Character sums, automorphic forms, equidistribution, and Ramanujan graphs ^{1}

Ching-Li Chai^{2} and Wen-Ching Winnie Li^{3}

Version 3.1, January 4, 2001

Abstract

According to the Langlands program, L-functions arising from families of exponential sums over global function fields are automorphic L-functions. We illustrate this general principle in examples related to Kloosterman sums and Ramanujan graphs.

Part I. The Kloosterman sum conjecture over function fields

§1. Introduction

Let K be a function field of one variable with the field of constants 𝔽 being a finite field with q elements. Denote by p the characteristic of 𝔽. Let E be an elliptic curve defined over K. For the sake of simplicity, assume that p is at least 5. Then E can be defined by an equation y^{2} = g(x) for some polynomial g over K of degree three. At almost all places v of K, E has a good reduction and the resulting elliptic curve E_{v} has zeta function

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

Here Nv, the norm of v, is the cardinality of the residue field 𝔽_{v} of K at v. The Hasse-Weil L-function attached to E is

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

At a good place v the coefficient a_{v} is equal to 1 + Nv minus the number of points of E_{v} over 𝔽_{v}, which in turn can be expressed as the character sum

a_{v} = − ∑_{x∈𝔽_{v}} χ ∘ Nm_{𝔽_{v}/𝔽}(g_{v}(x)).

Here χ is the unique multiplicative character of 𝔽^{×} of order 2 and g_{v} denotes g mod v, which is a polynomial over 𝔽_{v}. Due to the work of Grothendieck and Deligne [5], one can apply the converse theorem for GL_{2} proved by Jacquet and Langlands [14] to conclude that L(s, E) is an automorphic L-function for GL_{2} over K. Notice that each Euler factor 1 − a_{v}Nv^{−s} + Nv^{1−2s} is in fact the L-function of the idele class character η_{v} attached to the quadratic extension of the rational function field 𝔽_{v}(t) defined by y^{2} = g_{v}(x). Furthermore, Yoshida [36] has shown that the Sato-Tate conjecture holds for E when the j-invariant of E is not in 𝔽. In this case, we get examples of automorphic forms for GL_{2} over K for which the Sato-Tate conjecture holds.

12000 MSC: 11G40, 11T23, 22E55, 05C99

2partially supported by grant DMS 9800609 from the National Science Foundation

3partially supported by grant DMS96-22938 and DMS 99-70651 from the National Science Foundation

The above theme holds more generally for character sums over the rational points of a variety defined over a finite field. Given a family of character sums, often one can find a morphism π : X → B of schemes over 𝔽 and a (complex of) \overline{ℚ_{ℓ}}-sheaf L on X, such that a typical sum in the given family is the sum of the traces of the action on (the geometric fiber of) L of the Frobenii of all 𝔽-rational points of X lying over a fixed rational point b ∈ B(𝔽); similarly for finite extension fields of 𝔽. We assume that the sheaf L is a mixed sheaf in the sense of [7], as this is the case for most applications. For a family of character sums as above, one can apply the method of ℓ-adic cohomology to obtain information about the character sums. We recall some basic facts.

(1) The cohomology sheaves R^{i}π_{!}(L) are mixed \overline{ℚ_{ℓ}}-sheaves by Deligne [7]: especially it is a successive extension of smooth pure \overline{ℚ_{ℓ}}-sheaves over a dense open subscheme U of B. The weights of R^{i}π_{!}(L) provides sharp estimates of the character sums in question.

(2) By Grothendieck, the L-function attached to this family of character sums is a rational function (in q^{−s}); Poincaré duality yields a functional equation for the L-function.

(3) The local constants for the functional equation coincides with the local constants used in the theory of automorphic representations by Deligne [5] and Laumon [21].

(4) In examples one often can compute the vanishing cycles of the cohomology sheaves R^{i}π_{!}(L) at the “bad” places of the base scheme B. That is, the local L-factors at the bad places often can be determined explicitly.

(5) The traces of Frobenii of closed points of the base scheme B on the cohomology sheaves R^{i}π_{!}(L) is equidistributed; the distribution law is governed by the geometric monodromy group of R^{i}π_{!}(L), i.e. the Zariski closure of the action of the geometric fundamental group of U. The vanishing cycle sheaves often help one determine the geometric monodromy group.

The L-function attached to R^{i}π_{!}(L) is, up to finitely many Euler factors, a finite product of Lfunctions L(ρ_{j}, s) attached to irreducible \overline{ℚ_{ℓ}}-representations ρ_{j} of the Galois group of the function field 𝔽(B) of B, where the ρ_{j}’s are the irreducible subquotients of the action of the Galois group of 𝔽(B) on the geometric generic fiber of R^{i}π_{!}(L). For each irreducible \overline{ℚ_{ℓ}}-representation ρ of the Galois group of 𝔽(B), the L-function attached to ρ_{j} ⊗ ρ satisfies the “correct” functional equation by Grothendieck, Deligne and Laumon, just as stated in (2), (3) above. One deduces that L(ρ_{j}, s) is an automorphic L-function attached to GL(n_{j}) if the degree n_{j} of ρ_{j} is two by the converse theorem for GL(2) proved by Hecke- Weil-Jacquet-Langlands [14], and if n_{j} = 3 by the converse theorem for GL(3) established by Jacquet, Piatetski-Shapiro and Shalika [15]. The general converse theorem for GL(n) has been established by Cogdell and Piatetski-Shapiro [3], [2]. This combined with the global Langlands correspondence for GL(2) over function fields proved by Drinfel’d [9] implies that L(ρ_{j}, s) is an automorphic L-function attached to GL(4) if the degree of ρ_{j} is 4. Recently Lafforgue announced the proof of the Langlands correspondence over function fields for GL(n), see [18] and [19]. So the L function attached to a family of character sums in the above setting is a quotient of finite products of automorphic L-functions.

In this two-part paper we illustrate the above theme by exhibiting several examples, with character sums arising from eigenvalues of Ramanujan graphs. More precisely, in Part I we consider Kloosterman sums, and in Part II the eigenvalues of Terras graphs. Part I of this paper is organized as follows. In §2 we give two “prototype theorems”, Theorem 2.2 and Theorem 2.3, where the base scheme B is a smooth algebraic curve C over 𝔽, X = C × ℙ^{1} and f is the projection C × ℙ^{1} → C. In §3 we prove the Kloosterman sum conjecture over a function field K, which asserts the existence of automorphic forms of GL(2) over K whose Fourier coefficients are Kloosterman sums. It is interesting to note that recent computations by A. Booker [1] based on an idea of Sarnak seem to suggest that the analogous statement over ℚ, as questioned by Katz in [17], does not hold. More precisely, he showed that if an automorphic form for GL_{2} over ℚ with prescribed Fourier coefficients given by Kloosterman sums were to exist, then either the level or the eigenvalue at infinity of the form would be very large. The sharp contrast between the number field case and the function field case of this problem is very interesting. We also show the equidistribution of Kloosterman sums by computing the geometric monodromy group. The connection with a family of Ramanujan graphs, called norm graphs, is discussed in §4, where the base curve C is specified to ℙ^{1}. For this case one has detailed information about the vanishing cycles and local factors at bad places, which in turn enable us to show that the automorphic forms occurring in the Kloosterman sum conjecture actually live on a quaternion group. In other words, they can be viewed as functions on certain Ramanujan graphs constructed by Morgenstern, arising from lifting functions on norm graphs.

