Character sums, automorphic forms, equidistribution, and Ramanujan graphs

Part II. Eigenvalues of Terras graphs^{1}

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

Abstract

We study two types of character sums related to Terras graphs using the method of ℓ-adic cohomology. These character sums also arise as traces of Frobenii of some two dimensional linear representations of a global function field. Detailed information about these Galois representations at ramified places are obtained from analysis of vanishing cycles. Consequently we give a complete description of the automorphic forms of which these character sums appear as Fourier coefficients. These character sums are shown to be equidistributed with respect to the Sato-Tate measure.

§1. Introduction

Let 𝔽 be a finite field with q elements. For convenience we assume its characteristic p is odd although similar results for p = 2 also hold. Denote by 𝔽′ a quadratic extension of 𝔽. Choosing a nonsquare element δ ∈ 𝔽, we embed the multiplicative group of 𝔽′ into GL_{2}(𝔽) as the subgroup K_{δ}

consisting of matrices (\begin{matrix}a & bδ \\ b & a \end{matrix}) with a, b ∈ 𝔽. The coset space GL_{2}(𝔽)/K_{δ} may be represented by

the subgroup

H = \{(\begin{matrix}y & x \\ 0 & 1 \end{matrix}) : y ∈ 𝔽^{×}, x ∈ 𝔽\} ,

which resembles the classical Poincaré upper-half plane. Let S be a K_{δ}-double coset of GL_{2}(𝔽)

with cardinality greater than that of K_{δ}. It can be shown that S has coset representatives (\begin{matrix}y & x \\ 0 & 1 \end{matrix}),

where (x, y) runs through all 𝔽-rational points of an ellipse x^{2} = δ(ay + (y − 1)^{2}) for some a ∈ 𝔽. Further, there are q − 2 such double cosets, parametrized by the elements a in 𝔽 other than 0 and 4, which we denote by S_{aδ}.

The Terras graph X_{a} is the Cayley graph Cay(GL_{2}(𝔽)/K_{δ}, S_{aδ}/K_{δ}). Different choices of δ result in isomorphic graphs. It is a (q + 1)-regular graph whose eigenvalues can be explicitly expressed in terms of character sums of the following two types. The first type is associated to multiplicative characters χ of 𝔽:

λ_{a,χ} = ∑_{{x,y∈𝔽 \atop δ(ax+(x−1)^{2})=y^{2}}} χ(x),

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

2partially supported by grants DMS98-00609 and DMS01-00441 from the National Science Foundation

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

while the second type is associated to regular multiplicative characters ω of 𝔽′, that is, characters which are nontrivial on the kernel of Nm:

λ_{a,ω} = ∑_{z∈𝔽′, Nm(z)=1}ε(a − 2 − Tr(z))ω(z),

where Tr(z) and Nm(z) are the trace and norm of z from 𝔽′ to 𝔽, and ε(x) is 1, 0, -1 according to x ∈ (𝔽^{×})^{2}, x = 0 or x ∈ 𝔽 ∖ 𝔽^{2}. We remark that λ_{a,χ} is independent of the choice of δ. Using the Riemann hypothesis for curves, one can show that the nontrivial eigenvalues have absolute value majorized by 2\sqrt{q}, hence the Terras graphs X_{a} are Ramanujan graphs. The reader is referred to [2], [1], and Chap 9 of [9] for more details on Terras graphs.

As in Part I, let H be the quaternion algebra over the rational function field K with the field of constants 𝔽 which is ramified only at 0 and ∞, let D be the multiplicative group of H divided by its center, and let X_{𝒦} be the Ramanujan graph with vertices the double coset space D(K)\D(A_{K})/D(K_{∞})𝒦 for some congruence subgroup 𝒦 of ∏_{v≠∞} D(𝒪_{v}) and the edges inherited from the tree structure of D(K_{0})/D(𝒪_{0}) at place 0. Here D(𝒪_{v}) is defined by a fixed maximal order of the quaternion algebra H. The connection between Terras graphs and Ramanujan graphs constructed by Morgenstern [11] based on D was proved in [10] as follows.

(1.1) Proposition For b ∈ 𝔽, b ≠ 0, 1, the Terras graph X_{4(b−1)/b} is a quotient of the Morgenstern graph X_{𝒦_{b}} with 𝒦_{b} = κ_{b} ∏_{v≠b,∞} D(𝒪_{v}), where κ_{b} is the compact subgroup of D(𝒪_{b}) consisting of all elements in D(𝒪_{b}) congruent to the identity element modulo t − b.

Therefore the eigenvalues λ_{4(b−1)/b,χ} and λ_{4(b−1)/b,ω} of Terras graphs are among the eigenvalues of the Hecke operator T_{0} at place 0 acting on automorphic forms on the double coset space X_{𝒦_{b}} . Two questions arise naturally :

(1.2) Questions (i) Find automorphic forms on X_{𝒦_{b}} whose Fourier coefficients are eigenvalues of Terras graphs.

(ii) Find the distribution of eigenvalues of Terras graphs.

(1.3) Let K be a (not necessarily rational) function field with 𝔽 as its field of constants. The purpose of this paper is to construct automorphic forms of GL_{2} over K whose Fourier coefficients are given by the two types of eigenvalues of Terras graphs, analogous to what we did for norm graphs in Part I. These forms are parametrized by nonzero elements in K. We also show that the Sato-Tate conjecture holds for these forms when the parameter is not a constant. In particular, when K is the rational function field 𝔽(t), for special choices of the parameter, we compute the associated L-functions explicitly and show that they are also L-functions attached to automorphic forms on X_{𝒦_{b}} via Jacquet-Langlands correspondence. This answers question (i). For question (ii), the numerical data given by Terras in [12] suggests that the normalized eigenvalues of Terras graphs are uniformly distributed with respect to the Sato-Tate measure. We give a theoretic explanation of this phenomenon, as a consequence of the stronger result that the relevant automorphic forms satisfy the Sato-Tate conjecture.

Our approach is geometrical, using the theory of ℓ-adic cohomology. We study character sums of the second type in section 2 and the first type in section 3, by investigating the actions of Gal(K^{sep}/K) on certain sheaves, computing vanishing cycles and local factors at bad places, and determining the geometric monodromy group. The reformulation of the results in terms of automorphic forms is given in section 4, where applications to Terras graphs are also explored.

For readers primarily interested in the aspects of automorphic forms and graph theory, § 2 and § 3 may look overly technical. Here is an outline of how one uses informations from geometry. Once one knows that a family of character sum comes from a rank-two smooth \overline{ℚ_{ℓ}}-sheaf ℳ of geometric origin on an algebraic curve C over a finite field , one deduces from Weil’s converse theorem that the L-function attached to this family of character sum comes from an automorphic representation π of GL(2). If the neutral component of the geometric monodromy group of ℳ is SL(2), then the automorphic representation π is cuspidal, and the eigenvalues of Hecke operators are equidistributed according to the Sato-Tate law. The weight information about ℳ gives estimates of Hecke eigenvalues at unramified places. If one knows that there are at least two ramified places where the local components of π are either special or supercuspidal, then the automorphic representation π can be obtain from from an automorphic representation of a quaternion division algebra over the function field of C via the Jacquet-Langlands correspondence. Our calculation of the vanishing cycles in §§2–3 provides detailed information about the local Galois representations at the ramified places, which are local Langlands parameters of the ramified components of π. Readers who are more geometrically inclined may regard §§2–3 as examples in the theory of ℓ-adic cohomology whose singularities are explicitly computable.

§2. A family of character sums of the second type

The goal of this section is to study the family of exponential sums of the second type, also known as “Soto-Andrade type” in the literature,

SA_{b,ε,ω} := ∑_{u∈U(𝔽)} ε(Tr(u) + b) ω(u) ,

where b is an element of 𝔽, U(𝔽) is the norm-one subgroup of the quadratic extension 𝔽′ of 𝔽, and ω (resp. ε) is a character of U(𝔽) (resp. 𝔽^{×}).

(2.1) In this section, we follow the notations of [7] closely.

• Let 𝔽 be a finite field with q elements, where q is a power of an odd prime number p. Let 𝔽′ be a quadratic extension field of 𝔽.

• Let \underline{𝔽′^{×}} = Res_{𝔽′/𝔽}𝔾_{m} (resp. \underline{𝔽′} = Res_{𝔽′/𝔽}𝔾_{a}) be the Weil restriction of scalars of 𝔾_{m} (resp. 𝔾_{a}) for the extension 𝔽′/𝔽. For any 𝔽-algebra R, we have \underline{𝔽′^{×}}(R) = (𝔽′ ⊗_{𝔽} R)^{×} and \underline{𝔽′}(R) = 𝔽′ ⊗_{𝔽} R. Let Nm : \underline{𝔽′^{×}} → 𝔾_{m} and Tr : \underline{𝔽′} → 𝔾_{a} be the (𝔽′ ⊗_{𝔽} R/R)-norm and the (𝔽′ ⊗_{𝔽} R/R)-trace respectively.

• Denote by U the kernel of Nm : \underline{𝔽^{′×}} → 𝔾_{m}; it is a one-dimensional torus over 𝔽 such that for every 𝔽-algebra R, U(R) consists of all elements u ∈ (𝔽′ ⊗_{𝔽} R)^{×} with Nm(u) = 1. Especially U(𝔽) is the norm-one subgroup of 𝔽′.

• Let ω : U(𝔽) → \overline{ℚ_{ℓ}}^{×} be a character of U(𝔽), and let ε : 𝔽^{×} → \overline{ℚ_{ℓ}}^{×} be a character of 𝔽^{×}.

• Let L_{ω} be the smooth rank-one \overline{ℚ_{ℓ}}-sheaf over U, given by the push-out of the Lang torsor Id · Fr_{q}^{−1} : U → U by ω. For every closed point x of U, the geometric Frobenius Fr_{x} acts on the geometric generic fiber L_{ω,η̄} via the scalar ω(Nm_{𝔽(x)/𝔽,U}(x)).

• Let L_{ε} be the smooth rank-one \overline{ℚ_{ℓ}}-sheaf on 𝔾_{m} obtained from the Lang torsor construction, using the character ε of 𝔽^{×} = 𝔾_{m}(𝔽).

• Let ε̃ be the composition of Nm_{𝔽′/𝔽,𝔾_{m}} : 𝔽′^{×} → 𝔽^{×} with ε : 𝔽^{×} → \overline{ℚ_{ℓ}}^{×}. Let ω̃ be the composition of Nm_{𝔽′/𝔽,U} : U(𝔽′) → U(𝔽) with ω : U(𝔽) → \overline{ℚ_{ℓ}}^{×}.

• Denote by Y the scheme U ×_{Spec 𝔽} 𝔸^{1}[\frac{1}{Tr +t}], where t is the regular function on U ×_{Spec 𝔽} 𝔸^{1} which corresponds to the projection pr_{2} : U ×_{Spec 𝔽} 𝔸^{1} → 𝔸^{1}.

• Denote by L_{ε(Tr +t)} the pull-back of L_{ε} by Tr +t : Y → 𝔾_{m}.

• Denote by F = F_{ε,ω} the smooth rank-one \overline{ℚ_{ℓ}}-sheaf L_{ε(Tr +t)} ⊗ pr_{1}^{∗}L_{ω} on Y. For every closed point x of Y, the geometric Frobenius Fr_{x} acts on the geometric generic fiber F|η̄ of F_{ε,ω} via the scalar ε (Nm_{𝔽(x)/𝔽,𝔾_{m}}(Tr(u) + b)) · ω (Nm_{𝔽(x)/𝔽,U}(u)) , where (u, b) is the tautological point of Y (𝔽(x)) given by x, and Tr(u) ∈ 𝔾_{a}(𝔽(x)) is the image of u under Tr : U → 𝔾_{a}.

• Denote by π : Y → 𝔸^{1} the composition of the inclusion Y ↪ U × 𝔸^{1} and the projection U × 𝔸^{1} → 𝔸^{1}.

The theory of ℓ-adic cohomology yields the relative cohomology sheaves with compact support R^{i}π_{!}F on 𝔸^{1}; the alternating sum of the trace of the action of the geometric Frobenii of a closed point of 𝔸^{1} on these cohomology sheaves is a character sum of Soto-Andrade type. The following properties on sheaves are immediate from the definition.

(2.1.1) Lemma (i) The sheaf F_{ε^{−1},ω^{−1}} is naturally isomorphic to the dual of F_{ε,ω}.

(ii) The pull-back and the push-forward of the the sheaf F_{ε,ω} under the involution u ↦ u^{−1} of U are both naturally isomorphic to F_{ε,ω^{−1}}.

In the rest of this section we will analyze the sheaves R^{i}π_{!}F on 𝔸^{1}, and will obtain information on the action of the decomposition group at points where the sheaf R^{i}π_{!}F on 𝔸^{1} is ramified. These information are sufficient to determine local factors of the L-function attached to F at the ramified points of R^{i}π_{!}F on 𝔸^{1}.

(2.2) Since the torus U splits over 𝔽′, it is convenient to extend the base field from 𝔽 to 𝔽′. The scheme U ×_{Spec 𝔽} Spec 𝔽′ is isomorphic to 𝔾_{m} = Spec 𝔽′[z, z^{−1}]. The regular function z on U ×_{Spec 𝔽} Spec 𝔽′ is the character of U which induces the identity when restricted to U(𝔽) ⊂ 𝔽′^{×}. Moreover we have

Y ×_{Spec 𝔽} Spec 𝔽′ ≅ 𝔾_{m} ×_{Spec 𝔽′} 𝔸^{1} [\frac{1}{z + z^{−1} + t}] = 𝔾_{m} ×_{Spec 𝔽′} 𝔸^{1} [\frac{1}{z^{2} + tz + 1}] ,

since z is an invertible function on Y ×_{Spec 𝔽} Spec 𝔽′.

