for curves can be derived from estimating the first coefficient of an Lfunction of GL_{1}. Character sum estimates have imoprtant applications to coding theory. Judy Walker’s paper touches this subject. The character sums considered in her paper can be re-interpreted as the first coefficient of an L-function of GL_{1} over a function field. So a common thread of hers and this paper is L-functions of GL_{1} although quite different aspects of such L-functions are explored. By class field theory, L-functions of GL_{1} over K are also L-functions attached to degree one representations of Gal(K̄/K), or of the Galois group of the abelian closure of K over K. Hence such L-functions can be viewed as dual to abelian extensions of K. Kristin Lauter’s paper concerns constructing curves with many rational points. Put in the context of field extensions, she searches for abelian extensions of function fields with controlled ramification and splitting. This is the third aspect of L-functions of GL_{1}.

2. Zeta functions of a curve over a finite field

Given an irreducible nonsingular projective curve C of genus g defined over a finite field F of q elements, denote by K the field of F -rational functions on C, called a function field. Let F̄ denote an algebraic closure of F. The Galois group Gal(F̄/F) acts on C(F̄), the F̄-rational points on C. A Galois orbit is called a closed point or a place of C or K.

For example, if C is the projective line ℙ^{1} over F, then K = F (t). The set ℙ^{1}(F̄) can be represented by {(α : 1)|α ∈ F̄} ∪ {(1 : 0)}. In this expression, points in the first set may be regarded as the affine points of the projective line, while the only point in the second set is the point at infinity, or the ∞ place. The Galois orbit of an affine point (α : 1), called a place v, consists of points (β : 1), where β runs through the roots of the monic irreducible polynomial π_{v} of α over F. Thus the place v may be represented by the polynomial π_{v}, or, algebraically, the prime ideal (π_{v}) in F [t]. Call π_{v} a uniformizer of v. At place ∞, choose π_{∞} to be 1/t.

For a general curve C, the degree of a place denotes the number of points contained in the orbit. Define the norm of a place v of C to be Nv = q^{deg v}. Hence for C = P^{1} we have Nv = q^{deg π_{v}} if v ≠ ∞, and N∞ = q. The concept of norm allows us to define a zeta function attached to the function field K of C as an infinite product analogous to the Riemann zeta function:

ζ_{K}(s) = ∏_{v} \frac{1}{1 − Nv^{−s}} ,

where v runs through all places of K. As a function in the complex variable s, the zeta function converges absolutely for ℜs large. On the other hand, ζ_{K} affords another interpretation related to the arithmetic of the curve C, given by

ζ_{K}(s) = exp( ∑_{m≥1} \frac{N_{m}}{m} q^{−sm}),

where N_{m} is the number of rational points C has over a degree m extension of F. The latter expression, often called the zeta function of the curve C and denoted by Z(s, C), was shown to be a rational function in q^{−s} by F. K. Schmidt in 1931. More precisely, one has

ζ_{K}(s) = \frac{P (q^{−s}, C)}{(1 − q^{−s})(1 − q^{1−s})}.

Here P (x, C) is a polynomial in x of degree 2g. Similar to Riemann zeta function, the zeta function of C satisfies the functional equation

ζ_{K}(s) = ±(q^{1/2−s})^{2g−2}ζ_{K}(1 − s).

Analogous to the Dirichlet L-function attached to a character χ of ℤ of conductor N defined as

L(s, χ) = ∑_{n≥1} χ(n)n^{−s} = ∏_{p∤N} \frac{1}{1 − χ(p)p^{−s}} ,

to each idele class character χ of K of conductor f(χ), we attach the L-function

L(s, χ) = ∏_{v∤f(χ)} \frac{1}{1 − χ(π_{v})Nv^{−s}} .

Here π_{v} is a uniformizer at the place v and the product is over all places not occurring in the conductor f(χ), which measures the ramification of χ. Again, the L-function converges absolutely for ℜs large. It can be shown that, when expanded as a power series in q^{−s}, L(s, χ) is actually a polynomial in q^{−s} with degree equal to deg f(χ) + 2g − 2 provided that χ is not principal. Furthermore, like Dirichlet L-function, it satisfies a functional equation

L(s, χ) = ε(χ)q^{(1/2−s)(2g−2+deg f(χ))}L(1 − s, χ^{−1}),

with some constant ε(χ). Note that ζ_{K} is equal to L(s, χ) with χ being the trivial character.