§2. Two prototype families of automorphic L-functions

Observe that an idele class character of a function field over a finite field 𝔽 with q elements, if not principal, has the associated L-function a polynomial in q^{−s} with coefficients being character sums. 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. Given an additive or multiplicative character of 𝔽 and a rational function f ∈ K, we construct in this section an automorphic L-functions L(s) by taking a product of appropriate L-functions of GL_{1}, that is, L-functions attached to idele class characters of function fields.

(2.1) Let f(x) be a rational function with coefficients in K. Only finitely many places of K occur as zeros or poles of the coefficients of f. These are the ‘possibly 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 K 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}.

To facilitate our statement, we use the notation L_{1} ∼ L_{2} to mean that two Euler products L_{1} and L_{2} over places of K have the same Euler factors at almost all places. Since the L-functions attached to automorphic representations of GL_{n} over K satisfy the multiplicity one theorem, and all L-functions considered in this paper come from representations, it suffices to know the Euler factors for almost all places. Moreover, an L-function L(s, ρ) attached to a representation ρ is said to behave nicely if it is a rational function in q^{−s} and it satisfies a functional equation

L(s, ρ) = ε(s)L(1 − s, ρ̌),

where ρ̌ is the contragredient of ρ and ε(s) is a constant times a positive power of q^{−s}. An automorphic L-function for GL_{n} over K is known to behave nicely, as proved by Hecke [12] [13] and Tate [33] for n = 1, Jacquet and Langlands [14] for n = 2 and Godement and Jacquet [11] for n ≥ 3. Moreover, the L-functions attached to cuspidal representations are in fact polynomials in q^{−s}.

We fix a choice of a nontrivial additive character of a finite field as follows. For the prime field ℤ/pℤ, we choose ψ_{p} to be ψ_{p}(x) = e^{2πix/p} and for any finite field 𝔽 of characteristic p, choose ψ = ψ_{p} ∘ Tr_{𝔽/(ℤ/pℤ)}.

(2.2) Theorem A Let f(x) ∈ K(x) be a non-constant rational function over K.

(A1) 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, its value at a uniformizer π_{w} is given by the character sum

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.

(A2) Suppose that f(x) is not of the form h_{1}(x)^{p} − h_{1}(x) + h_{2} for any h_{1}(x) ∈ K(x) and h_{2} ∈ K. Then the L-function L(s, η_{ψ,f_{v}}) attached to η_{ψ,f_{v}} is a polynomial in Nv^{−s} of degree n equal to the degree of the conductor of η_{ψ,f_{v}} minus 2, and n is the same for almost all v.

(A3) With the same assumption as in (A2), there exists a compatible family of ℓ-adic representations of Gal(K^{sep}/K), depending on ψ and f, such that its associated L-function L(s, ψ, f) satisfies

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 K and by all ℓ-adic representations of Gal(K^{sep}/K) behave nicely.

(2.3) Theorem B Let g(x) ∈ K(x) be a non-constant rational function over K and let χ be a nontrivial character of 𝔽^{×} of order d.

(B1) 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, its value is given by the character sum

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

where α runs through all roots of π_{w} in 𝔽̄.

(B2) Suppose that g(x) is not of the form h_{1}h_{2}(x)^{d} for any h_{2}(x) ∈ K(x) and any h_{1} ∈ K. Then the L-function L(s, η_{χ,g_{v}}) attached to η_{χ,g_{v}} is a polynomial in Nv^{−s} of degree n equal to the degree of the conductor of η_{χ,g_{v}} minus 2, and n is the same for almost all places v.

(B3) Under the same assumption as in (B2), there exists a compatible family of ℓ-adic representations of Gal(K^{sep}/K)), depending on χ and g, such that the associated L-function L(s, χ, g) satisfies

L(s, χ, g) ∼ ∏_{v good} L(s, η_{χ,g_{v}})^{−1}.

Further L(s, χ, g) and its twists by idele class characters of K and by all ℓ-adic representations of Gal(K^{sep}/K) behave nicely.

(2.3.1) Remarks (1) From the motivic point of view, the L-functions in (A3) and (B3) arise from the first cohomology group. Here we normalize them by taking the inverse so that formally they appear like automorphic L-functions. In other words, we forget the degree of the cohomology and use only the Galois representation as the Langlands parameter of the automorphic representation.

(2) The function L(s, ψ, f) in (A3) is a factor of the Hasse-Weil L-function attached to the Artin-

Schreier curve defined by

y^{p} − y = f(x),

while L(s, χ, g) in (B3) is a factor of the Hasse-Weil L-function attached to the curve defined by

y^{d} = g(x).

In particular, the Hasse-Weil L-function L(s, E) attached to the elliptic curve E defined by y^{2} = g(x) at the beginning of the Introduction is equal to L(s, χ, g) with χ being the quadratic character of 𝔽^{×}.

(3) Given m rational functions f_{1}(x), ..., f_{m}(x) ∈ K(x) as in (A2), and/or n rational functions g_{1}(x), ..., g_{n}(x) ∈ K(x) as in (B2) and n nontrivial multiplicative characters χ_{1}, ..., χ_{n} of 𝔽^{×}, quite often there exists a compatible family of ℓ-adic representations of Gal(K^{sep}/K), depending on f_{1}(x), ..., f_{m}(x), ψ, and/or g_{1}(x), χ_{1}, ..., g_{n}(x), χ_{n}, such that its associated Lfunction has its local factor at a good place v equal to the reciprocal of the L-function attached to the product of the idele class characters η_{ψ,f_{1v}} , ..., η_{ψ,f_{mv}} and/or η_{χ_{1},g_{1v}} , ..., η_{χ_{n},g_{nv}}. Example 2.5.2 below falls in this case.

As explained in the Introduction, the two “prototype Theorems” 2.2 and 2.3 are consequences of Grothendieck’s geometric interpretation of L-functions as envisioned by Weil, and Deligne’s spectacular theorems in [7].

The next statement is a special case of the Langlands correspondence for GL(n) over global function fields, proved by Lafforgue.

(2.4) Theorem The L-functions L(s, ψ, f) and L(s, χ, g) in (A3) and (B3) respectively are automorphic L-functions for GL_{n} over the function field K. Moreover, these L-functions satisfy the Ramanujan-Petersson conjecture.

(2.5) Exhibited below are specific examples of the automorphic L-functions of GL(n) constructed above; some are also automorphic L-functions on quaternion groups and verify the Sato-Tate conjecture, cf. §4 of this part and §4 of Part II.

(2.5.1) Example Let f(x) = x+a/x with any nonzero element a in K. We obtain an automorphic L-function L(s, ψ, f) for GL_{2} over K whose local factor at a good place v is

L(s, η_{ψ,f_{v}}) = 1 + Kl(𝔽_{v}; a)Nv^{−s} + Nv^{1−2s},

where Kl(𝔽_{v}; a) is the Kloosterman sum

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

As we shall see in §4 that this form actually lives on a quaternion group, and it has close ties to eigenvalues of norm graphs.