We identify U ×_{Spec 𝔽} Spec 𝔽′ with 𝔾_{m} over 𝔽′ using the regular function z, and consequently we identify ω̃ with the character

ω̃ : (𝔽′)^{×} → \overline{ℚ_{ℓ}}^{×}, x ↦ ω̃(x) = ω(x/τ(x))

on (𝔽′)^{×}, where τ denotes the non-trivial element of Gal(𝔽′/𝔽).

Over Y_{/𝔽′} = Spec 𝔽′[z, z^{−1}, t, (z^{2} + tz + 1)^{−1}], the sheaf F becomes

L_{ε̃(z+z^{−1}+t) ω̃(z)} = L_{ε̃(z+z^{−1}+t)} ⊗ pr_{1}^{∗}Lω̃ .

(2.3) Proposition Assume either that the character ε^{2} of 𝔽^{×} is not trivial, or that the character ω^{2} of U(𝔽) is not trivial.

(i) If i ≠ 1, then R^{i}π_{!}F|_{𝔸^{1}} = 0.

(ii) The restriction R^{1}π_{!}F|_{𝔸^{1}−{2,−2}} of R^{1}π_{!}F to the open subscheme 𝔸^{1} − {2, −2} of 𝔸^{1} is a smooth \overline{ℚ_{ℓ}}-sheaf of rank two over 𝔸^{1} − \{2, −2\}.

(iii) The stalks of R^{1}π_{!}F at a geometric point b̄ = ±2 of 𝔸^{1} is a one-dimensional vector space over \overline{ℚ_{ℓ}}.

(iv) Assume moreover that the character ε of 𝔽^{×} is not trivial. Then the smooth rank-two \overline{ℚ_{ℓ}}-sheaf R^{1}π_{!}F|_{𝔸^{1}−{2,−2}} is pure of weight one.

(v) The sheaf R^{1}π_{!}F_{ε,ω}|_{𝔸^{1}−{2,−2}} is naturally isomorphic to R^{1}π_{!}F_{ε,ω^{−1}}|_{𝔸^{1}−{2,−2}}.

(vi) If the character ε of 𝔽^{×}is non-trivial, then the dual of R^{1}π_{!}F_{ε,ω}|_{𝔸^{1}−{2,−2}} is naturally isomorphic to R^{1}π_{!}F_{ε^{−1},ω^{−1}}|_{𝔸^{1}−\{2,−2\}}(1). If ε has order two, then det(R^{1}π_{!}F_{ε,ω}|_{𝔸^{1}−\{2,−2\}}) = \overline{ℚ_{ℓ}}(−1).

Proof. This proposition is proved in [7, Thm. 1]. We reproduce the proof for the convenience of the reader.

Some remark on the hypothesis is in order. The hypothesis on ε and ω means that the characters ε̃^{2} and ω̃^{2} of (𝔽′)^{×} cannot be both trivial, or equivalently that the characters ε̃ · ω̃ and ε̃ · ω̃^{−1} of (𝔽′)^{×} cannot both be trivial. Since the order of ε divides q − 1, and the order of ω divides q + 1, ε̃ = ω̃ if and only if ε̃ = ω̃^{−1}.

We claim that the above hypothesis on the characters ε, ω implies that for any geometric point b̄ ∈ 𝔸^{1}, the sheaf F|Yb̄ on Yb̄ ⊂ 𝔾_{m} ⊂ ℙ^{1} is ramified at both 0 and ∞. Consider first the point ∞. We have

F|_{Yb̄} = L_{ε̃(1+ \frac{b̄}{z} + \frac{1}{z^{2}} )} ⊗ L_{(ε̃·ω̃)(z)} .

Moreover L_{ε̃(1+ \frac{b̄}{z} + \frac{1}{z^{2}} )} is unramified at ∞. Hence F|_{Yb̄} is unramified at ∞ if and only if ε̃· ω̃ is equal

to the trivial character of (𝔽′)^{×}, which is ruled out by our hypothesis. Similarly we conclude from

F|_{Yb̄} = L_{ε̃(z^{2}+b̄z+1)} ⊗ L_{(ε̃^{−1}·ω̃)(z)}

that F|_{Yb̄} is unramified at 0 ∈ ℙ^{1} if and only if ε̃^{−1} · ω̃ is equal to the trivial character of (𝔽′)^{×}, which again is ruled out by our hypothesis. The claim is proved.

By the proper base change theorem, the first statement that R^{i}π_{!}F = 0 for i ≠ 1 can be checked fiber by fiber. Over the fiber Yb̄ of a geometric point b̄ ∈ 𝔸^{1}, the sheaf F|_{Yb̄} can be thought of as a sheaf on U ×_{Spec 𝔽} Spec 𝔽^{sep}. The sheaf F|_{Yb̄} is ramified at the zero locus of Tr +b̄ in U ×_{Spec 𝔽} Spec 𝔽^{sep}, if ε is not trivial. This zero locus is finite of degree two over b̄, and is finite étale of degree two over b̄ if b̄ ≠ ±2.

The cohomology group H_{c}^{0}(Yb̄, F|_{Yb̄}) vanishes because Yb̄ is not complete. The cohomology group

H_{c}^{2}(Yb̄, F|_{Yb̄}) ≅ (F\overline{η_{Yb̄}})_{π_{1}^{geom}(Yb̄)}(−1)

vanishes because F_{Yb̄} is ramified at both z = 0 and z = ∞.

The statement that the stalks of R^{1}π_{!}F over the open subscheme 𝔸^{1} − {2, −2} are twodimensional vector spaces over \overline{ℚ_{ℓ}} is a consequence of Grothendieck’s Euler-Poincaré characteristic formula. The point is that the sheaf L_{ε} on 𝔾_{m} is tamely ramified at 0 and ∞, and the sheaf L_{ω} on U is also tamely ramified at the complement of U in the smooth projective completion of U. Hence the sheaf F|_{Yb̄} is tamely ramified at the complement of Yb̄ in the smooth projective completion of Yb̄. Suppose that b̄ is a geometric point of 𝔸^{1} − {2, −2}. The sheaf F|_{Yb̄} being tamely ramified, we have

χ_{c}(Yb̄, F|_{Yb̄}) = χ_{c}(Yb̄, \overline{ℚ_{ℓ}}) = χ_{c}(U, \overline{ℚ_{ℓ}}) − 2 = −2 .

Therefore dim\overline{ℚ_{ℓ}}(H^{1}(Yb̄, F|_{Yb̄})) = 2. This argument actually shows that for any geometric point b̄ ∈ 𝔸^{1}(𝔽^{sep}),

−χ_{c} (Yb̄, F|Yb̄) = −χ_{c} (Yb̄, \overline{ℚ_{ℓ}})

is equal to the number of zeros of the polynomial Z^{2}+b̄Z+1 in (𝔽^{sep})^{×}, counted without multiplicity. In particular if b̄ = ±2, then

−χ_{c} (Yb̄, F) = 1 = dim\overline{ℚ_{ℓ}} H^{1}(Yb̄, F|_{Yb̄}) .

We have proved statement (iii).

The smoothness of the \overline{ℚ_{ℓ}}-sheaf R^{1}π_{!}F on Y follows from [8, Cor. 2.1.2]: It suffices to verify the smoothness after base change to 𝔽′. The scheme Y ×_{Spec 𝔽} Spec 𝔽′|_{𝔸^{1}−{2,−2}} is an open subscheme of ℙ^{1} × (𝔸^{1} − {2, −2}), whose complement Z is finite étale over 𝔸^{1} − {2, −2} of degree 4. Moreover the sum of the Swan conductor at the “missing points” is constant (being zero) for every point b in the base 𝔸^{1} − {2, −2}. Therefore the theorem [8, Thm. 2.1.1] of Deligne applies.

In order to prove (iv), we may and do extend the base field from 𝔽 to 𝔽′, and we identify U ×_{Spec 𝔽}Spec 𝔽′ with 𝔾_{m} as in 2.2. Let b̄ be a geometric point over a closed point b of 𝔸^{1} −{2, −2}. Let j_{b} be the injection of 𝔾_{m} [\frac{1}{z^{2}+bz+1}] = Y_{b} into ℙ^{1}. The assumption that ε is non-trivial implies that F|Yb̄ is ramified at the zero locus of z^{2} + b̄z + 1 in 𝔾_{m}, and we have seen that F|Yb̄ is ramified at 0 and ∞. Hence

j_{b}_{∗}F|_{Y_{b}} = j_{b}_{!}F|_{Y_{b}} .

for any closed point b of 𝔸^{1} − {2, −2}. The statement (iv) follows from [3, Thm. 3.2.3].

The statement (v) follows immediately from Lemma 2.1.1 (ii). By [4, dualité, Thm. 1.3, 2.1], we have a perfect pairing

R^{1}π_{!}F_{ε,ω} × R^{1}π_{!}F_{ε^{−1},ω^{−1}} → R^{2}π_{!}\overline{ℚ_{ℓ}} ⟶^{∼} \overline{ℚ_{ℓ}}(−1)

induced by the cup product. When ε is the character of order two of 𝔽^{×}, this pairing is isomorphic to the determinant map for R^{1}π_{!}F_{ε,ω}. This proves (vi).

We would like to compute the Swan conductor of the smooth \overline{ℚ_{ℓ}}-sheaf R^{1}π_{!}F|_{𝔸^{1}−\{2,−2\}} over 𝔸^{1} − {2, −2} at the “missing points” {2, −2, ∞} ⊂ ℙ^{1}. Our strategy is to compute the cohomology of F using the “first projection” f : Y → U of Y ⊂ U ×_{Spec 𝔽} 𝔸^{1}.

(2.4) Lemma Let f : Y → U be the morphism given by the restriction of the projection U ×_{Spec 𝔽} 𝔸^{1} ⊃ Y to the first factor U. Then R^{i}f_{!}F = (0) for all i ≥ 0.

Proof. For any point a ∈ U, the fiber f^{−1}(a) ⊂ 𝔸^{1} is equal to Spec 𝔽(a) [t, \frac{1}{Tr(a)+t}]. Hence the complement W of Y in U ×_{Spec 𝔽} ℙ^{1} is finite étale over U of degree 2. Moreover by the same argument used in the proof of 2.3, the smooth \overline{ℚ_{ℓ}} sheaf F is tamely ramified along the relative Cartier divisor W ⊂ U ×_{Spec 𝔽} ℙ^{1}. The Euler-Poincaré characteristic formula gives

χ_{c} ((f^{−1}(b̄), F|_{f^{−1}(b̄)}) = χ_{c} (f^{−1}(b̄), \overline{ℚ_{ℓ}}) = 0

for any geometric point b of U. On the other hand, the smooth sheaf F|_{f^{−1}(b̄)} is ramified at the divisor W |_{f^{−1}(b̄)} ⊂ ℙ^{1} of degree two, hence neither F|_{f^{−1}(b̄)} nor its dual can be geometrically constant. Therefore

H_{c}^{0} ((f^{−1}(b̄), F|_{f^{−1}(b̄)}) = H_{c}^{2} ((f^{−1}(b̄), F|_{f^{−1}(b̄)}) = (0) .

We conclude that

H_{c}^{1} ((f^{−1}(b̄), F|_{f^{−1}(b̄)}) = (0)

as well since the Euler-Poincaré characteristic is equal to 0.

(2.5) Proposition The rank-two smooth \overline{ℚ_{ℓ}}-sheaf R^{1}π_{!}|_{𝔸^{1}−\{2,−2\}}F on 𝔸^{1}−\{2, −2\} is tamely ramified at {2, −2, ∞} ⊂ ℙ^{1}.

Proof. Recall from 2.3 (iii) that the stalk of R^{1}π_{!}F at 2 and −2 are both one-dimensional. From the short exact sequence

0 → (R^{1}π_{!}F) |_{𝔸^{1}−{2,−2}} → R^{1}π_{!}F → (R^{1}π_{!}F) |_{{2,−2}} → 0

we get

χ_{c} (𝔸^{1} − {2, −2}_{/𝔽^{sep}}, R^{1}π_{!}F) = χ_{c} (𝔸_{/𝔽^{sep}}^{1}, R^{1}π_{!}F) − 2 .

On the other hand, from the Leray spectral sequence we have

−χ_{c} (𝔸_{/𝔽^{sep}}^{1}, R^{1}π_{!}F) = χ_{c} (Y ×_{Spec 𝔽} Spec 𝔽^{sep}, F)
= ∑_{i}(−1)^{i} χ_{c} (U ×_{Spec 𝔽} Spec 𝔽^{sep}, R^{i}f_{!}F) = 0

by Lemma 2.4. Therefore χ_{c} (𝔸^{1} − {2, −2}_{/𝔽^{sep}}, R^{1}π_{!}F|_{𝔸^{1}−{2,−2}}) = −2.

The Euler-Poincaré characteristic χ_{c} (𝔸^{1} − {2, −2}_{/𝔽^{sep}}, R^{1}π_{!}F|_{𝔸^{1}−{2,−2}}) = −2 can also be computed by Grothendieck’s formula, which gives

−2 = χ_{c} (𝔸^{1} − {2, −2}_{/𝔽^{sep}}, R^{1}π_{!}F|_{𝔸^{1}−{2,−2}})
= 2 χ_{c} (𝔸^{1} − \{2, −2\}_{/𝔽^{sep}}, \overline{ℚ_{ℓ}}) − ∑_{b∈\{2,−2,∞\}} Sw_{b} (R^{1}π_{!}F|_{𝔸^{1}−\{2,−2\}})

Since χ_{c} (𝔸^{1} − \{2, −2\}_{/Spec 𝔽^{sep}}, \overline{ℚ_{ℓ}}) = −1, the sum of the Swan conductor at 2, −2, ∞ is equal to zero, hence the Swan conductor at these three missing points are all equal to zero. We have proved the tameness of R^{1}π_{!}F|_{𝔸^{1}−{2,−2}}.