3. Automorphic L-functions of GL_{n} over K

The functions ζ_{K}(s) and L(s, χ) are automorphic L-functions of GL_{1} over K. Before going to automorphic functions of groups of higher rank, we recall a wellknown situation over the field of rational numbers. A classical modular form f of weight k for a congruence subgroup, say, Γ_{0}(N), of SL_{2}(ℤ) has a Fourier expansion

f(z) = ∑_{n≥0} a_{n}e^{2πinz}.

The first natural question towards the study of the arithmetic of the Fourier coefficients of f is to ask when the function n ↦ a_{n} is multiplicative. Hecke has shown that this function is multiplicative with respect to powers of a prime p ∤ N if and only if f is an eigenfunction of the Hecke operator T_{p}. Further, these Hecke operators are symmetric and commute to each other. As the space of modular forms with weight k for Γ_{0}(N) is finite-dimensional, it can be simultaneously diagonalized with respect to all T_{p} with p ∤ N. Forms in a 1-dimensional common eigenspace are called a newform of level N. It can be shown that a common eigenspace with dimension greater than 1 contains a newform g of level dividing N, and this eigenspace is generated by g as well as its ”push-ups”. This is the theory of newforms initiated by Atkin and Lehner [AL]. Hence we shall restrict our attention to newforms of any level. It was shown that if f is a newform, then a_{1} is nonzero, which we shall assume to be 1. Consequently f is an eigenfunction of the Hecke operator T_{p} with eigenvalue being the p-th Fourier coefficient a_{p} of f for all p not dividing the level of f. It turns out that for a newform f of weight k and level N, the associated L-function has Euler product everywhere:

L(s, f) = ∑_{n≥1} a_{n}n^{−s} = ∏_{p∤N} \frac{1}{1 − a_{p}p^{−s} + p^{k−1−2s}} ∏_{p|N} \frac{1}{1 − a_{p}p^{−s}} .

There is a similar theory for other types of congruence subgroups, only in the Euler product for L(s, f) the term p^{k−1−2s} should be multiplied by χ(p) for some Dirichlet character χ, known as the central character of f. See [L1] for more detail.

Modular forms are automorphic forms for GL_{2} over ℚ. Observe that except for finitely many places, the Euler factor at a place p of the L-function attached to a modular form has degree two in p^{−s}, while the Euler factor at p of a Dirichlet Lfunction has degree one in general. There is a parallel theory for automorphic forms of GL_{n} over a function field K, and similar phenomena hold. For convenience, we shall consider only ”nice” automorphic forms f for GL_{n} over K whose associated L-functions have the form

L(s, f) = ∏_{v} \frac{1}{P (v, s)},  (1)

where v runs through all places of K, and P (v, s) is a polynomial in Nv^{−s} with constant term 1 and of degree ≤ n with equality achieved for almost all v. These are also the L-functions attached to automorphic representations of the same group. To keep the exposition more accessible to nonspecialists, we choose not to state results in terms of automorphic representations at the cost of minor occasional imprecision. The infinite product converges absolutely for ℜs large enough. The analytic properties of these L-functions are summarized as follows.

Theorem A. (Jacquet-Langlands [JL] for n = 2, Godement-Jacquet [GJ] for arbitrary n) The L-function attached to a nice automorphic form f for GL_{n} over K is a rational function in q^{−s}, which is a polynomial in q^{−s} if f is a cuspidal automorphic form. Moreover, it satisfies the functional equation

L(s, f) = ε(s, f)L(1 − s, f̄)

for some nice automorphic form f̄. Here ε(s, f) is a positive power of q^{−s} times a constant.

For an idele class character χ of K unramified at v, the twist of P (s, v) by χ is the polynomial obtained from P (s, v) by replacing Nv^{−s} by χ(π_{v})Nv^{−s}. Note that if f is a nice automorphic form, so is its twist f_{χ} by any idele class character χ of K, and the inverse of the Euler factors of L(s, f_{χ}) at places where χ is unramified are the twist by χ of those of L(s, f). Also, the multiplicity one theorem holds for automorphic forms for GL_{n}, that is, if f and g are two nice automorphic forms such that if L(s, f) and L(s, g) have the same Euler factors at almost all places, then f = g. For this reason, sometimes we shall be vague by giving the Euler factors for all but finitely many places and write ”∼” in lieu of equality.