(2.5.2) Example Suppose p is odd. Let g_{1}(x) = (x − 1)^{2} + ax for a ∈ K^{×}, χ_{1} the quadratic character of 𝔽^{×}, g_{2}(x) = x and χ_{2} a nontrivial character of 𝔽^{×}. Then the product over good places v of K of the reciprocal of L(s, η_{χ_{1},g_{1v}} η_{χ_{2},g_{2v}}) yields an automorphic L-function for GL_{2} over K; see Theorem 4.2 of Part II for the precise statement. The relation to Terras graphs is also explained in §4 of Part II.

(2.5.3) Example Let f_{1}(x) = x^{2} + \frac{a}{x} and f_{2}(x) = x + \frac{a}{x^{2}} for a nonzero element a in K. If p is not 2, then L(s, ψ, f_{1}) and L(s, ψ, f_{2}) are automorphic L-functions for GL_{3} over K.

(2.5.4) Example L(s, ψ, f) is an automorphic L-function for GL_{4} over K if f(x) = x^{3} + \frac{a}{x} and p is not equal to 3, or f(x) = x^{2} + \frac{a}{x^{2}} and p is not equal to 2. Here a is any nonzero element in K as before.

Since the Hasse-Weil zeta function of the smooth projective curve arising from the affine equation y^{p} − y = f(x) (resp. y^{d} = g(x)) is a product of L-functions of type L(s, ψ, f) (resp. L(s, χ, g)), hence it is also an automorphic L-function. In other words, we have shown

(2.5.5) Corollary The Hasse-Weil L-function attached to a smooth projective curve of genus r defined over K with affine equation y^{p} − y = f(x) (resp. y^{d} = g(x)) is an automorphic L-function for GL(2r) over K.

The first two statements of Theorems A and B are proved in [23], Chap. 6, Theorem 4, and in [26]. They can also be seen geometrically. Since the third statement of each theorem is geometrical, using results of Grothendieck and Deligne, we shall give a unified geometric argument for all statements. The basic geometric settings below mostly follow Deligne and Katz in [5], [7], [17], [16].

(2.6) The basic method for the proof of Theorems A and B have been outlined in the Introduction. In the remainder of this section we will sketch the proof of Theorem A; the proof of Theorem B is similar.

(2.6.1) Described below is the geometric set-up for Theorem 2.2, following the notation of §1.

• The base scheme B is the smooth geometrically irreducible algebraic curve over 𝔽_{q} such that K is the field of rational functions on B.

• The source scheme X is B ×_{Spec 𝔽_{q}} ℙ^{1}, and π : X → B is the projection to the first factor,

• Let U ⊂ X be the largest open subscheme of X such that the rational function f(x) ∈ K(x) defines an 𝔽_{q}-morphism from U to 𝔾_{a}.

• Let L_{ψ} be the smooth rank-one \overline{ℚ_{ℓ}}-sheaf on 𝔾_{a}_{/Spec 𝔽_{q}}, the push-forward of the fiber 𝔽_{q} of the Lang torsor Id − Fr_{q} : 𝔾_{a} → 𝔾_{a} by ψ : 𝔽_{q} → \overline{ℚ_{ℓ}}^{×}.

• The sheaf L = L_{ψ,f} on X is (U ↪ X)_{!} (f^{∗}L_{ψ}), the extension-by-zero of the smooth rank-one \overline{ℚ_{ℓ}}-sheaf f^{∗}L_{ψ} on U.

(2.6.2) The hypothesis in (A2) ensures that R^{i}π_{∗}L = 0 unless i = 1. The assertion about the degree in (A2) now follows from Grothendieck’s Euler-Poincaré formula for constructible \overline{ℚ_{ℓ}}-sheaves over possibly open smooth curves. The rest of the statements of Theorem A all follow from the machinery of ℓ-adic cohomology. This concludes the sketch of the proof of Theorem A. The proof of Theorem B is similar, therefore omitted.

§3. The Kloosterman sum conjecture over function fields

(3.1) In this section we discuss a special occasion where Kloosterman sums arise as eigenvalues of the Hecke operators on certain automorphic forms over GL_{2}. To fix our notation, for each prime p we choose the additive character ψ_{p} of ℤ/pℤ to be ψ_{p}(x) = e^{2πix/p}. The Kloosterman sum attached to a finite field k of characteristic p and a nonzero element b in k is defined as

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

Let K be a function field with the field of constants 𝔽 of characteristic p as in §1 and let a be a nonzero element in K. Let f be the rational function f(x) = x + a/x and choose the nontrivial additive character ψ of 𝔽 to be ψ_{p} ∘ Tr_{𝔽/(ℤ/pℤ)}. At a place v which is neither a zero nor a pole of a, the function f_{v}(x) ∈ 𝔽_{v}(x) is regular everywhere except for a simple pole at 0 and a simple pole at ∞, hence the associated idele class character η_{ψ,f_{v}} of 𝔽_{v}(x) has conductor equal to 2 · 0 + 2 · ∞ and the L-function of η_{ψ,f_{v}} has degree 2 in Nv^{−s} given by

L(s, η_{ψ,f_{v}}) = 1 + Kl(𝔽_{v}; a)Nv^{−s} + Nv^{1−2s}

(cf. [26]). Here, by passing to the residue field of K at v, we regard a as a nonzero element in 𝔽_{v}. The coefficient of Nv^{−s} is obvious. We proceed to compute the coefficient of Nv^{−2s}, which we denote by b. It follows from the definition of L(s, η_{ψ,f_{v}}) that

b = ∑_{v_{1},v_{2}} ψ(π_{v_{1}})ψ(π_{v_{2}}) + ∑_{w} ψ(π_{w}),

where v_{1}, v_{2} run through places of 𝔽_{v}(x) of degree 1 other than 0 and ∞, and w runs through the places of degree 2. In view of the explicit description of η_{ψ,f_{v}} given in Theorem A, (A1), the first sum is over nonzero elements c, d in 𝔽_{v} of ψ(c + a/c)ψ(d + a/d) = ψ((c + d)(1 + \frac{a}{cd})), while the second sum is over monic irreducible polynomials π_{w} of degree 2 over 𝔽_{v} of ψ(z + a/z + z′ + a/z′) = ψ((z + z′)(1 + \frac{a}{zz′})) with z, z′ being the two roots of π_{w}. Viewed this way, the two parts of b have a uniform expression as sums over split and nonsplit degree two polynomials x^{2} − Ax + B over 𝔽_{v} with B nonzero, respectively. In other words, we may rewrite b as

b = ∑_{A∈𝔽_{v}} ∑_{B∈𝔽_{v}^{×}} ψ(A(1 + \frac{a}{B})).

By separating the term with A = 0, which sums to Nv − 1, and then summing over B with a fixed nonzero A, which yields −ψ(A), we arrive at

b = Nv − 1 − ∑_{A∈𝔽_{v}^{×}} ψ(A) = Nv,

as asserted.