(2.6) So far we have seen that G = G_{ε,ω} := R^{1}π_{!}F|_{𝔸^{1}−\{2,−2\}} is a smooth rank-two \overline{ℚ_{ℓ}}-sheaf which is tamely ramified at 2, −2, ∞. This sheaf corresponds to a linear representation ρ of the Galois group Gal(𝔽(ℙ^{1})^{sep}/𝔽(ℙ^{1})) on a two dimensional vector space G_{η\overline{𝔸^{1}}} over \overline{ℚ_{ℓ}}, the geometric generic fiber of G. We would like to understand the restriction of the representation ρ to a decomposition group D_{b}, b = 2, −2, ∞, using the theory of vanishing cycles.

We recall some standard notation about vanishing cycles. Let S be a smooth curve over a finite field 𝔽, and let Y be a scheme of finite type over S. Let K ∈ D_{c}^{b}(Y, \overline{ℚ_{ℓ}}) be a “constructible complex of \overline{ℚ_{ℓ}}-sheaves” on Y. Let s be a closed point of the base scheme S. Let S_{(s)} be the henselization of S at s, and let η_{s} be the generic point of S_{(s)}. Let \overline{η_{s}} = Spec(κ(η_{s})^{sep}) be the geometric point lying over η_{s}. Let \overline{S_{(s)}} be the spectrum of the normalization of S in κ(η_{s})^{sep}. Let

j\overline{η_{s}} : Y\overline{η_{s}} = Y ×_{S} \overline{η_{s}} ↪ Y\overline{S_{(s)}} = Y ×_{S} \overline{S_{(s)}}

be the natural open embedding, and let

is̄ : Ys̄ = Y ×_{S} s̄ ↪ Y\overline{S_{(s)}} = Y ×_{S} \overline{S_{(s)}}

be the natural closed embedding. The complex of near-by cycles over s for K is defined to be the object

RΨ_{s}(K) := i∗ ; s̄ Rj\overline{η_{s}}_{∗} K

in D_{c}^{b}(Ys̄, \overline{ℚ_{ℓ}}), endowed with the natural action of the decomposition group

D_{s} := Gal (κ(η_{s})^{sep}/κ(η_{s})) .

The complex of vanishing cycles RΦ_{s}(K) over s for K is defined as the mapping cone of the natural arrow i∗ ; s̄K → RΨ_{s}(K), endowed with the natural action of D_{s}. The cohomology sheaves of RΨ(K) and RΦ(K) are denoted by Ψ^{i}(K) and Φ^{i}(K) respectively.

(2.7) Denote by \overline{U} the projective completion of U, geometrically isomorphic to ℙ^{1}. Define \overline{Y} to be \overline{U} ×_{Spec 𝔽} ℙ^{1}, which contains Y = U ×_{Spec 𝔽} 𝔸^{1} as a dense open subscheme. Let j : Y ↪ \overline{Y} be the natural open embedding.

Recall that F is the rank-one \overline{ℚ_{ℓ}}-sheaf L_{ε(Tr +t)} ⊗ L_{ω} on Y. Let \overline{π} : \overline{Y} = \overline{U} ×_{Spec 𝔽} ℙ^{1} → ℙ^{1} be the natural projection.

Let \overline{F} be the sheaf j_{!}F, the sheaf F on Y extended by 0 to \overline{Y}. We would like to compute the vanishing cycle complex RΦ_{b}\overline{F} of \overline{F} for \overline{π}, where b ∈ \{2, −2, ∞\} is one of the “bad points” of the base scheme ℙ^{1}.

(2.8) Theorem Assume that ε is non-trivial, and that the order of ω is not equal to two if ε has order two. Let b be either 2 or −2.

(i) The vanishing cycle complex RΦ_{b}(\overline{F}) is concentrated at one geometric point z_{0} of Y_{b}, where z_{0} = −1 if b = 2, and z_{0} = 1 if b = −2.

(ii) The i-th cohomology sheaf Φ_{b}^{i}(\overline{F})_{z_{0}} of the stalk at z_{0} of RΦ_{b}(\overline{F}) is zero if i ≠ 1, and Φ_{b}^{1}(\overline{F})_{z_{0}} is one-dimensional.

(iii) Suppose that ε is non-trivial and has order two, and ω is not the non-trivial character of U(𝔽) of order two. Then the inertia group I_{b} operates trivially on Φ_{b}^{1}(\overline{F})_{z_{0}}, and Φ_{b}^{1}(\overline{F})_{z_{0}} is pure of weight two as a module of the decomposition group D_{b}.

(iv) Suppose that ε^{2} is non-trivial. Then Φ_{b}^{1}(\overline{F})_{z_{0}} is pure of weight one as a module of the decomposition group D_{b}.

Proof of (i) and (ii). The statement (i) follows from [8, Cor. 2.1.2]. It is easy to see that Φ_{b}^{−1}(\overline{F})_{z_{0}} = 0 and that Φ_{b}^{0}(\overline{F})_{z_{0}} = 0. The long exact sequence

⋯ → H^{i}(\overline{Y} b̄, i∗ ; b̄\overline{F}) → H_{c}^{i}(Y\overline{η_{b}}, j∗ ; \overline{η_{b}}F) → Φ_{b}^{i}(\overline{F})_{z_{0}} → H^{i+1}(\overline{Y} b̄, i∗ ; b̄\overline{F}) → ⋯

gives Φ_{b}^{i}(\overline{F})_{z_{0}} = 0 for all i ≥ 2, and it reduces to the following short exact sequence

0 → H^{1}(\overline{Y} b̄, i∗ ; b̄\overline{F}) → H_{c}^{1}(Y\overline{η_{b}}, j∗ ; \overline{η_{b}}F) → Φ_{b}^{1}(\overline{F})_{z_{0}} → 0 .

By the same argument as in the proof of Proposition 2.3, essentially a Euler characteristic calculation, we see that H^{1}(\overline{Y} b̄, i∗ ; b̄\overline{F}) = H_{c}^{1}(Yb̄, F|Yb̄) is one dimensional. This proves the statement (ii). Also notice that the inertia group I_{b} operates trivially on the one-dimensional subspace H^{1}(\overline{Y} b̄, i∗ ; b̄\overline{F}) of Gη̄.

Proof of (iv). From the proof of (i), (ii) above, we see that H_{c}^{1}(Yb̄, F|Yb̄) is a one-dimensional D_{b}-submodule of Gη̄. Let j_{b} : Y_{b} ↪ \overline{Y}_{b} be the canonical inclusion. Extending the base to 𝔽′, the map j_{b} becomes the inclusion of 𝔾_{m} − {z_{b}} in ℙ^{1}, where z_{b} = 1 (resp. z_{b} = −1) if b = −2 (resp. b = 2); the sheaf F_{b} becomes the sheaf L_{ε̃^{2}(z−z_{b})} ⊗ L_{ε̃^{−1}·ω̃} on 𝔾_{m} − {z_{b}}.

The assumption that ε^{2} is nontrivial implies that ε̃ ≠ ω̃^{±1}. Therefore F_{b} is ramified at the points z_{b}, 0 and ∞ of ℙ^{1}. Hence (j_{b})_{!}F_{b} = (j_{b})_{∗}F_{b}. By Theorem 3.2.3 in [3], the action of Fr_{b} on H_{c}^{1}(Yb̄, Fb̄) is pure of weight one. By Deligne’s theorem on the monodromy weight filtration, Theorem 1.8.4 of [3], we conclude that the D_{b}-module Gη̄ is pure of weight one.

To prove (iii), it suffices to show that the D_{b}-module H_{c}^{1}(Yb̄, F|Yb̄) is unramified and pure of weight zero, by Deligne’s theorem on the monodromy weight filtration [3, 1.8.4]. That the inertia group I_{b} operates trivially on H_{c}^{1}(Yb̄, F|Yb̄) is obvious. As before we extend the base field to 𝔽′. Since ε^{2} is trivial, the sheaf F|Yb̄ on 𝔾_{m} − {z_{b}} becomes the restriction to 𝔾_{m} − {z_{b}} of the smooth rank-one sheaf L_{ε̃·ω̃} on 𝔾_{m}.

We have H_{c}^{i}(𝔾_{m}_{/𝔽′^{sep}}, L_{ε̃·ω̃}) = (0) for i = 0, 2 since L_{ε̃·ω̃} is ramified at 0 and ∞. Furthermore χ_{c}(𝔾_{m}_{/𝔽′^{sep}}, L_{ε̃·ω̃}) = 0 by Grothendieck’s Euler-Poincaré characteristic formula. This implies that H_{c}^{i}(𝔾_{m}_{/𝔽′^{sep}}, L_{ε̃·ω̃}) = (0) for all i.

From the short exact sequence

0 → L_{ε̃·ω̃}|_{𝔾_{m}−{z_{b}}} → L_{ε̃·ω̃} → L_{ε̃·ω̃}|_{b} → 0

we see that H_{c}^{1}(Yb̄, L_{ε̃·ω̃}) ≅ L_{ε̃·ω̃}|b̄ as D_{b}-modules. This finishes the proof of Theorem 2.8.

(2.8.1) Remark In the case (iv) of Theorem 2.8, the inertia group I_{b} acts via a non-trivial finite cyclic quotient μ_{m} on Φ_{b}^{1}(\overline{F})_{z_{0}} with (m, p) = 1. The order, m, of this cyclic group depends only on the order of the character ε and is given as follows. Let n be the order of ε^{2}; write n = 2^{a}n_{1}, a ≥ 0, and (2, n_{1}) = 1. Then m = 2n if a = 0, m = n_{1} = \frac{n}{2} if a = 1, and m = n if a ≥ 2.

(2.9) Theorem Assume that p is an odd prime number. Suppose that ε is the non-trivial character of 𝔽^{×} of order two, and ω is not the non-trivial character of U(𝔽) of order two.

(i) Suppose that b = −2. Then the decomposition group D_{b} operates via the unramified character of order two λ_{2} : D_{b} ↠ μ_{2} of D_{b} on the one-dimensional \overline{ℚ_{ℓ}}-space (G|\overline{η})^{I_{b}} consisting of all inertia invariants in the geometric generic fiber of G. In other words every geometric Frobenius element Fr_{q} operates as −1.

(ii) Suppose that b = 2. Then the one-dimensional D_{b}-module (G|\overline{η})^{I_{b}} is unramified, on which every geometric Frobenius element Fr_{q} ∈ D_{b} operates as −ε(−1)ω(−1).

(iii) The geometric monodromy group for G is equal to SL(2).

Proof of (i). Suppose that b = −2, and denote by z_{1} a geometric point lying above (1, −2) ∈ U × 𝔸^{1}(𝔽) = U(𝔽) × 𝔽 This is the only point in the fiber Y_{b} where the sheaf F is ramified. Denote by b̄ a geometric point lying above b.

In the proof of Theorem 2.8 (iii) we saw that there is a short exact sequence

0 → H^{1}(\overline{Y} b̄, \overline{F}|_{\overline{Y} b̄}) → G|b̄ → Φ_{b}^{1}(\overline{F})_{z_{1}} → 0 .

The one-dimensional \overline{ℚ_{ℓ}} space H^{1}(\overline{Y}_{b}, \overline{F}|_{\overline{Y}_{b}}) is equal to (G|_{b})^{I_{b}}, the inertia invariants in the twodimensional D_{b}-module G|_{b}. The sheaf \overline{F}|_{\overline{Y}_{b}} on \overline{Y}_{b} ≅ \overline{U} is canonically isomorphic to the extension by zero to \overline{U} of the sheaf L_{ε(Tr −2)} ⊗ L_{ω} on U − \{1\}. Let j_{U−\{1\}} : U − \{1\} ↪ \overline{U} and j_{U} : U ↪ \overline{U} be the natural inclusions. Over 𝔽′, U ×_{Spec 𝔽}Spec 𝔽′ is isomorphic to 𝔾_{m}, and the sheaf L_{ε(Tr −2)} ⊗L_{ω}is isomorphic to

L_{}_{ε̃^{2}}_{(z−1)} ⊗ L_{ε̃^{−1}·ω̃} = L_{ε̃^{−1}·ω̃}|_{U−{1}} ,

since ε has order two. Clearly (U −\{1\} ↪ U)_{∗}L_{ε̃^{−1}·ω̃} is equal to the smooth rank-one \overline{ℚ_{ℓ}}-sheaf L_{ε̃^{−1}·ω̃} on U_{/𝔽′}, which is tamely ramified at 0 and ∞. By Grothendieck’s Euler-Poincaré characteristic formula, χ_{c}(U ×_{Spec 𝔽} Spec 𝔽^{sep}, L_{ε̃^{−1}·ω̃}) = 0, hence H_{c}^{i}(U ×_{Spec 𝔽} Spec 𝔽^{sep}, L_{ε̃^{−1}·ω̃}) = (0) for all i.

From the above we know that (U − {1} ↪ U)_{∗}L_{ε(Tr −2)} ⊗ L_{ω} is a smooth rank-one sheaf on U, and we have a short exact sequence

\begin{matrix}0 → (U − \{1\} ↪ U)_{!}(L_{ε(Tr −2)} ⊗ L_{ω}) → (U − \{1\} ↪ U)_{∗}(L_{ε(Tr −2)} ⊗ L_{ω}) \\ → (U − \{1\} ↪ U)_{∗}(L_{ε(Tr −2)} ⊗ L_{ω})|_{u=1} → 0 . \end{matrix}

From the associated long exact sequence of cohomologies we get an isomorphism

H^{1}(\overline{Y}_{b}, \overline{F}|_{\overline{Y}_{b}}) ≅ (U − \{1\} ↪ U)_{∗}(L_{ε(Tr −2)} ⊗ L_{ω})|_{u=1}

of D_{b}-modules.