How does one recognize an automorphic L-function? More precisely, suppose that we are given, at each place v of K, a polynomial P (v, s) in Nv^{−s} with constant term 1 and degree ≤ n so that the degree is n for almost all v, when is the infinite product as in (1) equal to L(s, f) for a nice automorphic form f of GL_{n} over K? Back to the case of classical modular forms, this amounts to saying that given the eigenvalues a_{p}, does there exist a modular form f so that it is an eigenfunction of the Hecke operator T_{p} with the prescribed eigenvalue a_{p}? This question was first considered by Hecke who answered it for modular forms for SL_{2}(ℤ). Then Weil [W1], [W2] gave a criterion for modular forms with level. An answer to such kind of question is called a converse theorem for GL_{n}. Over a function field K, an analytic criterion for GL_{2} was given by Jacquet and Langlands [JL] and for GL_{3} was given by Jacquet, Piatetskii-Shapiro and Shalika [JPS], following the spirit of Hecke and Weil.

Theorem B (Converse theorem for GL_{2} and GL_{3} over K). Let P (v, s) be as given above with degree ≤ n, where n = 2, 3. Form the infinite product

L(s) = ∏_{v} \frac{1}{P (v, s)}.

Then there is a nice automorphic form f for GL_{n} over K such that L(s, f) = L(s) if and only if L(s) and its twists by all idele class characters χ of K have the analytic properties described in Theorem A above with the constants ε(s) and ε(s, χ) occurring in the functional equations related in a prescribed way.

In terms of representations, the converse problem amounts to asking when a given collection of local representations piece together nicely to yield a global automorphic representation of GL_{n}. In this case the compatibility of global ε(s) and ε(s, χ) is already built in, resulting from properties of local ε-factors.

The converse theorem for GL_{n} with n ≥ 4 was proved by Cogdell and Piatetskii-Shapiro [CP1], requiring good analytic behavior for L(s) and its twists by cuspidal automorphic forms for GL_{m} with m ≤ n − 1. For n = 4, they showed in [CP2] that the condition can be reduced to m ≤ 2.

4. The Hasse-Weil zeta function of a curve over a function field

A curve C of genus g defined over the function field K has a good reduction at all but finitely many places of K. At a place v with good reduction, we obtain a curve C_{v} of the same genus defined over the residue field of v, which is an extension of F of degree equal to deg v. The numerator of the zeta function of C_{v} is a polynomial in Nv^{−s} of degree 2g. The Hasse-Weil zeta function of C is the product of the reciprocal of the numerator of the zeta function of C_{v} over places v with good reductions together with product over places with bad reductions. While we shall not elaborate on the Euler factors at bad places, suffice it to say that they are reciprocal of polynomials in Nv^{−s} of degree less than 2g. This general construction is illustrated in more detail in the following example of an elliptic curve, which also serves as a good example of the problems we shall address later.

Assume that K = F (t) is a rational function field of characteristic at least 5. An elliptic curve E over K can be defined by an equation y^{2} = h(x), where h ∈ K[x] has degree 3 and h has distinct roots. Let v be a good place for E, that is, the coefficients of h have no poles at v and y^{2} ≡ h(x) mod v defines an elliptic curve E_{v} over the residue field F_{v} of K at v. Note that Nv is the cardinality of F_{v}. The zeta function of E_{v} is

Z(s, E_{v}) = \frac{1 − a_{v}Nv^{−s} + Nv^{1−2s}}{(1 − Nv^{−s})(1 − Nv^{1−s})}

and the Hasse-Weil L-function attached to the elliptic curve E is defined as

L(s, E) = ∏_{v good} \frac{1}{1 − a_{v}Nv^{−s} + Nv^{1−2s}} ∏_{v bad} \frac{1}{1 − a_{v}Nv^{−s}} .

Grothendieck has shown that L(s, E) and its twists by idele class characters of K have the analytic behavior as described in Theorem A. Later Deligne [D1] showed that the ε-factors occurring in the functional equation for L(s, E) and its twists are related in the desired way so that one can apply Theorem B to conclude that there is a cuspidal automorphic form f for GL_{2} over K such that L(s, f) = L(s, E). In particular, f is an eigenfunction of the Hecke operator T_{v} with eigenvalue a_{v} at all good places v.