As a consequence of Theorem A and the converse theorem for GL_{2}, we know that L(s, ψ, f) is an automorphic L-function for GL_{2} over K. Alternatively, one can simply quote [34] 4.5, since the Kloosterman L-function L(s, ψ, f) comes from a compatible system of two-dimensional ℓ-adic representations of Gal(K^{alg}/K), as shown in §2. We summarize this result in

(3.2) Theorem (Kloosterman sum conjecture over a function field) Let K be a function field with the field of constants 𝔽 a finite field. 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}} .

(3.2.1) Remark When K = 𝔽(t) is a rational function field, the conjecture for the case p = 2 and a = t was proved by C. Moreno [30]. The case of p = 2 and 3 was proved in [25] with some help from J. Teitelbaum.

It is interesting to compare this with its counterpart over the field ℚ, which is a question raised by Katz in [17].

(3.3) Question (Kloosterman sum question over ℚ) Given a nonzero integer b, does there exist 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), that is, with

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

(3.3.1) Remark Based on an idea of Sarnak, A. Booker [1] did some computations showing that if f existed, then either its level or its eigenvalue at infinity would be very large. He also explained the impossibility to give a counter-example by numerical computations because there always exists a modular form for SL_{2}(ℤ) whose Fourier coefficients at the finitely many prescribed places are arbitrarily close to the prescribed numbers (satisfying the Ramanujan-Peterson conjecture) provided that either the weight (for holomorphic forms) or the eigenvalue at infinity (for Maass wave forms) is allowed to grow unboundedly.

Weil [35] showed that the Kloosterman sum Kl(k; b) has absolute value bounded by 2\sqrt{|k|}. On the other hand, each Kloosterman sum is in fact real. An interesting question is to study the distribution of a family of normalized sums Kl(k; b)/\sqrt{|k|}, which lie in the interval [-2, 2]. It follows from the work of Deligne [7] and our proof of the above theorem that the Sato-Tate conjecture holds for the automorphic form f in Theorem 3.2 associated to a non-constant a since f in this case does not arise from a quadratic extension of K.

(3.4) Corollary If the element a in the statement of Theorem 3.2 is not a constant in K, then the normalized Kloosterman sums Kl(𝔽_{v}; a)/\sqrt{|𝔽_{v}|} are uniformly distributed with respect to the Sato-Tate measure

μ_{ST} (x) = \frac{1}{π} \sqrt{1 − \frac{x^{2}}{4}} dx

as Nv tends to infinity.

Proof. The equidistribution statement is a special case of the general equidistribution theorem in [7] 3.5.3 for \overline{ℚ_{ℓ}}-representations of the Weil group of K attached to a smooth Weil \overline{ℚ_{ℓ}}-sheaf pure of some weight on an open subset of C. See also [17] 3.6 for several forms of equidistribution statements and a self-contained exposition.

In our case the sheaf in question is the restriction to U of the sheaf R^{1} pr_{1,U}_{!}L_{ψ,f}, where

• U is the complement in B of the zeroes and poles of a,

• pr_{1,U} : U × 𝔾_{m} → U is the first projection,

• f = x + \frac{a}{x}, regarded as an 𝔽_{q}-morphism from U × 𝔾_{m} → 𝔸_{1},

• L_{ψ,f} is the pull-back by f of the smooth rank-one \overline{ℚ_{ℓ}}-sheaf L_{ψ} on 𝔾_{a}.

This sheaf is smooth of rank two on U and is pure of weight one, for it is well known that the product of the two eigenvalues of Fr_{v} is equal to q_{v}. We know that an equidistribution law is satisfied, and it is governed by a (possibly disconnected) semisimple group, the Zariski closure of the image of the geometric monodromy group of R^{1} pr_{1,U}_{!}L_{ψ,f}.

The cup product gives a perfect pairing

R^{1} pr_{1,U}_{!}L_{ψ,f} × R^{1} pr_{1,U}_{!}L_{ψ,−f} ⟶ R^{2} pr_{1,U}_{!}\overline{ℚ_{ℓ}} ⟶^{∼} \overline{ℚ_{ℓ}}(−1) .

On the other hand, the involution x ↦ −x on 𝔸^{1} sends L_{ψ,f} to L_{ψ,−f}, and induces a natural isomorphism

R^{1} pr_{1,U}_{!}L_{ψ,f} ⟶^{∼} R^{1} pr_{1,U}_{!}L_{ψ,f} .

The duality pairing above then becomes an isomorphism det(R^{1} pr_{1,U}_{!}L_{ψ,f}) ⟶^{∼} \overline{ℚ_{ℓ}}(−1).

Since det(R^{1} pr_{1,U}_{!}L_{ψ,f})= \overline{ℚ_{ℓ}}(−1), the geometric monodromy group of R^{1} pr_{1,U}_{!}L_{ψ,f} is a subgroup of SL(2), hence either it is equal to SL(2) or it is finite. We will be done if we rule out the second possibility, because the Sato-Tate measure given in the statement is exactly the pushforward of the Haar measure of the compact form of SL(2) to its conjugacy classes. The case that it is final is forbidden by the assumption that a is not constant. Otherwise it would mean that the sheaf R^{1} pr_{1,U}_{!}L_{ψ,f} is geometrically trivial after a finite base change, or more concretely that the two eigenvalues of Fr_{v} are equal up to a root of unity of bounded order. That will violate the elementary fact that Kloosterman sums are the inverse Fourier transform of Gauss sums. One can also quote [6], which says among other things that the sheaf R^{1} pr_{1,U}_{!}L_{ψ,f} is tamely ramified at 0, and is unipotent with exactly one Jordan block there. This of course rules out the possibility that the geometric monodromy group is finite.

Let K be the rational function field 𝔽(t) and a be the element t. The same argument as in the proof of Corollary 3.4 in [25] combined with Corollary 3.4 gives

(3.5) Corollary (Katz [17]) To each finite field k attach the measure

μ_{k} = \frac{1}{|k| − 1} ∑_{b∈k^{×}} δ_{Kl(k;b)/\sqrt{|k|}},

where δ_{x} denotes the Dirac measure supported at x. Then as |k| tends to infinity, the measures μ_{k} converge weakly to the Sato-Tate measure μ_{ST}.

(3.6) We examine the special case K = 𝔽(t) and f(x) = x + \frac{t}{x}. By Theorem 3.2, L(s, ψ, f) is automorphic for GL_{2} over 𝔽(t). Call π the associated automorphic representation. The only bad places for f are at 0 and ∞. As discussed above, the curve C defined by

y^{p} − y = f(x)

has semistable reduction at the place 0, hence the local component of π at 0 is a special representation (arising from inducing unramified characters) with the local L-factor equal to 1/(1 − q^{−s}), q being the cardinality of 𝔽. At ∞, the Jacobian of C has potentially good reduction, so the local L-factor of the component of π at ∞ is 1. It follows from Remark 3.7.3 (i) that the exponent of the conductor of π at ∞ is 3, therefore local components of π are discrete series at exactly two places of 𝔽(t), namely, 0 and ∞, and the conductor of π, written additively, is 0 + 3∞. By a result of Gelbart and Jacquet [11], for any quaternion algebra H over 𝔽(t) ramified at 0 and ∞, there is an automorphic representation σ of H^{×} over 𝔽(t) corresponding to π locally and globally according to Langlands philosophy. In particular, the component of σ at 0 is the trivial representation, and σ and π have the same L-functions. We record this result in

(3.6.1) Theorem Let f(x) = x+\frac{t}{x} be as above and let H be a quaternion algebra over 𝔽(t) ramified at 0 and ∞. Then there is an automorphic representation σ of H^{×} over 𝔽(t) whose L-function is

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

Here q is the cardinality of 𝔽.