To finish (i) we have to compute the action of D_{b} on (U − {1} ↪ U)_{∗}(L_{ε(Tr −2)} ⊗ L_{ω})|_{u=1}. By definition, the geometric Frobenius element Fr_{1} as ω(1) = 1 on the restriction of the smooth rank-one sheaf L_{ω} to the point 1 ∈ U(𝔽), so we only need to compute the Galois action on the restriction of (U − {1} ↪ U)_{∗}L_{ε(Tr −2)} to 1 ∈ U(𝔽). Since ε is the non-trivial character of 𝔾_{m}(𝔽) of order two, the sheaf L_{ε} on 𝔾_{m} is the push-forward of the short exact sequence

1 → μ_{2} → 𝔾_{m} ⟶^{[2]} 𝔾_{m} → 1

by the non-trivial character of μ_{2}. Hence the action of D_{b}/I_{b} ≅ Fr_{q}^{ℤ} on the one-dimensional \overline{ℚ_{ℓ}}- vector space (U − {1} ↪ U)_{∗}L_{ε(Tr −2)}|_{u=1} can be computed as follows. Pick a generator π_{1} of the maximal ideal of the local ring O_{U,1} and consider the leading term a_{m} · π_{1}^{m} in expansion of the element Tr −2 ∈ O_{U,1} as a power series in π_{1} with coefficients in 𝔽. Then every Frobenius element Fr_{q} operates as ε(a_{m}) · (−1)^{m} on the one-dimensional space above.

We still need to perform the power series expansion of Tr −2 ∈ O_{U,1}. Pick an element a ∈ 𝔽′, a ∉ 𝔽. Let z be the standard coordinate function of 𝔾_{m}_{/𝔽′} ≅ U ×_{Spec 𝔽} Spec 𝔽′. Then the function π_{1} = a z + a^{q} z^{−1} − a − a^{q} ∈ Γ(O_{U}) ⊗_{𝔽} 𝔽′ = Γ(𝔾_{m}_{/𝔽′}, 𝒪_{𝔾_{m}_{/𝔽′}}) is a regular function on U and gives a uniformizing element of O_{U,1}. Expanded as a series in the monomials of (z − 1), the leading term of a z + a^{q} z^{−1} − a − a^{q} is (a − a^{q}) (z − 1). On the other hand, the leading term of the expansion of Tr −2 in powers of (z − 1) is (z − 1)^{2}. Hence the leading term of the expansion of Tr −2 in the powers of π_{1} is (a − a^{q})^{−2} π_{1}^{2}. Since ε((a − a^{q})^{2}) = −1, we have proved the assertion in (i).

Proof of (ii). The case b = 2 is similar to the case b = −2 before. The only difference is the power series expansion of Tr +2. Pick an element a ∈ 𝔽′, a ∉ 𝔽 as before. Then π_{−1} = a z+a^{q} z^{−1}+a+a^{q} is a uniformizing element for O_{U,−1}, and the leading term of the expansion of Tr +2 in the monomials of π_{−1} is −(a − a^{q})^{2} π_{−1}^{2}.

Proof of (iii). The geometric monodromy group is contained in SL(2) by Prop. 2.3 (vi). On the other hand, we know by [3, Cor. 1.3.9] that the neutral component of the geometric monodromy group is semisimple, hence it is either trivial or is equal to SL(2) in the present case. According to the theory of monodromy weight filtration in [3, §1.8], the geometric monodromy group contain the exponential of the logarithm of N of the inertia action at b = 2 and b = −2. By (i) and (ii), the monodromy weight filtration is non-trivial both for b = 2 and b = −2, hence the logarithm N of the inertia action is nontrivial at b = ±2 according to Deligne’s theory in [3, §1.8]. In particular the geometric monodromy group cannot be finite. This finishes the proof of Theorem 2.9.

(2.9.1) Remarks (1) Suppose that ε has order two. From (i) and (ii) one sees that det(G) is unramified at ±2. We already know that it is tamely ramified at ∞, therefore det(G) is unramified since the tame fundamental group of 𝔸^{1} is trivial. This means that det(G) is the pull-back from Spec 𝔽. Then we can determine det(G) from its stalk at 2 or −2 using (i), (ii); or one use the fact that the characteristic polynomials of Frobenii for G are real to deduce that det(G) = \overline{ℚ_{ℓ}}(−1). This gives an alternative proof of the fact that det(G) is \overline{ℚ_{ℓ}}(−1).

(2) If ε^{2} is not trivial, then det(G) is ramified, hence is not equal to \overline{ℚ_{ℓ}}(−1).

(2.10) Theorem Assume that ε is not the trivial character of 𝔽^{×}, that ω is not the trivial character of U(𝔽), and that ε^{2} and ω^{2} are not both trivial. Consider the geometric fiber G|\overline{η} of G := R^{1}π_{!}F|_{𝔸^{1}−{2,−2}} as a module of the decomposition group D_{∞} at ∞. Then the wild inertia group P_{∞} acts trivially, and I_{∞} operates via a finite quotient with no non-zero fixed element in G|\overline{η}. As a D_{∞}-module, G|\overline{η} is a direct sum of two one-dimensional D_{∞}-submodules corresponding to two characters of D_{∞}; each character is ramified (tamely) and pure of weight one.

(2.11) Over 𝔽′, the complement of Y ×_{Spec 𝔽} Spec 𝔽′ in

\overline{Y} ×_{Spec 𝔽} Spec 𝔽′ = \overline{U} ×_{Spec 𝔽} ℙ^{1} ×_{Spec 𝔽} Spec 𝔽′

is the union of four divisors:

• S_{0} = {0} × ℙ^{1},

• S_{∞} = {∞} × ℙ^{1},

• \overline{Y}_{v} = \overline{U} × \{∞\},

• D_{h}, the Zariski closure in \overline{Y} of the zero locus of Tr +t in Y. The Frobenius element Fr_{q} interchanges S_{0} and S_{∞}. The divisors \overline{Y}_{v} and D_{h} are defined over 𝔽. The three divisors S_{0}, D_{h}, \overline{Y}_{v} intersect at the point y_{0} = (0, ∞); the three divisors S_{∞}, D_{h}, \overline{Y}_{v} intersect at the point y_{∞} = (∞, ∞).

Let X → \overline{Y} be the blowing-up of \overline{Y} at the two points \{y_{0}, y_{∞}\}. The inverse image in Z of the union of the four divisors in \overline{Y} above is the union of the following six divisors in Z.

• E_{0}, the exceptional divisor above y_{0},

• E_{∞}, the exceptional divisor above y_{∞},

• D̃_{0}, the strict transform of S_{0},

• D̃_{∞}, the strict transform of S_{∞},

• D̃_{h}, the strict transform of S_{h},

• Z_{v}, the strict transform of \overline{Y}_{v}. Let Z be the union of six divisors above; it is a reduced divisor with normal crossings. This

divisor has six singularities:

• x_{0,0}, the intersection of D̃_{0} and E_{0},

• x_{0,h}, the intersection of D̃_{h} and E_{0},

• x_{0,v}, the intersection of Z_{v} and E_{0},

• x_{∞,∞}, the intersection of D̃_{∞} and E_{∞},

• x_{∞,h}, the intersection of D̃_{h} and E_{∞},

• x_{∞,v}, the intersection of Z_{v} and E_{∞} Denote by j̃ = j̃_{Y,X} : Y ↪ X the open immersion of Y in X, and let F̃ := j̃_{!}F. Let π̃ : X → ℙ^{1}

be the projection from X to the base scheme ℙ^{1}, and let X_{∞} be the divisor E_{0} ∪ Z_{v} ∪ E_{∞}, the inverse image of ∞ under π̃. We would like to understand the vanishing cycle complex RΦ_{∞}(F̃) with respect to π̃. (2.12) Theorem Assume as in the statement of 2.10 that ε is not the trivial character of 𝔽^{×}, that ω is not the trivial character of U(𝔽), and that ε^{2} and ω^{2} are not both trivial.

(i) The i-th cohomology sheaf Φ_{∞}^{i}(F̃) of RΦ_{∞}(F̃) is equal to zero for all i ≥ 1.

(ii) The stalk of Φ_{∞}^{0}(F̃)_{x} is equal to zero if x ∈ {x_{0,0}, x_{0,h}, x_{0,v}, x_{∞,∞}, x_{∞,h}, x_{∞,v}}.

(iii) The sheaf Φ_{∞}^{0}(F̃)|_{Z_{v}−{x_{0,v},x_{∞,v}}} is smooth of rank one, tamely ramified at the two points {x_{0,v}, x_{∞,v}} with respect to Z_{v} − {x_{0,v}, x_{∞,v}} ↪ Z_{v}. Moreover

\begin{matrix}(Z_{v} − \{x_{0,v}, x_{∞,v}\} ↪ Z_{v})_{!} (Φ_{∞}^{0}(F̃)|_{Z_{v}−\{x_{0,v},x_{∞,v}\}}) \\ = (Z_{v} − \{x_{0,v}, x_{∞,v}\} ↪ Z_{v})_{∗} (Φ_{∞}^{0}(F̃)|_{Z_{v}−\{x_{0,v},x_{∞,v}\}}) \end{matrix}

The inertia group I_{∞} acts on this sheaf via a one-dimensional tame character whose order is equal to the order of ε.

(iv) The sheaf Φ_{∞}^{0}(F̃)|_{E_{0}−{x_{0,v},x_{0,h},x_{0,0}}} is smooth of rank one, tamely ramified at the three point {x_{0,v}, x_{0,h}, x_{0,0}} with respect to E_{0} − {x_{0,v}, x_{0,h}, x_{0,0}}. Moreover

\begin{matrix}(E_{0} − \{x_{0,v}, x_{0,h}, x_{0,0}\})_{!} (Φ_{∞}^{0}(F̃)|_{E_{0}−\{x_{0,v},x_{0,h},x_{0,0}\}}) \\ = (E_{0} − \{x_{0,v}, x_{0,h}, x_{0,0}\})_{∗} (Φ_{∞}^{0}(F̃)|_{E_{0}−\{x_{0,v},x_{0,h},x_{0,0}\}}) \end{matrix}

The inertia group I_{∞} acts on this sheaf via a tame character whose order is equal to the order of ε̃^{−1} · ω̃.

(v) The sheaf Φ_{∞}^{0}(F̃)|_{E_{∞}−{x_{∞,v},x_{∞,h},x_{∞,∞}}} is smooth of rank one, tamely ramified at the three points {x_{∞,v}, x_{∞,h}, x_{∞,∞}} with respect to E_{∞} − {x_{∞,v}, x_{∞,h}, x_{∞,∞}}. Moreover

\begin{matrix}(E_{∞} − \{x_{∞,v}, x_{∞,h}, x_{∞,∞}\})_{!} (Φ_{∞}^{0}(F̃)|_{E_{∞}−\{x_{∞,v},x_{∞,h},x_{∞,∞}\}}) \\ = (E_{∞} − \{x_{∞,v}, x_{∞,h}, x_{∞,∞}\})_{∗} (Φ_{∞}^{0}(F̃)|_{E_{∞}−\{x_{∞,v},x_{∞,h},x_{∞,∞}\}}) \end{matrix}

The inertia group I_{∞} acts on this sheaf via a tame character whose order is equal to the order of ε̃· ω̃.

Deduction of Theorem 2.10 from Theorem 2.12. Since the restriction of F̃ to X_{∞} is zero, Gη̄ is isomorphic to H^{1}(X\overline{∞}, RΦ_{∞}(F̃)) as representations of the decomposition group D_{∞}, where X\overline{∞} is the geometric fiber of X over the geometric point \overline{∞} above ∞ ∈ ℙ^{1}. From the statements (i), (ii) of Theorem 2.12, we see that H^{1}(X\overline{∞}, RΦ_{∞}(F̃)) is isomorphic to the direct sum of

H_{c}^{1}(Z_{v} × Spec 𝔽^{sep} − {x_{0,v}, x_{∞,v}}, Φ_{∞}^{0}(F̃)),

H_{c}^{1}(E_{0} × Spec 𝔽^{sep} − {x_{0,v}, x_{0,h}, x_{0,0}}, Φ_{∞}^{0}(F̃)), and

H_{c}^{1}(E_{∞} × Spec 𝔽^{sep} − {x_{∞,v}, x_{∞,h}, x_{∞,∞}}, Φ_{∞}^{0}(F̃)).

From the Euler-Poincaré characteristic formula we see that first of the three cohomology groups is zero according to 2.10 (iii). Again by the Euler-Poincaré formula, the latter two cohomology groups are both one dimensional, and are pure of weight one as representations of the inertia group D_{∞} by [3, 3.2.3].

(2.13) Lemma Let j : 𝔾_{m} ×_{Spec 𝔽} 𝔾_{m} ↪ 𝔸^{1} ×_{Spec 𝔽} 𝔸^{1} be the natural open immersion. Let χ_{1}, χ_{2} : 𝔾_{m}(𝔽) → \overline{ℚ_{ℓ}}^{×} be characters of 𝔾_{m}(𝔽) = 𝔽^{×}. We assume that χ_{1} is non-trivial. Let (x, y) be the standard coordinates on 𝔸^{1}×𝔸^{1} and on 𝔾_{m}×𝔾_{m}. Denote by L_{χ_{1}(x)χ_{2}(y)} the smooth rank-one

\overline{ℚ_{ℓ}}-sheaf pr_{1}^{∗}L_{χ_{1}}⊗ pr_{2}^{∗}L_{χ_{2}} on 𝔾_{m} ×_{Spec 𝔽} 𝔾_{m}. Let RΦ (j_{!}L_{χ_{1}(x)χ_{2}(y)}) be the complex of vanishing

cycle for j_{!} (L_{χ_{1}(x)χ_{2}(y)}) over 𝔸^{1} × {0} with respect to the second projection pr_{2} : 𝔸^{1} × 𝔸^{1} → 𝔸^{1}.