Now take a closer look of the coefficient a_{v} at a good place v. Let χ_{v} be the multiplicative character of F_{v} of order 2. Extend it to a function on F_{v} so that χ_{v}(0) = 0, its value at a square in F_{v}^{×} is 1 and its value at a nonsquare is −1. Denote by χ_{1} the all 1 function on F_{v}. Write h_{v} for the polynomial h mod v. Given x ∈ F_{v}, the number of y ∈ F_{v} such that (x, y) satisfies the equation y^{2} = h_{v}(x) is χ_{1}(h_{v}(x)) + χ_{v}(h_{v}(x)). Hence the number of solutions of E_{v} over F_{v} is

1 + ∑_{x∈F_{v}} χ_{1}(h_{v}(x)) + χ_{v}(h_{v}(x)) = 1 + Nv + ∑_{x∈F_{v}} χ_{v}(h_{v}(x)) = 1 + Nv − a_{v}

by the definition of Z(s, E_{v}). In other words, we find a_{v} = − ∑_{x∈F_{v}} χ_{v}(h_{v}(x)). Since F_{v}^{×} has only one character of order 2, the character χ_{v} is equal to the quadratic character χ of F^{×} composed with the norm map N_{F_{v}/F}. Thus we have

a_{v} = − ∑_{x∈F_{v}} χ ∘ N_{F_{v}/F} (h_{v}(x)).  (2)

In summary, we have an automorphic form f for GL_{2} over K which is an eigenfunction of the Hecke operator T_{v} with eigenvalue a_{v} given by the character sum (2). Observe that as v varies, the character χ and the polynomial h remain fixed. We shall discuss more examples of this sort in next sections. Another point we would like to address is that the denominator of the Euler factor of L(s, f) at each good place v is in fact the L-function of an idele class character η_{v} of the function field F_{v}(t). In other words, we have an L-function for GL_{2} which is constructed from taking suitable product of L-functions for GL_{1}!

5. Kloosterman conjectures

In this section we shall see another occasion where eigenvalues of Hecke operators acting on an automorphic form for GL_{2} over K = F (t) are given by character sums parametrized in a similar way. This connection arose from recent research on eigenvalues of explicitly constructed Ramanujan graphs [L2]. Briefly speaking, from one construction one gets eigenvalues of Hecke operators on forms of GL_{2} over K as eigenvalues of certain Ramanujan graphs. These graphs are later shown to be isomorphic to some graphs with eigenvalues given by generalized Kloosterman sums. Naturally one would like to find out the automorphic forms having the generalized Kloosterman sums as eigenvalues of Hecke operators. The so-called Kloosterman conjecture over K provides an answer.

To facilitate our discussion, we fix some notation. For a prime p, denote by ψ_{p} the additive character of ℤ/pℤ given by ψ_{p}(x) = e^{2πix/p}. For a finite field F of characteristic p and a nonzero element a ∈ F, the Kloosterman sum is defined as

Kl(F; a) = ∑_{x∈F^{×}} ψ_{p} ∘ Tr_{F/(ℤ/pℤ)}(x + a/x).

A generalized Kloosterman sum is the nonsplit analogue of a Kloosterman sum, and it was shown by Deligne [D2] to equal to the opposite of a Kloosterman sum. Hence we shall consider only the Kloosterman sums.

The question of a possible connection between Kloosterman sums and eigenvalues of Hecke operators was first raised by N. Katz [K]. Its precise statement is as follows:

Kloosterman conjecture over ℚ. Given a nonzero integer b, there exists a Maass wave form f which is an eigenfunction of the Hecke operator T_{p} for all p ∤ b with eigenvalue −Kl(ℤ/pℤ; b). In other words,

L(s, f) ∼ ∏_{p∤b} \frac{1}{1 + Kl(ℤ/pℤ; b)p^{−s} + p^{1−2s}} .