(3.7) Appendix: Kloosterman curves and their Jacobians

(3.7.1) First we consider a Kloosterman curve over a field k of characteristic p > 2. Such a curve has an affine part given by the equation

y^{p} − y = x + \frac{a}{x} , a ∈ k^{×}

and we denote its projective smooth model by C_{a}. One quickly finds that the one-form \frac{dx}{x} is a regular differential on C_{a}, with exactly two zeroes at (x = 0, y = ∞) and (x = ∞, y = ∞), both of order p − 2. So we conclude that C_{a} has genus p − 1.

Our curve C_{a} is a hyperelliptic curve; “the projection to y” makes C_{a} a double cover of ℙ^{1}, ramified at two points. The hyperelliptic involution is given by (x, y) ↦ (\frac{a}{x}, y). Besides the hyperelliptic involution, C_{a} has two automorphisms, defined by σ : (x, y) ↦ (x, y + 1), τ : (x, y) ↦ (−x, −y) respectively. The two automorphisms σ and τ generate a dihedral group D_{2p} with 2p elements. Its integral group ring therefore operates on the Jacobian J_{a} of C_{a}. (Of course the hyperelliptic involution operates as [−1] on J_{a}.) So we have a homomorphism ℚ[D_{2p}] → End^{0}(J_{a}) := End(J_{a}) ⊗_{ℤ}ℚ.

We study the semi-simple algebra ℚ[D_{2p}]. From elementary representation theory for the dihedral group, one knows that it decomposes as a product

ℚ[D_{2p}] ≅ ℚ × ℚ × B

where B is a quaternion algebra over F := ℚ(μ_{p})^{+}, the maximal totally real subfield of the cyclotomic field ℚ(μ_{p}). Actually the quaternion algebra B spits: B ≅ M_{2}(F). One only has to write down the following representation of D_{2p} over F:

σ ↦ (\begin{matrix}cos(\frac{2π}{p}) & − sin(\frac{2π}{p}) \\ sin(\frac{2π}{p}) & cos(\frac{2π}{p}) \end{matrix}) , τ ↦ (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) .

One verifies without difficulty that the image of ℤ[D_{2p}] in M_{2}(F) generated by these two elements is maximal away from the primes above 2, but not maximal there.

From the structure of ℚ[D_{2p}] above we conclude that J_{a} is isogenous to a product X_{1} × X_{2} of two abelian varieties over k, each of dimension \frac{p−1}{2}. For convenience we take X_{1} to be the image of 1 + τ, and X_{2} the image of 1 − τ. Both X_{1} and X_{2} have multiplication by 𝒪_{F}, and they are isogenous to each other.

We can say a little bit about the slope condition of J_{a}, namely J_{a} is an ordinary abelian variety. To check this, first recall the general fact that the Cartier operator on H^{0}(C_{a}, Ω^{1}) is dual to the Frobenius action on H_{1}(C_{a}, 𝒪_{C_{a}}). It is immediate from the definition that the inverse Cartier operator sends the regular one-form \frac{dx}{x} to itself. Therefore the p-rank of J_{a} is strictly positive. Now the ring ℤ[μ_{p}] ⊗_{ℤ} ℤ_{p} operates on the p-adic Tate module of the maximal étale quotient of the Barsotti-Tate group of J_{a}. This module is non-zero since the p-rank is positive, and note that ℚ(μ_{p}) is totally ramified above p. This forces the p-adic Tate module to have rank p − 1, meaning that J_{a} is ordinary.

Let us consider the family {C_{t}} of Kloosterman curves over 𝔾_{m}, where t is the coordinate function of 𝔾_{m} = Spec k[t, t^{−1}]. We would like to know about the behavior of this family as t → 0 and t → ∞. By elementary calculation, one sees that the family {J_{t}} has multiplicative reduction. In fact the family of curves {C_{t}} has semistable reduction at t = 0, the fiber at t = 0 being two smooth projective lines intersecting each other at p points with normal crossings. The calculation at t = ∞ is messy. All we want to say is that the family {J_{t}} has bad reduction at t = ∞, as one can check easily. On the other hand it has potentially good reduction at t = ∞. One quick way to see this is by Raynaud’s trick, see [32] Th. 5, p. 62. Assume otherwise that {J_{t}} does not have potentially good reduction. We recall that X_{1,t} and X_{2,t} has multiplication by 𝒪_{ℚ(μ_{p})}_{+}. This forces {J_{t}} to have totally multiplicative reduction above t = ∞ after a finite base change. Now we remember that C_{t} is ordinary for t ∈ 𝔾_{m}, and it has semistable totally degenerate reduction above t = 0. Taking the p-torsion subgroup of the semiabelian reduction of the family {J_{t}} over a finite cover of ℙ^{1}, we obtain a trivialization of the Hodge line bundle \underline{ω} of the semiabelian scheme. By the positivity of the Hodge line bundle, we deduce that the family {J_{t}} is isotrivial, a contradiction. This proves the assertion. The same argument also shows that the good reduction fibers above t = ∞ of the potentially good reduction must have strictly positive slopes.

There is a mildly interesting question in this aspect, namely one would like to know the slopes of the reduction of {J_{t}} above t = ∞ after realizing the potentially good reduction by making a finite base change. From the consideration above, one deduces that either there are two distinct positive slopes with equal multiplicity, or that \frac{1}{2} is the only slope and the reduction is supersingular. The authors have not determined what the answer is, although it is quite conceivable that this is already known in the literature.

Of course, all of the above are compatible with the basic properties of Kloosterman sums given in [6], Th.7.8, p. 221. Each nontrivial irreducible character ψ : ℚ(μ_{p}) ↪ \overline{ℚ_{ℓ}} cuts out a ranktwo piece of the \overline{ℚ_{ℓ}}-representation given by the ℓ-power torsion points of J_{t}. This establishes the connection between our elementary consideration above and Deligne’s more sophisticated point of view. There one sees that the monodromy is unipotent with one Jordan block at t = 0, while wildly ramified at t = ∞.

We summarize the above discussion in

(3.7.2) Proposition Let k be a field of characteristic p > 2. For each a ∈ k^{×}, let C_{a} be the projective smooth curve over k with affine model given by the equation y^{p} − y = x + \frac{a}{x}.

(1) The curve C_{a} is a hyperelliptic curve of genus p − 1.

(2) There is a dihedral group D_{2p} acting on C_{a}.