Then the stalk RΦ (j_{!}L_{χ_{1}(x)χ_{2}(y)})_{(0,0)} of the complex of vanishing cycles at (0, 0) is acyclic.

Proof. According to the definition of vanishing cycles we have a natural isomorphism

RΦ (j_{!}L_{χ_{1}(x)χ_{2}(y)}) = (L_{χ_{2}} |\overline{η}) ⊗ RΦ (j_{!}L_{χ_{1}(x)}) ,

where L_{χ_{2}} |\overline{η} is the stalk of the sheaf L_{χ_{2}} over a geometric generic point \overline{η} of 𝔾_{m}, regarded as a representation of the decomposition group D_{0} over the point 0 of the base 𝔸^{1}. Therefore we may and do assume that χ_{2} is trivial.

Let k : 𝔾_{m} ×_{Spec 𝔽} 𝔸^{1} ↪ 𝔸^{1} ×_{Spec 𝔽} 𝔸^{1} be the standard natural inclusion. Denote by pr_{1}^{∗}L_{χ_{1}} the smooth rank-one \overline{ℚ_{ℓ}}-sheaf on 𝔾_{m} ×_{Spec 𝔽} 𝔸^{1} obtained by pulling back the sheaf L_{χ_{1}} on 𝔾_{m}_{/𝔽} via the first projection. We have a short exact sequence

0 → j_{!} (pr_{1}^{∗}L_{χ_{1}} ) → k_{!} (pr_{1}^{∗}L_{χ_{1}} ) → k_{!} (pr_{1}^{∗}L_{χ_{1}} ) |_{𝔸^{1}×{0}} → 0 .

By SGA 7 II, exposé XIII, Lemma 2.1.11,

RΦ (k_{!} (pr_{1}^{∗}L_{χ_{1}} )) = 0 .

It follows that

RΦ (j_{!}L_{χ_{1}(x)χ_{2}(y)})_{(0,0)} = k_{!} (pr_{1}^{∗}L_{χ_{1}} ) |_{(0,0)}[−1] = 0 .

(2.14) Proposition Let j : 𝔾_{m} ×_{Spec 𝔽} 𝔾_{m} ↪ 𝔸_{/𝔽}^{2} be the natural inclusion. Let f : 𝔸^{2} = Spec 𝔽[x, y] → 𝔸^{1} = Spec 𝔽[t] be the morphism over 𝔽 given by the 𝔽-algebra homomorphism which sends t to xy. Let

χ_{1}, χ_{2} : 𝔾_{m}(𝔽) = 𝔽^{×} → \overline{ℚ_{ℓ}}^{×}

be two characters of 𝔽^{×}, χ_{1} ≠ χ_{2}. Let L_{χ_{1}(x)·χ_{2}(y)} = pr_{1}^{∗}L_{χ_{1}} ⊗ pr_{2}^{∗}L_{χ_{2}} , where L_{χ_{i}} is the smooth rank-one \overline{ℚ_{ℓ}}-sheaf on 𝔾_{m} given by pushing out the Lang torsor by χ_{i}, i = 1, 2. Then the stalk RΦ(j_{!}L_{χ_{1}(x)·χ_{2}(y)})_{(0,0)} at (0, 0) of the complex RΦ(j_{!}L_{χ_{1}(x)·χ_{2}(y)}) of vanishing cycles with respect to f is an acyclic complex.

Proof. After tensoring with f^{∗}L_{χ_{2}^{−1}}, we may and do assume that χ_{2} is trivial, therefore χ_{1} is non-trivial.

We partially compactify the morphism f involved as follows. Let U = 𝔾_{m} ×_{Spec 𝔽}𝔾_{m}, contained in ℙ^{1} ×_{Spec 𝔽} 𝔸^{1}. Let (x, s) be the standard coordinate functions for 𝔾_{m} × 𝔾_{m}, and let y = \frac{s}{x}. Let X be the blowing-up of ℙ^{1} ×_{Spec 𝔽} 𝔸^{1} at (0, 0). Let \overline{f} : X → 𝔸^{1} be the composition of the blowing-up morphism X → ℙ^{1} × 𝔸^{1} and the second projection ℙ^{1} × 𝔸^{1} → 𝔸^{1}. This is compatible with the geometric set-up in the statement of Theorem 2.14, since X contains Spec𝔽[x, y] as an open subscheme. The fiber \overline{f}^{−1}(0) is the union of the strict transform D̃ of ℙ^{1} × \{0\} ⊂ ℙ^{1} × 𝔸^{1} and the exceptional divisor E of \overline{f}. Let x̃_{0} be the intersection of D̃ with E. Let x_{∞} ∈ D̃ be the intersection of D̃ with the strict transform of {∞} × 𝔸^{1}. Let x′_{∞} ∈ E be the point on E which lies outside the affine open subscheme Spec 𝔽[x, y] of X. Let k : 𝔾_{m} × 𝔾_{m} ↪ X be the inclusion map. Over 𝔾_{m} × 𝔾_{m} = Spec [x, x^{−1}, s, s^{−1}] = Spec [x, x^{−1}, y, y^{−1}] we have the smooth rank-one \overline{ℚ_{ℓ}} sheaf L_{χ_{1}(x)}. Let G = k_{!}L_{χ_{1}(x)}. Let D′ = D − {x̃_{0}, x_{∞}}, and let E′ = E − {x̃_{0}, x′_{∞}}. We compute the restriction to D′ ⊔ E′ of the complex of vanishing cycles for G.

The open subscheme 𝔾_{m} × 𝔸^{1} of ℙ^{1} × 𝔸^{1} contains D′. Over this open subscheme the sheaf L_{χ_{1}(x)} on Spec 𝔽[x, x^{−1}, y, y^{−1}] extends to a smooth rank-one smooth sheaf, the pull-back of the sheaf L_{χ_{1}} on 𝔾_{m} to 𝔾_{m} × 𝔸^{1} via the second projection 𝔾_{m} × 𝔸^{1} → 𝔾_{m}. From this we deduce that the restriction of RΦ(G) to D′ = Spec[x, x^{−1}] ≅ 𝔾_{m} is isomorphic to L_{χ_{1}} [1], with trivial action by the decomposition group.

On the other hand, E′ is contained in the open subscheme Spec 𝔽[s, y, y^{−1}]. Over the open subscheme Spec 𝔽[s, s^{−1}, y, y^{−1}] ⊂ Spec 𝔽[s, y, y^{−1}], we have L_{χ_{1}(x)} = L_{χ_{1}(s)χ_{1}^{−1}(y)}. From the definition of vanishing cycles we get

RΦ(G)|_{E′=Spec 𝔽[y,y^{−1}]} = (L_{χ_{1}} )\overline{η} ⊗ L_{χ_{1}^{−1}(y)}[1] ,

where the second factor L_{χ_{1}^{−1}} is the smooth rank-one \overline{ℚ_{ℓ}}-sheaf on E′ = Spec 𝔽[y, y^{−1}] = 𝔾_{m} attached to the character χ−1_{1}, while the first factor (L_{χ_{1}} )\overline{η} is the geometric generic fiber of the smooth rank-one \overline{ℚ_{ℓ}}-sheaf L_{χ_{1}} regarded as a linear representation of the decomposition group D_{0} at 0.

From Grothendieck’s Euler-Poincaré characteristic formula and the computation above, we deduce that

ℍ_{c}^{i}(D′, RΦ(G)|_{D′}) = 0 ∀i ∈ ℤ

and

ℍ_{c}^{i}(E′, RΦ(G)|_{E′}) = 0 ∀i ∈ ℤ .

Similarly from the Euler-Poincaré formula we have

H_{c}^{i}(𝔾_{m}, L_{χ_{1}} ) = 0 ∀i ∈ ℤ .

We have a short exact sequence

0 → (D′ ⊔ E′ ↪ D ∪ E)_{!} (RΦ(G)|_{D′⊔E′}) → RΦ(G)
→ RΦ(G)|x̃_{0} ⊕ RΦ(G)|x̃_{∞} ⊕ RΦ(G)|_{x′_{∞}} → 0

Let \overline{η} be a geometric generic fiber of the base scheme 𝔸^{1}. The theory of vanishing cycles gives

0 = H_{c}^{i}(𝔾_{m}\overline{η}, L_{χ_{1}} ) = H^{i}(ℙ1 ; \overline{η}, G\overline{η}) = ℍ_{c}^{i}(D ∪ E, RΦ(G)) ∀i .

From the long exact sequence we deduce that the stalk at x̃_{0} RΦ(G)|x̃_{0} of the complex of vanishing cycles RΦ(G) is acyclic. This proves Proposition 2.14 since RΦ(G)|x̃_{0} is isomorphic to RΦ(j_{!}L_{χ_{1}(x)})|_{(0,0)}.

Proof of Theorem 2.12. The proof is a little tedious, consisting of computation of the restriction of the complex of vanishing cycles RΦ_{∞}(F̃) to the disjoint open subsets E′_{0} := E_{0} −{x_{00}, x_{0,h}, x_{0,v}}, Z′_{v} = Z_{v} − {x_{0,v}, x_{∞,v}}, E′_{∞} = E_{v} − {x_{∞,∞}, x_{∞,h}, x_{∞,v}} of X_{∞}, and the stalks of the complex of vanishing cycles at the six points x_{00}, x_{0,h}, x_{0,v}, x_{∞,∞}, x_{∞,h}, x_{∞,v}.

Notice that the automorphism (u, t) ↦ (u^{−1}, t) of U × 𝔾_{m} sends L_{ε(Tr +t)} ⊗ L_{ω(t)} to L_{ε(Tr +t)} ⊗ L_{ω^{−1}(t)}, and interchanges y_{0} and y_{∞}. Therefore it suffices to do the calculation for E′_{0}, Z′_{v}, and x_{00}, x_{0,h}, x_{0,v}.

(a) RΦ_{∞}(F̃)|_{E′_{0}}.

Let s = \frac{1}{t}, a coordinate for ℙ^{1} at ∞ ∈ ℙ^{1}. Let u = \frac{z}{s}. The affine open subscheme Spec 𝔽′[u, u^{−1}, \frac{1}{s^{2}u^{2}+1+u}, s] ⊂ X contains E′_{0}. On Spec 𝔽′[u, u^{−1}, \frac{1}{s^{2}u^{2}+1+u}, s] ⊂ X, the sheaf F̃ is equal to

L_{ε̃(s^{2}u^{2}+1+u)} ⊗ L_{(ω̃·ε̃^{−1})(u)} ⊗ L_{(ω̃·ε̃^{−1})(s)} .

Since L_{ε̃(s^{2}u^{2}+1+u)} ⊗ L_{(ω̃·ε̃^{−1})(u)} is smooth on Spec 𝔽′[u, \frac{1}{s^{2}u^{2}+1+u}, s], we get

RΦ_{∞}(F̃)|_{E′_{0}} = L_{ε̃(1+u)} ⊗ L_{(ω̃·ε̃^{−1})(u)} ⊗ RΦ_{t=∞}(j̃_{!}pr_{2}^{∗}L_{(ω̃·ε̃−1)(s)})|_{E′_{0}} .

By the definition of the vanishing cycles,

RΦ_{t=∞}(j̃_{!}pr_{2}^{∗}L_{(ω̃·ε̃^{−1})(s)})|_{E′_{0}} ≅ RΨ_{t=∞}(j̃_{!}pr_{2}^{∗}L_{(ω̃·ε̃^{−1})(s)})|_{E′_{0}}

is equal to pr_{2}^{∗}RΨ_{t=∞}((𝔾_{m} ↪ ℙ^{1})_{!}(L_{(ω̃^{−1}ε̃)(t)}))|_{E′_{0}}. By definition, RΦ_{t=∞}(j̃_{!}L_{(ω̃^{−1}ε̃)(t)}) is represented by the D_{∞}-module L_{(ω̃^{−1}ε̃)(t)}|\overline{η_{∞}}, “concentrated at degree zero”. In other words, RΦ_{∞}(F̃)|_{E′_{0}} “is” the smooth sheaf L_{ε̃(1+u)} ⊗ L_{(ω̃·ε̃^{−1})(u)}, with D_{∞} action via the character for the D_{∞}-module L_{(ω̃^{−1}ε̃)(t)}|\overline{η_{∞}}.

(b) RΦ_{∞}(F̃)|_{Z′_{v}}.

Near Z′_{v}, we have coordinates z, s, with s = \frac{1}{t}: Z′_{v} ⊂ Spec 𝔽′[z, z^{−1}, s] ⊂ X. Over this open subscheme Spec 𝔽′[z, z^{−1}, s, s^{−1}, (sz^{2} + s + z)^{−1}], F̃ is equal to

L_{ε̃(sz^{2}+s+z)} ⊗ L_{(ω̃·ε̃^{−1})(z)} ⊗ L_{ε̃^{−1}(z)} .

Hence

RΦ_{∞}(F̃)|_{Z′_{v}} = L_{ω̃(z)} ⊗ RΦ_{∞}(j̃_{!}pr_{2}^{∗}L_{ε^{−1}(s)})|_{Z′_{v}} .

In other words, RΦ_{∞}(F̃)|_{Z′_{v}} “is” equal to the smooth sheaf L_{ω̃(z)} on Z′_{v}, with the decomposition group D_{∞} operating via the character for the D_{∞}-module L_{ε(t)}|_{\overline{η}_{∞}}.

(c) RΦ_{∞}(F̃)_{x_{0,0}} .