A few words are in order here. There are two kinds of automorphic forms for GL_{2} over ℚ, corresponding to (holomorphic) modular forms and (real analytic) Maass wave forms. A holomorphic modular form has the property that its Fourier coefficients generate a finite extension of ℚ. Since no finite extensions of ℚ can contain all Kloosterman sums −Kl(ℤ/pℤ; b), p ∤ b, the conjectured form f, if exists, has to be a Maass wave form. Unfortunately, Maass forms are little understood. At present, there is no supporting evidence for this conjecture.

Regarding ℤ/pℤ as the residue field of the completion of ℚ at p, one can ask a similar question over a function field.

Kloosterman conjecture over a function field. Fix a rational function field K = F (t) over a finite field F of characteristic p. Given a nonzero element a ∈ K, there exists an automorphic form f of GL_{2} over K which is an eigenfunction of the Hecke operator T_{v} at all places v of K, which is neither a zero nor a pole of a, with eigenvalue −Kl(F_{v}; a). Here F_{v} denotes the residue field of the completion of K at v. In other words,

L(s, f) ∼ ∏_{v good} \frac{1}{1 + Kl(F_{v}; a)Nv^{−s} + Nv^{1−2s}} .

This conjecture was first proved by C. Moreno [M] for the case p = 2 and a = t. The case p = 2 and 3 was proved in [L3] with some help from J. Teitlebaum. The general case was proved in [CL].

Theorem C. The Kloosterman conjecture over a rational function field holds.

Notice that the eigenvalues of f with respect to the Hecke operators are given by the Kloosterman sums, in which the element a and the additive character ψ_{p} remain fixed when the place v varies. Further, the denominator of a general Euler factor of L(s, f) is a global L-function attached to an idele class character η_{v} of the function field F_{v}(t), similar to the L-function attached to an elliptic curve. It turns out that there is a way to construct global L-functions for groups of higher rank by piecing together suitable global L-functions for GL_{1}. This will be discussed in the next section.

6. Constructing automorphic L-functions from character sums

Let f(x) be a rational function with coefficients in F (t), where, as before, F is a finite field of q elements. Only finitely many places of F (t) occur as zeros or poles of the coefficients of f. These are the bad places, and the remaining places are called good places of f. At a good place v, the coefficients of f are units in the completion of F (t) at v, by passing to the residue field F_{v}, we get a rational function f_{v} := f mod v with coefficients in F_{v}.

Theorem D. Let f be a nice rational function over F (t) and let ψ be a nontrivial additive character of F. (D1) At each good place v of f, there exists an idele class character η_{ψ,f_{v}} of F_{v}(t)

such that at the places w of F_{v}(t) where η_{ψ,f_{v}} is unramified, we have

η_{ψ,f_{v}}(π_{w}) = ψ ∘ Tr_{F_{v}/F} (∑_{α} f_{v}(α)),

where α runs through all roots of π_{w} in an algebraic closure F̄ of F. (D2) The L-function L(s, η_{ψ,f_{v}}) attached to η_{ψ,f_{v}} is a polynomial in Nv^{−s} of

degree equal to the degree of the conductor of η_{ψ,f_{v}} minus 2, which remains the same for almost all v.

(D3) There is a compatible family of ℓ-adic representations of Gal(\overline{F (t)}/F (t)) with L-function

L(s, ψ, f) ∼ ∏_{v good} L(s, η_{ψ,f_{v}})^{−1}.

Further L(s, ψ, f) and its twists by idele class characters of F (t) and ℓadic representations of Gal(\overline{F (t)}/F (t)) behave nicely, namely, are rational functions in q^{−s} and satisfy functional equations as described in Theorem A.

Remark. Of the three statements in Theorem D, the first one is proved in [L4], and the remaining ones in [CL]. The assumption f being nice is used to assure that η_{ψ,f_{v}} is nonprincipal so that its L-function is a polynomial. This is true for a generic rational function f.

By combining the work of Deligne [D1] and the converse theorem for GL_{2} and GL_{3} (Theorem B), we get Corollary 1. If the degree of L(s, η_{ψ,f_{v}}) is 2 (resp. 3) for good v, then L(s, ψ, f) is an automorphic L-function for GL_{2} (resp. GL_{3}) over F (t).