(3) The Jacobian J_{a} of C_{a} is isogenous to a product X_{1} ×X_{2} of two abelian varieties of dimension \frac{p−1}{2}. The two abelian varieties X_{1} and X_{2} are isogenous to each other. Each X_{i} has endomorphisms by the ring of integers of the maximal totally real subfield ℚ(μ_{p})^{+} in the cyclotomic field ℚ(μ_{p}).

(4) The Jacobian J_{a} is an ordinary abelian variety.

(5) Suppose that the base scheme is 𝔾_{m}, and consider the family {C_{t}} of Kloosterman curves. This family has totally degenerate reduction at t = 0, the closed fiber being two copies of smooth projective lines intersecting transversally at p distinct points.

(6) At t = ∞ the family {J_{t}} has potentially good reduction. The fiber of the good reduction either has two distinct slopes, both strictly positive, or is supersingular.

(3.7.3) Remarks (i) Suppose that the base scheme is 𝔾_{m} as above. At t = 0, both the Swan conductor and the Artin conductor of the first cohomology of {J_{t}} is zero. This follows from (5) above and remains true after base change, or one can simply cite [6], p. 222, Thm. 7.8 (iii). When transformed into a representation of the Weil-Deligne group, the Artin conductor at t = 0 is 1 due to the nontrivial nilpotent action N. At t = ∞, the Swan conductor of {J_{t}} is p − 1, while the Artin conductor is 3(p − 1). More precisely, each nontrivial additive character ϕ of ℤ/pℤ cuts out from the first cohomology of {J_{t}} a rank-two smooth sheaf ℱ_{ϕ} over 𝔾_{m}; and ℱ_{ϕ} is isomorphic to ℱϕ̄. Each ℱ_{ϕ} has Swan conductor 1 and Artin conductor 3 at t = ∞; see [6], p. 222, Thm. 7.8 (iv).

(ii) When p = 2, the curve C_{a} is an elliptic curve. Statements (3) – (6) of Proposition 3.7.2 remain true, with the same proof, and Remark (i) above was proved in [30].

§4. Connections with Ramanujan graphs

A Ramanujan graph is a connected k-regular graph whose eigenvalues other than ±k (called nontrivial eigenvalues) are majorized by 2\sqrt{k − 1}. Such graphs are close to being random regular graphs, and hence are good communication networks. We refer to [24] and papers therein for more details concerning such graphs. Explicit constructions of infinite families of Ramanujan graphs with valency k equal to 1 plus a prime power are initiated by Margulis [29] and, independently, by Lubotzky, Phillips and Sarnak [28] based on quaternion groups over ℚ, and later followed by Morgenstern [31] based on quaternion groups over function fields. In this construction the nontrivial eigenvalues of graphs are in fact eigenvalues of Hecke operators, which are known to satisfy the Ramanujan-Petersson conjecture, giving rise to the desired eigenvalue estimate. We review the essential ingredients relevant to our concern.

Denote by K the rational function field 𝔽(t) with |𝔽| = q, an odd integer. Fix a non-square δ ∈ 𝔽. Let H be the quaternion algebra over K with basis 1, i, j, ij satisfying i^{2} = δ, j^{2} = t − 1, ij = −ji. Then H ramifies only at 1 and ∞, and it has class number one. Denote by D the algebraic group H^{×} mod its center and write 𝒪_{v} for the ring of integers of the completion of K at the place v. The vertices of a Morgenstern graph X_{𝒦} are double cosets of adelic points of D depending on the choice of an open subgroup 𝒦 of ∏_{v≠∞} D(𝒪_{v}). More precisely,

X_{𝒦} = D(K) ∖ D(A_{K})/D(K_{∞})𝒦.

To get the graph structure on X_{𝒦} , we find double coset representatives of X_{𝒦} at the place 0 of K, namely,

X_{𝒦} ≈ Γ(𝒦) ∖ D(K_{0})/D(𝒪_{0}) = Γ(𝒦) ∖ P GL_{2}(K_{0})/P GL_{2}(𝒪_{0}),

where Γ(𝒦) = D(K) ∩ D(K_{∞})𝒦. Since P GL_{2}(K_{0})/P GL_{2}(𝒪_{0}) has a natural structure as (q + 1)- regular infinite tree, this interpretation shows that X_{𝒦} is a quotient graph of the tree, and hence is a (q +1)-regular finite graph. The first expression of X_{𝒦} shows that the functions on the vertices of X_{𝒦} are in fact automorphic functions of the quaternion group D(A_{K}), and the second expression of X_{𝒦} implies that the action of the adjacency matrix of the graph X_{𝒦} is nothing but the Hecke operator T_{0} at the place 0 acting on automorphic forms, and the nontrivial eigenvalues of X_{𝒦} are the eigenvalues of T_{0} at the nonconstant automorphic forms on X_{𝒦} . Drinfel’d [10] has shown that the nontrivial eigenvalues have absolute value majorized by 2\sqrt{q}. Hence X_{𝒦} is a Ramanujan graph.

On the other hand, for a finite group G and a symmetric subset S (that is, s^{−1} lies in S whenever s does) of generators of G, the Cayley graph Cay(G, S) is a graph with vertices being the elements in G and the neighbors of a vertex x being xs, s ∈ S. Denote by 𝔽′ a quadratic extension of 𝔽, and for a ∈ 𝔽^{×} let S_{a} consist of elements in 𝔽′ whose norm to 𝔽 is a. It was shown in [22] that for any a ∈ 𝔽^{×} the Cayley graph Cay(𝔽′, S_{a}) is a Ramanujan graph, called a norm graph. Different choices of a result in isomorphic norm graphs. More precisely, the nontrivial eigenvalues of Cay(𝔽′, S_{a}) are the generalized Kloosterman sums

kl(𝔽; b) = ∑_{x∈S_{b}} ψ_{p} ∘ Tr_{𝔽′/(ℤ/pℤ)}(x)

for all b ∈ 𝔽^{×}, each occurring with multiplicity q + 1. Deligne in [6] showed that the generalized Kloosterman sum kl(𝔽; b) is the opposite of the Kloosterman sum Kl(𝔽; b), which Weil [35] has shown to have absolute value majorized by 2\sqrt{q}.

These norm graphs are related to the Morgenstern graphs as follows.

(4.1) Proposition (Proposition 4.1[27]) The Cayley graph Cay(𝔽′, S_{−1}) is isomorphic to the Morgenstern graph X_{𝒦_{0}} with 𝒦_{0} = (1 + 𝒫_{1}^{2}) ∏_{v≠1,∞} D(𝒪_{v}). Here 𝒫_{1} denotes the maximal ideal of the integral elements 𝒪_{1} in the quaternion algebra H(K_{1}).

Consequently, for each a ∈ 𝔽^{×}, there exist q + 1 linearly independent automorphic forms on X_{𝒦_{0}} = D(K) ∖ D(A_{K})/D(K_{∞})(1 + 𝒫_{1}^{2}) ∏_{v≠1,∞} D(𝒪_{v}) which are eigenfunctions of the Hecke operator T_{0} at 0 with eigenvalue kl(𝔽; a) = −Kl(𝔽; a). We proceed to describe these forms.