For u = \frac{z}{s}, the affine scheme V_{0,0} = Spec 𝔽′[u, s, \frac{1}{s^{2}u^{2}+1+u}] is an open neighborhood of x_{0,0}. Let U_{0,0} = Spec 𝔽′[u, s, \frac{1}{(s^{2}u^{2}+1+u)su}], and let j_{0,0} : U_{0,0} ↪ V_{0,0} be the natural inclusion. Over V_{0,0} we have

F̃ = j_{0,0}_{!} (L_{ε̃(s^{2}u^{2}+1+u)} ⊗ L_{(ω̃·ε̃^{−1})(s)} ⊗ L_{(ω̃·ε̃)^{−1}(u)}) .

Hence

RΦ_{∞}(F̃)_{x_{0,0}} = L_{ε̃(1+u)}|_{u=0} ⊗ L_{(ω̃^{−1}·ε̃)(t)}|\overline{η_{∞}} ⊗ RΦ_{∞} (j_{0,0}_{!}(L_{(ω̃·ε̃^{−1})(u)}))_{x_{0,0}}

which is acyclic by Lemma. 2.13, since ω̃ · ε̃^{−1} is non-trivial.

(d) RΦ_{∞}(F̃)_{x_{0,h}}.

Let V_{0,h} = Spec 𝔽′[u, s, \frac{1}{u}], an affine open subscheme which contains x_{0,h}, and let U_{0,h} = Spec 𝔽′[u, s, \frac{1}{su(s^{2}u^{2}+1+u)}]. Let j_{0,h} : U_{0,h} ↪ V_{0,h} be the open immersion. The elements (s, s^{2}u^{2} + 1 + u) form a regular system of parameters at x_{0,h}. Over V_{0,h} we have

F̃ = j_{0,h}_{!}(L_{(ω̃·ε̃^{−1})(u)} ⊗ L_{ε̃(s^{2}u^{2}+1+u)} ⊗ L_{(ω̃·ε̃^{−1})(s)}) .

So

RΦ_{∞}(F̃)_{x_{0,h}} = L_{(ω̃·ε̃^{−1})(u)}|_{u=−1} ⊗ L_{(ω̃^{−1}·ε̃)(t)}|\overline{η_{∞}} ⊗ RΦ_{∞} (j_{0,h}_{!}(L_{ε̃(s^{2}u^{2}+1+u)}))_{x_{0,h}} ,

which is acyclic by Lemma 2.13, since ω̃^{−1} · ε̃ is non-trivial.

(e) RΦ_{∞}(F̃)_{x_{0,v}}.

Let v = \frac{s}{z}. The scheme Spec 𝔽′[s, v, (z^{2}v + v + 1)^{−1}] is an open neighborhood of the point x_{0,v}, given by s = v = 0. Over Spec 𝔽′[s, v, (z^{2}v + v + 1)^{−1}] we have

L_{ε̃(z+z^{−1}+s^{−1})} ⊗ L_{ω̃(z)} = L_{ε̃(z^{2}v+v+1)} ⊗ L_{(ε̃^{−1}·ω̃)(z)} ⊗ L_{ε̃^{−1}(v)}

By Prop. 2.14 we conclude that RΦ(F̃)_{x_{0,v}} is acyclic, because ω̃ is non-trivial.

§3. A family of character sums of the first type

In this section we study the family of character sums denoted by λ_{a,χ} in §1.

(3.1) The geometric set-up

Let 𝔽 be a finite field with q elements, where q is a power of an odd prime number p. Let B be the base curve ℙ^{1} with rational function field 𝔽(t). Let 𝔛 be the surface contained in B × ℙ^{2} defined by the equation

SY^{2} = S(X − Z)^{2} + 4T XZ .

Here S, T are the homogeneous coordinates of B, while X, Y, Z are the homogeneous coordinates of ℙ^{2}; let t, x, y denote the rational functions \frac{T}{S} , \frac{X}{Z} , \frac{Y}{Z} respectively. Let f : 𝔛 → B be the natural projection. Let η be the generic point of B.

It is easy to see that 𝔛 is a proper smooth surface over 𝔽. Moreover, f : 𝔛 → B is smooth over V = B − {b_{0}, b_{1}, b_{∞}}, where b_{0}, b_{1}, b_{∞} are the points of B where the value of t is equal to 0, 1, ∞ respectively. The three degenerate fibers will be denoted by 𝔛_{0}, 𝔛_{1} and 𝔛_{∞} respectively; one checks easily that they are all reduced with normal crossings. Moreover, each of the three singular fibers is a union of two copies of ℙ^{1} intersecting transversally at exactly one 𝔽-rational point. The singular point s_{0} (resp. s_{1}, resp. s_{∞}) of f is given by equations t = 0, x = 1, y = 0 (resp. t = 1, x = −1, y = 0, resp. \frac{S}{T} = 0, \frac{X}{Y} = \frac{Z}{Y} = 0).

Let D_{0} (resp. D_{∞}) be the divisor of 𝔛 given by X = 0 (resp. Z = 0). Let U be the complement in 𝔛 of the union of D_{0} and D_{∞}. One checks that D_{0} is the union of a divisor D_{0}^{h} finite étale over B with degree two and the component E_{X,S} of 𝔛_{∞} where X vanishes. Similarly D_{∞} is the union of a divisor D_{∞}^{h} finite étale over B with degree two and the component E_{Z,S} of 𝔛_{∞} where Z vanishes. Below we give an explicit description of the divisors involved.

• D_{0}^{h} is defined by the equations {X = 0, Y^{2} − Z^{2} = 0}. It is the disjoint union of two divisors D_{Y +Z,X}^{h} and D_{Y −Z,X}^{h}, where the subscripts are the defining equations for the two divisors, each isomorphic to the base curve B under f.

• E_{X,S} is defined by the equations {X = 0, S = 0}.

• D_{∞}^{h} is defined by the equations {Z = 0, Y^{2} − X^{2} = 0}. It is the disjoint union of two divisors D_{X−Y,Z}^{h} and D_{X+Y,Z}^{h}, indexed by their defining equations, and each is isomorphic to B.

• E_{Z,S} is defined by the equations {Z = 0, S = 0}.

• 𝔛_{0} is defined by the equations {T = 0, Y^{2} − (X − Z)^{2} = 0}. It is the union of two divisors D_{Y −X+Z,T}^{v} and D_{Y +X−Z,T}^{v} , indexed by their defining equations.

• 𝔛_{1} is defined by the equations {S − T = 0, Y^{2} − (X + Z)^{2} = 0}. It is the union of two divisors D_{Y +X+Z,S−T}^{v} and D_{Y −X−Z,S−T}^{v} , indexed by their defining equations.

• 𝔛_{∞} is defined by the equations {S = 0, XZ = 0}; it is the union of E_{X,S} and E_{Z,S}.

The union of the horizontal divisors meet each of the three singular fibers 𝔛_{0}, 𝔛_{1}, 𝔛_{∞} transversally at four 𝔽-rational points. For instance the horizontal divisor D_{∞}^{h} meets E_{Z,S} transversally at two points x_{∞,∞}, x′_{∞,∞}, with projective coordinates ([X : Y : Z :] = [1 : 1 : 0], [S : T] = [0 : 1]) and ([X : Y : Z :] = [1 : −1 : 0], [S : T] = [0 : 1]) respectively; the horizontal divisor D_{0}^{h} meets E_{X,S} transversally at two points x_{0,0}, x′_{0,0}, with projective coordinates ([X : Y : Z] = [0 : 1 : 1], [S : T] = [0 : 1]) and ([X : Y : Z] = [0 : 1 : −1], [S : T] = [0 : 1]) respectively. Denote these four points on 𝔛_{∞} by s_{∞,i}, i = 1, 2, 3, 4. Similarly we denote by s_{0,i}, i = 1, …, 4 (resp. s_{1,i}, i = 1, 2, 3, 4) the four intersection points of 𝔛_{0} (resp. 𝔛_{0}) with the horizontal divisors; the actual numbering will be unimportant for us.

The rational function x = \frac{X}{S} on 𝔛 defines a morphism from U to 𝔾_{m}; denote by ℱ(χ) the pull-back x^{∗}ℒ_{χ} of the rank-one smooth sheaf ℒ_{χ} on 𝔾_{m} defined by the character χ of 𝔽^{×}. So ℱ(χ) is a smooth rank-one \overline{ℚ_{ℓ}}-sheaf on U. We assume that χ is non-trivial. Let j : U → 𝔛 be the inclusion map. Let \overline{F(χ)} = j_{!}F(χ). The complement of U in 𝔛 can be regarded as the ramification divisor of the \overline{ℚ_{ℓ}}-sheaf \overline{F(χ)}; it is the union of the horizontal divisors and 𝔛_{∞}.

(3.2) Proposition Assume that χ is a non-trivial character of 𝔽^{×}. Then

R^{i}(f|_{U})_{!} \overline{F(χ)} = 0 if i ≠ 1 .

The \overline{ℚ_{ℓ}}-sheaf G(χ) := R^{1}(f|_{U})_{!} \overline{F(χ)} on B − \{b_{0}, b_{1}, b_{∞}\} is smooth of rank two on V = B − {b_{0}, b_{1}, b_{∞}}, and is pure of weight one.

Proof. For every geometric point b̄ ∈ V, the restriction of the sheaf \overline{F(χ)} to the geometric fiber 𝔛_{b} is smooth outside the four intersection points with the horizontal divisors, and is tamely ramified at these four intersection points since χ is not trivial. This implies the vanishing statement, and the statement about the rank follows from the Euler-Poincaré characteristic formula. The purity of the sheaf G(χ) := R^{1}(f|_{U})_{!} \overline{F(χ)} follows from [3, Thm. 3.2.3]. The smoothness of G(χ) := R^{1}(f|_{U})_{!} \overline{F(χ)} follows from Lemma 2.13; it is also a consequence of the local acyclicity of the vanishing cycle complex for smooth morphisms, since F(χ) is tamely ramified along the horizontal divisors, which are smooth over B.

(3.3) Lemma The sheaf G(χ) := R^{1}(f|_{U})_{!} \overline{F(χ)} is canonically isomorphic to G(χ^{−1}), and the coefficients of the characteristic polynomial of the Frobenii on G(χ) are totally real algebraic numbers. If χ is not trivial, then det(G(χ)) is equal to \overline{ℚ_{ℓ}}(−1).

Proof. We have a B-involution ι of 𝔛, which interchanges the projective coordinates X, Z and leaves Y, S, T fixed. Moreover ι_{∗}(F(χ)) = F(χ^{−1}). When χ is non-trivial, the cup product gives a perfect pairing

R^{1}(f|_{U})_{!} \overline{F(χ)} × R^{1}(f|_{U})_{!} \overline{F(χ^{−1})} → R^{2}(f|_{U})_{!} \overline{ℚ_{ℓ}} ⟶^{∼} \overline{ℚ_{ℓ}}(−1)

by [4, dualité, Thm. 1.3, 2.1], and this pairing factors through det(G).

(3.4) Proposition Assume that χ is a non-trivial character of 𝔽^{×}. Denote by RΦ_{b_{0}}(\overline{F(χ)}) (resp. RΦ_{b_{1}}(\overline{F(χ)}) the complex of vanishing cycles for the sheaf \overline{F(χ)} with respect to the map f : 𝔛 → B over the point b_{0} ∈ B (resp. b_{1} ∈ B.)

(i) The stalks of the vanishing cycle complex at the intersection points with the horizontal divisors are acyclic:

\begin{matrix}Φ_{b_{0}}^{j}(\overline{F(χ)})_{s_{0,i}} = 0 ∀j ≥ 0, i = 1, 2, 3, 4, \\ Φ_{b_{1}}^{j}(\overline{F(χ)})_{s_{1,i}} = 0 ∀j ≥ 0, i = 1, 2, 3, 4. \end{matrix}

(ii) At the singular points s_{0}, s_{1} of 𝔛_{0}, 𝔛_{1}, we have for i = 0, 1

dim(Φ_{b_{i}}^{j}(\overline{F(χ)})_{s_{i}}) = \{ \begin{matrix}0 & if & j ≠ 1 \\ 1 & if & j = 1 \end{matrix}

(iii) The action of the decomposition group D_{b_{i}} on the one-dimensional space Φ_{b_{i}}^{1}(\overline{F(χ)})_{s_{i}} is unramified. The geometric Frobenius Fr_{b_{0}} operates as q on Φ_{b_{0}}^{1}(\overline{F(χ)})_{s_{0}}, while the geometric Frobenius Fr_{b_{1}} operates as χ(−1)q on Φ_{b_{1}}^{1}(\overline{F(χ)})_{s_{1}}.

Proof. The statement (i) follows from Lemma 2.13. The statements (ii) and (iii) are consequences of [3], (3.1.3) case (a), since the restriction of F(χ) to the 𝔽-rational point b_{0} (resp. b_{1}) is a rank-one \overline{ℚ_{ℓ}}-sheaf over Spec 𝔽 on which F r_{q} operates as χ(1) = 1 (resp. χ(−1).)

(3.5) Corollary Assume that χ is non-trivial. Then the representation of the decomposition group D_{b} on the geometric generic fiber Gη̄ of G is tamely ramified for b = b_{0} and b = b_{1}. Moreover

(1) At b_{0}, the local Galois representation corresponds to the representation sp(2) ⊗ \overline{ℚ_{ℓ}}(−1) of the Deligne-Weil group; i.e. it corresponds to the special representation σ(∥ · ∥^{−1}, 1) of GL(2) under the local Langlands correspondence. The local Artin conductor is equal to 1, and the local L-factor is (1 − q^{−s})^{−1}.

(2) At b_{1}, the local Galois representation corresponds to the representation sp(2)⊗χ_{−1}(−1), where χ_{−1}(−1) denotes the unramified character whose value at the geometric Frobenius is equal to χ(−1)q. Its local L-factor is equal to (1 − χ(−1)q^{−s})^{−1}.