Example 1. Let f(x) = x + \frac{1/t}{x}. Then a place v is good if v ≠ 0, ∞. Choose ψ = ψ_{p} ∘Tr_{F/(ℤ/pℤ)}. At a good place v, we find L(s, η_{ψ,f_{v}}) = 1+Kl(F_{v}; 1/t)Nv^{−s} + Nv^{1−2s} and

L(s, ψ, f) = ∏_{v≠0,∞} \frac{1}{1 + Kl(F_{v}; 1/t)Nv^{−s} + Nv^{1−2s}} · \frac{1}{1 + N∞^{−s}}

is an automorphic L-function for GL_{2} over F (t). This proves Theorem C.

Example 2. Let f_{1}(x) = x^{2} + \frac{1/t}{x} and f_{2}(x) = x + \frac{1/t}{x^{2}}. If the characteristic of F is not 2, then L(s, ψ, f_{1}) and L(s, ψ, f_{2}) are automorphic L-functions for GL_{3} over F (t).

Drinfeld in [Dr] and subsequent work showed that all automorphic L-functions for GL_{2} over F (t) come from L-functions attached to ℓ-adic representations of Gal(\overline{F (t)}/F (t)), hence by the converse theorem for GL_{4} and Theorem D, we have

Corollary 2. If the degree of L(s, η_{ψ,f_{v}}) is 4 for good v, then L(s, ψ, f) is an automorphic L-function for GL_{4} over F (t).

Example 3. L(s, ψ, f) is an automorphic L-function for GL_{4} over F (t) if f(x) = x^{3} + \frac{1/t}{x} and the characteristic of F is not equal to 3, or f(x) = x^{2} + \frac{1/t}{x^{2}} and the characteristic of F is not equal to 2.

We have a similar result using a multiplicative character instead of an additive character of F.

Theorem E. Let g be a nice rational function over F (t) and let χ be a nontrivial character of F^{×}.

(E1) At a good place v of g, there exists an idele class character η_{χ,g_{v}} of F_{v}(t) such that at the places w of F_{v}(t) where η_{χ,g_{v}} is unramified, we have

η_{χ,g_{v}}(π_{w}) = χ ∘ N_{F_{v}/F} (∏_{α} g_{v}(α)),

where α runs through all roots of π_{w} in an algebraic closure F̄ of F. Further, assertions (E2) and (E3) similar to (D2) and (D3) hold.

Moreover, we have a similar result by combining additive and multiplicative characters together.

Theorem F. Let f and g be nice rational functions over F (t), ψ be a nontrivial additive character of F and χ be a nontrivial multiplicative character of F^{×}.

(F1) At a place v good for f and g, there exists an idele class character η_{ψ,f_{v},χ,g_{v}} of F_{v}(t) such that at the places w of F_{v}(t) where η_{ψ,f_{v},χ,g_{v}} is unramified, we have

η_{ψ,f_{v},χ,g_{v}}(π_{w}) = ψ ∘ Tr_{F_{v}/F} (∑_{α} f_{v}(α))χ ∘ N_{F_{v}/F} (∏_{α} g_{v}(α)),

where α runs through all roots of π_{w} in an algebraic closure F̄ of F.

Further, assertions (F2) and (F3) similar to (D2) and (D3) hold.

Analogous remark applies to Theorems E and F, and consequences similar to Corollaries 1 and 2 hold.

Example 4. Let g(x) = (x−1)^{2} +ax for a ∈ F (t)^{×} and suppose the characteristic of F is odd. Then for any nontrivial character χ of F^{×}, L(s, χ, g) is an automorphic L-function for GL_{2} over F (t). The eigenvalues of the corresponding automorphic form with respect to the Hecke operators are also eigenvalues of certain Ramanujan graphs.

Corollary 3. The Hasse-Weil L-function attached to a hyperelliptic curve of genus 2 defined over F (t) is an automorphic L-function for GL_{4} over F (t).

Naturally, if the generalization of Drinfeld’s result to representations of GL_{n} were available for 3 ≤ n ≤ m, then one would be able to conclude that the L-functions occurring in Theorems D-F with the denominator of local Euler factor of general degree m + 1 ≥ 5 are automorphic L-functions.

References

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Department of mathematics, University of Pennsylvania, Philadelphia, PA 19104

E-mail address: chai@math.upenn.edu

Department of Mathematics, Pennsylvania State University, University Park, PA 16802

E-mail address: wli@math.psu.edu