The graph X_{𝒦_{0}} has q^{2} vertices. Instead of representing them using elements in H(K_{0}) as in Morgenstern graph, we choose double coset representatives from H(K_{1}). More precisely, by applying the strong approximation theory to the adelic double coset expression of X_{𝒦_{0}} , we see that automorphic functions on X_{𝒦_{0}} can be identified with functions on G = 𝒪_{1}^{×}/(1 + 𝒫_{1}^{2}) which are left invariant by the nonzero elements in the quadratic extension 𝔽(i) of 𝔽 since j and 𝔽(i)^{×}, viewed as K-rational elements, are integral and invertible at all places other than 1 and j is a uniformizing element in H(K_{1}). As 𝔽(i)^{×} ∖ G may be represented by 1 + bj with b ∈ 𝔽(i) viewed as elements in H(K_{1}), these elements represent the vertices of X_{𝒦_{0}} . The space V of automorphic functions has a basis f_{ϕ} defined by

f_{ϕ}(1 + bj) = ϕ(b)

as ϕ runs through all additive characters of 𝔽(i). As before, denote by ψ the nontrivial additive character ψ_{p} ∘ Tr_{𝔽(i)/(ℤ/pℤ)}, thus all characters of 𝔽(i) are given by ψ_{x}, x ∈ 𝔽(i), which maps b to ψ(bx). The group G acts on V by right translations. More precisely, a(1 + cj) ∈ G sends f_{ψ_{x}} to ψ(xc)f_{ψ_{xa′}}, where a′ = ā/a with ā being the image of a under the Frobenius automorphism of 𝔽(i) over 𝔽. Clearly, when x = 0, we obtain the one dimensional irreducible space V_{0} generated by f_{ψ_{0}}. For a nonzero x, the orbit of f_{ψ_{x}} under G is a (q + 1)-dimensional space spanned by f_{ψ_{xu}} as u runs through all elements in S_{1}. Thus this space depends only on the norm a of x, which we denote by V_{a}. It is an irreducible G-space by Clifford’s theorem. We have decomposed V into a direct sum of q irreducible subspaces V_{a}, parametrized by elements of 𝔽, and no two of them are equivalent. The action of G on V_{a} extends to an irreducible representation of D(K_{1}) on the same space via right translations (j ∈ D(K_{1}) sends f_{ψ_{x}} to f_{ψx̄}), which is the component at the place 1 of the global admissible irreducible automorphic representation σ_{a} of D(A_{K}) such that V_{a}, viewed as a space of functions on X_{𝒦_{0}} , consists of the automorphic forms in the space of σ_{a} invariant by the largest congruence subgroup.

Next we look at the action of the Hecke operators T_{v} on V_{a} for v ≠ 1, ∞. Here a ≠ 0 so that the representation is nontrivial. Given such a place v, choose the uniformizer at K_{v} to be the irreducible polynomial π_{v} in 𝔽[t] such that when expressed in term of a polynomial in t − 1 the constant term is 1. As the quaternion algebra H splits at v, the set of elements in H(K_{v})^{×} with reduced norm in π_{v}𝒪_{v}^{×} is a disjoint union of Nv + 1 right cosets γ_{v,k}H(𝒪_{v})^{×}. The Hecke operator T_{v} acts on V by sending a function f in V to the linear combination of right translations of f by γ_{v,k}. This action is independent of the choice of the coset representatives γ_{v,j} since functions in V are right invariant by H(𝒪_{v})^{×}, and it preserves each space V_{a}. As the double coset space X_{𝒦_{0}} is represented by 𝔽(i) ∖ G, there are K-rational elements γ_{k} in H(K) so that at the place v they form coset representatives defining the Hecke operator T_{v} and at other places w ≠ v, 1, ∞ they fall in H(𝒪_{w})^{×}. Hence γ_{k}’s are integral everywhere except possibly at 1 and ∞. Multiplying γ_{k} by a suitable power of j if necessary, we may assume that all γ_{k} = f_{k} + g_{k}j with f_{k}, g_{k} polynomials in t − 1 over 𝔽(i) and at least one of f_{k}, g_{k} has nonzero constant term. This implies that the reduced norm of γ_{k}, denoted by Nm_{rd} γ_{k}, is π_{v} times a nonzero element in 𝔽. Since the norm map from 𝔽(i) to 𝔽 is surjective, multiplying by a suitable element in 𝔽(i) if necessary, we may assume that

Nm_{rd} γ_{k} = f_{k}f̄_{k} − (t − 1)g_{k}ḡ_{k} = π_{v}.

Here f̄ = ∑_{m} ā_{m}(t − 1)^{m} if f = ∑_{m} a_{m}(t − 1)^{m} is a polynomial over 𝔽(i). The constant term of π_{v} is 1 by our choice, hence the constant term of f_{k} is nonzero and it has norm 1. We may further assume that the constant term of f_{k} is one for all k. To proceed, observe the following

(4.2) Lemma Let f, g be two polynomials in t − 1 over 𝔽(i) such that g ≠ 0 and the element γ = f + gj has reduced norm π_{v}. Let u be an element in S_{1} and δ = f + ugj. Then γH(𝒪_{v})^{×} = δH(𝒪_{v})^{×} if and only if u = 1.

Proof. Sufficiency is obvious. To see the converse, suppose γH(𝒪_{v})^{×} = δH(𝒪_{v})^{×}. Then γ^{−1}δ lies in H(𝒪_{v})^{×}. Since γ^{−1} = π_{v}^{−1}(f̄− gj), the above condition implies uf̄g ≡ gf̄ (mod π_{v}). Thus if we can show that f and g are both coprime to π_{v} over 𝔽(i)[t − 1], then we conclude u = 1. To see this, start with

Nm_{rd} γ = ff̄− gḡ(t − 1) = π_{v}.

If f and π_{v} are coprime, then so are f̄ and π_{v}, and hence so are g and π_{v}. If h = gcd(f, π_{v}) is not a constant, then h is irreducible over 𝔽(i), either π_{v} = h or π_{v} = hh̄. If π_{v} = h, then π_{v} also divides f̄, which in turn implies that π_{v} divides both g and ḡ. Consequently, π_{v}^{2} divides Nm_{rd} γ, a contradiction. If π_{v} = hh̄, then h̄ divides f̄, and hence π_{v} divides ff̄. This in turn implies that π_{v} also divides gḡ. Consequently, Nm_{rd} γ/π_{v} = ff̄/π_{v} − (t − 1)gḡ/π_{v} = 1, which is possible only if ff̄ = π_{v} and g = 0, contradicting the assumption that g ≠ 0.

Lemma 4.2 shows that if γ_{k} = f_{k} + g_{k}j with g_{k} ≠ 0, then for all u ∈ S_{1}, the q + 1 elements f_{k} + ug_{k}j, viewed as elements in H(𝒪_{v}), have reduced norm π_{v} and represent different H(𝒪_{v})^{×}- cosets, hence they can be chosen in the definition of the Hecke operator T_{v}. We shall assume this is the case. For a global element x ∈ H(K), denote by x_{w} its component at the place w.