(3.6) Theorem Assume that χ is not the trivial character of 𝔽^{×}. Let L_{χ(t)} (resp. L_{χ^{−1}(t)}) be the rank-one smooth sheaf on Spec 𝔽[t, t^{−1}] = B − {b_{0}, b_{∞}} attached to χ (resp. χ^{−1}), where t is the rational function \frac{T}{S} on B. Let s_{∞,1}, s_{∞,2} be the two intersection points of E_{Z,S} with the horizontal divisors, and let E′_{Z,S} = E_{Z,S} − {s_{∞}, s_{∞,1}, s_{∞,2}}. Similarly let s_{∞,3}, s_{∞,4} be the two intersection points of E_{X,S} with the horizontal divisors, and let E′_{Z,X} = E_{Z,X} − {s_{∞}, s_{∞,3}, s_{∞,4}}.

(i) Write E′_{Z,S} = Spec 𝔽[u, \frac{1}{u^{2}−1}], where u is the restriction to E′_{Z,S} of the rational function \frac{Y}{X} on 𝔛. The restriction of the vanishing cycle complex RΦ_{b_{∞}}(\overline{F(χ)}) to E′_{Z,S} is represented by the smooth rank-one \overline{ℚ_{ℓ}}-sheaf L_{χ^{−1}(4^{−1}(u^{2}−1))} in degree zero, and the decomposition group D_{b_{∞}} acts on this rank-one sheaf via its natural action on (L_{χ(t)})η̄.

(ii) Write E′_{X,S} = Spec 𝔽[y, \frac{1}{y^{2}−1}], where u is the restriction to E′_{X,S} of the rational function \frac{Y}{T} on 𝔛. The restriction of the vanishing cycle complex RΦ_{b_{∞}}(\overline{F(χ)}) to E′_{X,S} is represented by the smooth rank-one \overline{ℚ_{ℓ}}-sheaf L_{χ(4^{−1}(y^{2}−1))} in degree zero, and the decomposition group D_{b_{∞}} acts on this rank-one sheaf via its natural action on (L_{χ^{−1}(t)})η̄.

(iii) The stalk of the vanishing cycle complex RΦ_{b_{∞}}(\overline{F(χ)}) at s_{∞,i} is acyclic for i = 1, 2, 3, 4.

(iv) If χ^{2} is non trivial, then the stalk of the vanishing cycle complex RΦ_{b_{∞}}(\overline{F(χ)}) at s_{∞} is acyclic.

(v) If χ^{2} is trivial, then

dim(Φ_{b_{∞}}^{j}(\overline{F(χ)})_{s_{∞}}) = \{ \begin{matrix}0 & if & j ≠ 0, 1 \\ 1 & if & j = 0, 1 . \end{matrix}

The decomposition group D_{b_{∞}} operates on the one-dimensional \overline{ℚ_{ℓ}}-vector space

Φ_{b_{∞}}^{0}(\overline{F(χ)})_{s_{∞}} (resp. Φ_{b_{∞}}^{1}(\overline{F(χ)})_{s_{∞}} )

via its natural action on (L_{χ(t)}(−1))η̄ (resp. (L_{χ(t)})η̄.)

Proof. Since the proof of (i) and (ii) are essentially the same, we only give the proof of (i). Let u = \frac{Y}{X}, v = \frac{Z}{X} and s = \frac{S}{T}. Then near E′_{Z,S} the surface 𝔛 is defined by the equation

s (u^{2} − (1 − v)^{2}) = 4v .

We have

L_{χ(x)} = L_{χ(v^{−1})} = L_{χ(s^{−1})} ⊗ L_{χ^{−1}(\frac{u^{2}−(1−v)^{2}}{4} )} ,

and the sheaf L_{χ^{−1}(\frac{u^{2}−(1−v)^{2}}{4} )} is smooth at points of E′_{Z,S}. This proves (i).

The statement (iii) follows from Lemma 2.13. For (iv) and (v), let u_{1} = \frac{X}{Y}, v_{1} = \frac{Z}{Y}, s = \frac{S}{T}; u_{1}, v_{1} is a regular system of parameters for 𝒪_{𝔛,s_{∞}}. The equation for 𝔛 near s_{∞} is

s (1 − (u_{1} − v_{1})^{2}) = 4u_{1}v_{1} .

We have

L_{χ(x)} = L_{χ(u_{1})χ^{−1}(v_{1})} = L_{χ(s)} ⊗ L_{χ^{−2}(v_{1})} ⊗ L_{χ(\frac{1−(u_{1} −v_{1} )^{2}}{4} )} ,

and L_{χ(\frac{1−(u_{1} −v_{1} )^{2}}{4} )} is smooth at s_{∞}. The statement (v) follows directly from the above by the

definition of vanishing cycles, while we deduce (iv) from Proposition 2.14.

(3.6.1) Remarks It may be instructive to explain the calculation of vanishing cycles at s_{∞} in an analogous complex analytic situation.

(1) When the bad point c is the double point s_{∞}, the variety of vanishing cycles W is

W = \{(τ, u, v) ∈ ℍ × Δ^{∗} × Δ^{∗}| exp(2π\sqrt{−1}τ) = uv\} ,

where ℍ is the upper half-plane and Δ^{∗} is the punctured unit disk. We have a map f : W → Δ^{∗} which sends (τ, u, v) to \frac{u}{v}. Let 𝕃 be a non-trivial rank-one local system on Δ^{∗}. Then all cohomologies H^{i}(W, f^{∗}𝕃) vanish, for all i ≥ 0 if 𝕃^{⊗2} is non-trivial. When 𝕃^{⊗2} is trivial, both H^{0}(W, f^{∗}𝕃) and H^{1}(W, f^{∗}𝕃) are one-dimensional; all other cohomologies vanish. By homotopy invariance, the above statements quickly reduce to the fact that all cohomologies of a non-trivial rank-one local system on Δ^{∗} vanish, while the zeroth and the first cohomology of the trivial rank-one local system on Δ^{∗} are one-dimensional.

(2) When the bad point c is one of the four intersection points of the horizontal ramification divisors with 𝔛_{∞}, the situation is even easier. The variety of vanishing cycles is W = ℍ × Δ, where Δ is the unit disk. We have a map g : ℍ × Δ^{∗} → Δ^{∗}, which sends (τ, u) to exp(2π\sqrt{−1}τ)u. Let 𝕃 again be a non-trivial rank-one local system on Δ^{∗}. Let j : ℍ×Δ^{∗} → W be the inclusion. Then H^{i}(W, j_{!}g^{∗}𝕃) = 0 for all i ≥ 0. Homotopy invariance quickly reduces this to the fact that all cohomologies H^{i}(Δ, (Δ^{∗} ↪ Δ)_{!}𝕃) vanish.

(3.7) Proposition Let χ be a nontrivial character of 𝔽^{×}.

(1) The smooth rank-two sheaf G = R^{1}f_{∗}\overline{F(χ)} over V is tamely ramified at b_{∞} with local conductor equal to 2, and the local L-factor is trivial.

(2) The geometric monodromy group for G is equal to SL(2).

(3) When χ^{2} ≠ 1, the local Galois representation at b_{∞} is a direct sum of two one-dimensional characters λ_{1}, λ_{2} of weight one, and λ_{1}λ_{2} = ∥ · ∥^{−1}. Both λ_{1} and λ_{2} are tamely ramified, and the restriction of each λ_{i} to the tame ramification group is equal to χ^{±1} if we identify 𝔽^{×} as the canonical quotient of the tame ramification group of 𝔾_{m} at 0 or ∞ via the Lang torsor.

(4) When χ^{2} = 1, the local Galois representation at b_{∞} corresponds to the special representation σ(λ∥·∥^{−1}, λ) of GL(2) under the local Langlands correspondence, where λ is a tamely ramified character with λ^{2} = 1 whose restriction to the tame ramification group is the unique character of order two.

Proof. One deduces statements (1), (3), (4) from Theorem 3.6, using the natural isomorphism

Gη̄ = H^{1}(𝔛_{∞} ×_{Spec 𝔽} Spec \overline{𝔽}, RΦ_{b_{∞}}(\overline{F(χ)}))

of D_{b_{∞}}-modules. For instance if χ^{2} is not trivial, then RΦ_{b_{∞}}(\overline{F(χ)} is represented by the smooth rank-one sheaf on E′_{Z,S} ⊔ E′_{X,S} described in (i), (ii) of Theorem 3.6, extended by zero to 𝔛_{∞}. According to Thm. 3.6 (i), the restriction of RΦ_{b_{∞}}(\overline{F(χ)} to E′_{Z,S} is isomorphic to L_{χ^{−1}( \frac{u^{2}−1}{4} )},

which is ramified at u = 1, −1, ∞; similarly for the restriction to E′_{X,S}. Thus both H_{c}^{1}(E′_{Z,S} ×_{Spec 𝔽} Spec \overline{𝔽}, RΦ_{b_{∞}}(\overline{F(χ)})) and H_{c}^{1}(E′_{X,S}×_{Spec 𝔽}Spec \overline{𝔽}, RΦ_{b_{∞}}(\overline{F(χ)})) are one-dimensional, pure of weight one, with tame action by the decomposition group D_{∞} as described in (i), (ii) of Thm. 3.6. This proves (3) and the statement (1) when χ^{2} is not trivial; the proof of (4) and the rest of (1) is similar.

Since the determinant of G comes from the sheaf \overline{ℚ_{ℓ}}(−1) on Spec 𝔽, its geometric monodromy group is contained in SL(2). On the other hand we see from Prop. 3.4 that the geometric monodromy group of G is not finite, so this semisimple group must be equal to SL(2). This proves (2).

§4. Automorphic forms and applications to Terras graphs

We shall first reformulate the results in the previous two sections in terms of automorphic forms for GL(2), and then show their connections to automorphic forms on quaternion group D and eigenvalues of Terras graphs. Let K be a function field with the field of constants 𝔽. For convenience, the residue field of K at a place v will be denoted 𝔽_{v}, which has cardinality Nv.

(4.1) Let C be the underlying curve of K. Given a rational function a in K, it defines a morphism from C to ℙ^{1}. Denote by 𝔛_{a} the pull-back surface in C × ℙ^{2} of the surface 𝔛 in §3 via a and by ℱ_{a}(χ) the pullback sheaf on 𝔛_{a} of the sheaf ℱ(χ) on 𝔛. The sheaf R^{1}(pr_{1})_{!}ℱ_{a}(χ) is smooth of rank two when restricted to a sufficiently small open subscheme U_{χ,a} ⊂ C; it coincides with the pull-back by the morphism U_{χ,a} ↪ C ⟶^{a} ℙ^{1} of the sheaf 𝒢(χ) in the notation of Lemma 3.3. Let ρ_{a,χ} be the degree two representation of Gal(K^{sep}/K) attached to R^{1}(pr_{1})_{!}ℱ_{a}(χ)|_{U_{χ,a}}. By the results of Grothendieck, Deligne, and the converse theorem of GL(2), we know that the L-function attached to ρ_{a,χ} is an automorphic L-function for GL(2) over K. We examine its local factors.

Let g(x) = (x − 1)^{2} + 4ax. Denote by g_{v}(x) the polynomial g(x) (mod v). Our computation in the previous section shows that at a place v which is not a pole of a and where a ≢ 0, 1 (mod v), the local factor of L(s, ρ_{a,χ}) at v is a polynomial of degree 2 in Nv^{−s}, with the coefficient of Nv^{−s} being

−λ_{χ}(𝔽_{v}; a) = ∑_{{x,y∈𝔽v \atop y^{2}=gv(x)}} χ ∘ Nm_{𝔽_{v}/𝔽}(x),

and the coefficient of Nv^{−2s} being Nv by Lemma 3.3. Hence the local factor at v is

1 − λ_{χ}(𝔽_{v}; a)Nv^{−s} + Nv^{1−2s}.

We have shown

(4.2) Theorem Let K be a function field with the field of constants 𝔽. Given a nonzero element a in K and a nontrivial character χ of 𝔽^{×}, there is an automorphic form f_{a,χ} of GL_{2} over K which is an eigenfunction of the Hecke operator T_{v} at all places v of K, which is not a pole of a and where a ≢ 0, 1 (mod v), with eigenvalue λ_{χ}(𝔽_{v}; a) as defined above. In other words,

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

Here and later we use L_{1} ∼ L_{2} to mean that two Euler products L_{1} and L_{2} agree at all but finitely many factors.

At each place v choose a nonsquare δ in 𝔽_{v}. Observe (cf. [5]) that

−λ_{χ}(𝔽_{v}; a) = − ∑_{{x,y∈𝔽v \atop y^{2}=δgv(x)}} χ ∘ Nm_{𝔽_{v}/𝔽}(x)

since

∑_{{x,y∈𝔽v \atop y^{2}=gv(x)}} χ ∘ Nm_{𝔽_{v}/𝔽}(x) + ∑_{{x,y∈𝔽v \atop y^{2}=δgv(x)}} χ ∘ Nm_{𝔽_{v}/𝔽}(x) = 2 ∑_{x∈𝔽_{v}} χ ∘ Nm_{𝔽_{v}/𝔽}(x) = 0.

Note that the character sum is independent of the choice of δ. Therefore λ_{χ}(𝔽_{v}; a) is nothing but the eigenvalue λ_{4a,χ∘Nm_{𝔽v/𝔽}} of a Terras graph with base field 𝔽_{v}. This theorem shows that the eigenvalues of f_{a,χ} with respect to the Hecke operators are eigenvalues of Terras graphs of first type, parallel to Theorem 3.6.1 in Part I. Its existence for the case K = 𝔽(t) was conjectured in [10].