We are now ready to study the action of T_{v} on f ∈ V. Let y be any element in G. We have

T_{v}f(y) = ∑_{1≤k≤Nv+1} f(y(γ_{k})_{v}) = ∑_{1≤k≤Nv+1} f((γ_{k}^{−1})_{1}y)

since f is right invariant by ∏_{w≠1,v} D(𝒪_{w}) and left invariant by D(K). Further, since f is right invariant by 1 + 𝒫_{1}^{2}, each term in the above sum depends only on γ_{k} modulo t − 1, which has the form 1 + d_{k}j for some element d_{k} ∈ 𝔽(i) by our choice of γ_{k}. As f is invariant by the center of H(A_{K})^{×}, we may thus replace γ_{k}^{−1} by 1 − d_{k}j so that

T_{v}f(y) = ∑_{1≤k≤Nv+1} f((1 − d_{k}j)_{1}y).

Now take f to be f_{ψ_{x}} with x ∈ 𝔽(i). Given y = a_{1}(1 + bj)_{1} ∈ G with a, b ∈ 𝔽(i) and a ≠ 0, we have

\begin{aligned}T_{v}f_{ψ_{x}}(y) & = ∑_{1≤k≤Nv+1} f_{ψ_{x}}((1 − d_{k}j)_{1}a_{1}(1 + bj)_{1}) \\ & = ∑_{1≤k≤Nv+1} f_{ψ_{x}}(a_{1}(1 + (b − d_{k}a′)j)_{1}) = ∑_{1≤k≤Nv+1} ψ_{x}(−d_{k}a′)f_{ψ_{x}}(y), \end{aligned}

where a′ = ā/a is an element in S_{1}. Because of our choice of γ_{k}, d_{k} occurs if and only if all of its multiples by elements in S_{1} occur, the summation ∑_{1≤k≤Nv+1} ψ_{x}(−d_{k}a′) is independent of a and depends only on the norm of x. This implies that f_{ψ_{x}} is an eigenfunction of T_{v} for all x ∈ 𝔽(i), and those lie in the space V_{a} have the same eigenvalue with respect to T_{v}. Therefore V_{a} is a common eigenspace for all Hecke operators T_{v} with v ≠ 1.

To see the eigenvalues explicitly, we have to find explicit representatives occurring in T_{v} as described above. Let v be a place of degree one other than 1 and ∞. Then π_{v} = c(t − 1) + 1 for some c ∈ 𝔽^{×}. The representatives occurring in T_{v} are 1 + uj with u ∈ S_{−c}, and the eigenvalue at T_{v} for f_{ψ_{x}} ∈ V_{a} is

∑_{u∈S_{−c}} ψ_{x}(−u) = ∑_{u∈S_{−ac}} ψ(u) = −Kl(𝔽; −ac) = −Kl(𝔽_{v}; a/t − 1)

since 1/t − 1 ≡ −c (mod π_{v}).

Finally we show that no two eigenspaces V_{a}, a ∈ 𝔽, have the same eigenvalues with respect to all T_{v} with degree v equal to 1 and v not the place at 1. This is obvious for V_{0} since all eigenvalues are equal to q + 1, while eigenvalues for V_{a} with a ≠ 0 have absolute value at most 2\sqrt{q}. Suppose a and b are two nonzero elements in 𝔽 such that V_{a} and V_{b} have the same eigenvalues. Then we have Kl(𝔽, ac) = Kl(𝔽, bc) for all c ∈ 𝔽^{×}. Let χ be a character of 𝔽^{×} such that χ(a) ≠ χ(b). Consider

\begin{aligned}∑_{c∈𝔽^{×}} χ(c)Kl(𝔽, ac) & = ∑_{x∈𝔽^{×}} ψ(x) ∑_{c∈𝔽^{×}} χ(c)ψ(ac/x) = g(χ, ψ) ∑_{x∈𝔽^{×}} ψ(x)χ(x/a) \\ & = g(χ, ψ)^{2}χ(1/a), \end{aligned}

which is then equal to ∑_{c∈𝔽^{×}} χ(c)Kl(𝔽, bc) = g(χ, ψ)^{2}χ(1/b) by assumption on Kloosterman sums. Here g(χ, ψ) denotes the Gauss sum which has absolute value \sqrt{q}. Since χ(a) ≠ χ(b), we have a contradiction.

We have shown that the eigenspaces V_{a} with a ≠ 0 are distinguished by the eigenvalues −Kl(𝔽_{v}, a/t − 1) of T_{v} for places v of degree 1, v ≠ 1, ∞. This combined with Theorem 3.2 and 3.6.1 yields the conclusion that for any v ≠ 1, ∞, the eigenvalue of T_{v} on V_{a} for a ≠ 0 is −Kl(𝔽_{v}, a/t − 1). We summarize the above discussion in

(4.3) Theorem For each a ∈ 𝔽^{×}, let V_{a} be the (q + 1)-dimensional subspace of automorphic forms on D(K) ∖ D(A_{K})/D(K_{∞})(1 + 𝒫_{1}^{2}) ∏_{v≠1,∞} D(𝒪_{v}) spanned by the functions f_{ψ_{x}}, x ∈ S_{a}, as defined above. Then V_{a} is an eigenspace of all Hecke operators T_{v}, v ≠ 1, ∞, with eigenvalue −Kl(𝔽_{v}; a/(t−1)). Moreover, the L-function attached to the automorphic representation associated to the space V_{a} is

L_{a}(s) = ∏_{v≠1,∞} \frac{1}{1 + Kl(𝔽_{v}; a/(t − 1))Nv^{−s} + Nv^{1−2s}} · \frac{1}{1 − N∞^{−s}} .

(4.3.1) Remark Implicitly contained in the theorem above are many identities on character sums. For example, take a place v of degree 2 with π_{v} = c_{2}(t − 1)^{2} + c_{1}(t − 1) + 1, we may choose the representatives in T_{v} to consist of 1 + α(t − 1) + βj, where α, β ∈ 𝔽(i) satisfy the relations αᾱ = c_{2} and α + ᾱ − ββ̄ = c_{1}. Of these, two of α’s, namely, the roots of c_{2} − c_{1}x + x^{2}, have trace equal to c_{1} so that the corresponding β = 0, and the remaining α’s yield nonzero β. Note that replacing β in a representative by uβ for any u ∈ S_{1} yields another representative. In view of the action of the Hecke operators as analyzed above, we see that the eigenvalue of T_{v} on V_{a} is 2 + ∑_{b} ∑_{x∈S_{ab}} ψ(x), where b runs through all elements in 𝔽^{×} so that b = Tr_{𝔽(i)/𝔽}(α) − c_{1} for some α ∈ S_{c_{2}}. There are (q −1)/2 such b. This character sum is equal to −Kl(𝔽_{v}, a/(t−1)) by Theorem 4.3. In other words, kl(𝔽_{v}, a/(t − 1)) = 2 + ∑_{b} kl(𝔽, ab) with b as above.

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Ching-Li Chai  Wen-Ching Winnie Li
Department of Mathematics  Department of Mathematics
University of Pennsylvania  Pennsylvania State University
Philadelphia, PA 19104  University Park, PA 16802
U.S.A.  U.S.A.

         Email: chai@math.upenn.edu  Email: wli@math.psu.edu