(4.2.1) As a consequence of Proposition 3.7 (2), the geometric monodromy group of the ℓ-adic representation ρ_{a,χ} is SL(2) provided that a is not a constant. In other words, the Sato-Tate conjecture holds for f_{a,χ}.

(4.2.2) Corollary With the same notation as in Theorem 4.2, suppose that a is not a constant. Then the normalized eigenvalues λ_{χ}(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

when Nv tends to infinity.

(4.3) Let D be the algebraic group over K attached to the multiplicative group of the quaternion algebra H over K ramified exactly at 0 and ∞ as in §1. Given an element b ∈ 𝔽 not equal to 0 or 1, we specialize Theorem 4.2 to the case K = 𝔽(t) and a = (t − 1)/(t − b). The bad places for the automorphic form f_{a,χ} are 1, b and ∞, and the local components of the corresponding representation of GL(2) over K are special representations at the places 1 and ∞ with nontrivial local L-factors as shown in Corollary 3.5, and is either a principal series or a special representation at the place b with trivial local L-factor as shown in Proposition 3.7. Moreover, the local component of f_{a,χ} at the place b is a new vector for the congruence subgroup Γ_{0}((t − b)^{2}) with central character ∥ · ∥^{−1}.

Conjugation of f_{a,χ} by (\begin{matrix}t − b & 0 \\ 0 & 1 \end{matrix}) yields another automorphic form which has the same properties

as f_{a,χ} at all places except b, and at b it is invariant by the principal congruence subgroup Γ(t − b). By the global correspondence between automorphic representations of the quaternion group D over K and those of GL(2) over K proved in [6], there is an automorphic form f_{b,χ} on D(A_{K}) right invariant by 𝒦_{b} (as defined in Introduction) which has the same L-function as f_{a,χ}. In particular, f_{b,χ} is an eigenfunction of the Hecke operator T_{0} at the place 0 with eigenvalue λ_{χ}(𝔽; 1/b). We compare the eigenvalue λ_{χ}(𝔽; 1/b) with λ_{4(b−1)/b,χ}.

Let z = −x and note that

δ(x − 1)^{2} + 4δ(b − 1)x/b = δ(x + 1)^{2} − 4δx/b = δ(z − 1)^{2} + 4δz/b,

so we have

λ_{χ}(𝔽; 1/b) = χ(−1)λ_{4(b−1)/b,χ}.

Thus if χ is an even character, then f_{b,χ} is an eigenfunction of T_{0} with eigenvalue λ_{4(b−1)/b,χ}, while if χ is an odd character, then the eigenvalue of T_{0} is opposite to what we are looking for. Therefore, the unramified twist g_{b,χ} of f_{b,χ} defined by

g_{b,χ}(x) = χ(−1)^{deg Nm_{rd}(x)}f_{b,χ}(x)

for x ∈ D(A_{K}) has eigenvalue λ_{4(b−1)/b,χ} at place 0 in both cases. This proves

(4.4) Theorem Let K be the rational function field 𝔽(t). Given a nonzero element b ∈ 𝔽 not equal to 1 and a nontrivial character χ of 𝔽^{×}, there is an automorphic form g_{b,χ} on D(A_{K}), right invariant by 𝒦_{b} as in Introduction, which is an eigenfunction of Hecke operators T_{v} for all places v other than 1, b, ∞, and whose associated L-function is

L(s, g_{b,χ}) = \frac{1}{1 − χ(−1)q^{−s}} · \frac{1}{1 − q^{−s}} ∏_{v≠1,b,∞} \frac{1}{1 − χ(−1)^{deg v}λ_{χ}(𝔽_{v}; \frac{t−1}{t−b})Nv^{−s} + Nv^{1−2s}} .

In particular, g_{b,χ} is an eigenfunction of T_{0} with eigenvalue λ_{4(b−1)/b,χ}.

In the above expression of L-function, the first factor is at place 1, and the second is at place ∞, by Corollary 3.5.

(4.5) Next we reformulate the results from section 2. Recall that 𝔽 has cardinality q. As in the beginning of this section, a rational function a in K defines a morphism from the curve C to the projective line ℙ^{1}, which in turn yields a map from U ×_{Spec 𝔽} C to U ×_{Spec 𝔽} ℙ^{1}. This allows us to pull back the open subset Y of U ×_{Spec 𝔽} 𝔸^{1} to an open subset Y_{a} of U ×_{Spec 𝔽} C. Given a nontrivial character ε of 𝔽^{×} and a regular character ω of the quadratic extension 𝔽′ (which induces a nontrivial character of U(𝔽)), denote by F_{a,ε,ω} the sheaf on Y_{a} which is the pull-back of the sheaf F_{ε,ω} on Y. The restriction of the sheaf R^{1}(pr_{1})_{!}F_{a,ε,ω} to a sufficiently small open subset U_{a,ε,ω} ⊂ ℙ^{1} is smooth of rank two; it is equal to the pull-back of the sheaf G_{ε,ω} in 2.6 by the composition U_{a,ε,ω} ↪ C ⟶^{a} ℙ^{1}. Let ρ_{a,ε,ω} be the representation of Gal(K^{sep}/K) associated to the smooth rank-two sheaf R^{1}(pr_{1})_{!}F_{a,ε,ω}|_{U_{a,ε,ω}} on U_{a,ε,ω}, and write η_{a,ε,ω} for the determinant of ρ_{a,ε,ω}. As before, the global L-function attached to the representation ρ_{a,ε,ω} is an automorphic L-function for GL(2) over K with central character η_{a,ε,ω}. We examine the local factors of the L-function.

At each place v away from the poles of a and where a ≢ ±2 (mod v) , let

−λ_{ε,ω}(𝔽_{v}; a) = ∑_{u∈U(𝔽_{v})}ε ∘ Nm_{𝔽_{v}/𝔽}(Tr_{𝔽′_{v}/𝔽_{v}}(u) + a_{v})ω ∘ Nm_{𝔽′_{v}/𝔽′}(u),

where a_{v} denotes a (mod v) and 𝔽′_{v} = 𝔽_{v} ⊗_{𝔽} 𝔽′. When deg v is odd, 𝔽′_{v} is a quadratic extension of 𝔽_{v} and

−λ_{ε,ω}(𝔽_{v}; a) = ∑_{u∈𝔽′_{v}, Nm_{𝔽′_{v}/𝔽v}(u)=1}ε ∘ Nm_{𝔽_{v}/𝔽}(Tr_{𝔽′_{v}/𝔽_{v}}(u) + a_{v})ω ∘ Nm_{𝔽′_{v}/𝔽′}(u).

When deg v is even, 𝔽_{v} contains 𝔽′ so that 𝔽′_{v} = 𝔽_{v} ×𝔽_{v} and elements in U(𝔽_{v}) are the pairs (x, 1/x) as x runs through all nonzero elements in 𝔽_{v}. In this case

−λ_{ε,ω}(𝔽_{v}; a) = ∑_{u∈𝔽_{v}^{×}} ε ∘ Nm_{𝔽_{v}/𝔽}(u + \frac{1}{u} + a_{v})ω ∘ Nm_{𝔽_{v}/𝔽′}(\frac{u}{u^{q}})
= ∑_{u∈𝔽_{v}} ε ∘ Nm_{𝔽_{v}/𝔽}((u − 1)^{2} + (a_{v} + 2)u)(ε ∘ Nm_{𝔽_{v}/𝔽} · ω^{1−q} ∘ Nm_{𝔽_{v}/𝔽′})(u).

It follows from the computations in §2 that the L-factor at v is

1 − λ_{ε,ω}(𝔽_{v}; a)Nv^{−s} + η_{a,ε,ω}(π_{v})Nv^{−2s}.

Here π_{v} is a uniformizer at the place v. We summarize the above discussion in

(4.6) Theorem Let K be a function field with the field of constants 𝔽. Let a be a nonzero element in K, ε be a nontrivial character of 𝔽^{×}, and ω be a regular character of 𝔽′^{×}, such that either ε^{2} is a nontrivial character of 𝔽^{×}, or ω^{2} is a regular character of 𝔽′^{×}. Then there is an automorphic form f_{a,ε,ω} of GL_{2} over K, with central character η_{a,ε,ω}, which is an eigenfunction of the Hecke operator T_{v} at all places v of K, which is not a pole of a and where a ≢ ±2 (mod v), with eigenvalue λ_{ε,ω}(𝔽_{v}; a) as defined above. In other words,

L(s, f_{a,ε,ω}) ∼ ∏_{v good} \frac{1}{1 − λ_{ε,ω}(𝔽_{v}; a)Nv^{−s} + η_{a,ε,ω}(π_{v})Nv^{−2s}} .

Moreover, if ε has order two, then η_{a,ε,ω} is the unramified idele class character ∥ · ∥^{−1}.

The last statement follows from Proposition 2.3, (vi). For the remainder of this section, assume ε has order two. We remark that, up to sign, the eigenvalue λ_{ε,ω}(𝔽_{v}; a) of f_{a,ε,ω} at a good place v is an eigenvalue of a Terras graph with base field 𝔽_{v}. Specifically, the computation above shows that if the degree of v is odd, then it is equal to −ε(−1)λ_{2−a_{v},ω∘Nm_{𝔽′_{v}/𝔽′}}; while if the degree of v is even, it is equal to λ_{a_{v}+2,χ} with the character χ of 𝔽_{v}^{×} being ε ∘ Nm_{𝔽_{v}/𝔽} ω^{1−q} ∘ Nm_{𝔽_{v}/𝔽′}. Let g_{a,ε,ω} be the twist of f_{a,ε,ω} by the unramified character −ε(−1), that is,

g_{a,ε,ω}(x) = (−ε(−1))^{deg det(x)}f_{a,ε,ω}(x)

for all x ∈ GL_{2}(A_{K}). Then the eigenvalues of g_{a,ε,ω} at places of odd degree are eigenvalues of Terras graphs of the second type, while those at places of even degree are eigenvalues of Terras graphs of the first type.

(4.7) When ε has order two and a is not a constant, we know from Theorem 2.9 that the geometric monodromy of the representation ρ_{a,ε,ω} is SL(2). Therefore, the Sato-Tate conjecture holds for f_{a,ε,ω}.

(4.7.1) Theorem With the same notation as in Theorem 4.6, assume further that ε is a character of order two and that a is not a constant. Then the normalized Fourier coefficients λ_{ε,ω}(𝔽_{v}; a)/\sqrt{|𝔽_{v}|} of f_{a,ε,ω} are uniformly distributed with respect to the Sato-Tate measure μ_{ST} as Nv tends to infinity.

(4.8) We specify Theorem 4.6 to the case K = 𝔽(t) and a = \frac{2t−4+2b}{t−b} for an element b ∈ 𝔽 and b ≠ 0, 1. Assume that ε has order two and ω has order greater than two. The bad places for the automorphic form g_{a,ε,ω} are at 1, ∞ and b. The local components of the corresponding representation of GL(2) over K are special representations at 1 and ∞ with nontrivial local Lfactors by Theorem 2.9, and it is a principal series at place b with trivial local L-factor by Theorem 2.12. Moreover, the local component of g_{a,ε,ω} at the place b is a new vector for the congruence subgroup Γ_{0}((t − b)^{2}) with central character ∥ · ∥^{−1}. By the same argument as in the previous case, we obtain an automorphic form g_{b,ε,ω} of D(A_{K}) right invariant by 𝒦_{b} (as defined in Introduction) which has the same L-function as g_{a,ε,ω}. In particular, as remarked right after Theorem 4.6, g_{b,ε,ω} is an eigenfunction of the Hecke operator T_{0} at place 0 with eigenvalue −ε(−1)λ_{ε,ω}(𝔽; −2 + 4/b), which is equal to the eigenvalue λ_{4(b−1)/b,ω} of the Terras graph X_{4(b−1)/b} as defined in Introduction.

Combining the above discussion with Theorems 2.9 and 2.12, we obtain

(4.8.1) Theorem Let K be the rational function field 𝔽(t). Given a nonzero element b ∈ 𝔽 not equal to 1, a character ε of 𝔽^{×} of order two, and a regular character ω of 𝔽′ of order greater than two, there is an automorphic form g_{b,ε,ω} on D(A_{K}), right invariant by 𝒦_{b} as in Introduction, which is an eigenfunction of the Hecke operators T_{v} for all places v outside 1, b, ∞, and whose associated L-function is

L(s, g_{b,ε,ω}) = \frac{1}{1 − ω(−1)q^{−s}} · \frac{1}{1 − ε(−1)q^{−s}} ∏_{v≠1,b,∞} \frac{1}{1 − (−ε(−1))^{deg v}λ_{ε,ω}(𝔽_{v}; \frac{2t−4+2b}{t−b})Nv^{−s} + Nv^{1−2s}} .

In particular, g_{b,ε,ω} is an eigenfunction of T_{0} with eigenvalue λ_{4(b−1)/b,ω}.

(4.8.2) Remarks (1) The eigenvalues of g_{b,ε,ω} are eigenvalues of Terras graphs, as discussed in detail after Theorem 4.6.

(2) In writing L(s, g_{b,ε,ω}) as an Euler product above, the first factor is the local factor at the place ∞ and the second factor is at the place 1.

(4.9) Recall from Proposition 1.1 that the Terras graph X_{4(b−1)/b} is a quotient graph of the Morgenstern graph X_{𝒦_{b}} . For each nontrivial character χ of 𝔽^{×}, we have found an automorphic form g_{b,χ} on X_{𝒦_{b}} which realizes the eigenvalue λ_{4(b−1)/b,χ} of the Terras graph X_{4(b−1)/b}, and for each regular character ω of 𝔽′ of degree greater than two, we have found an automorphic form g_{b,ε,ω} on X_{𝒦_{b}} which realizes the eigenvalue λ_{4(b−1)/b,ω} of the Terras graph X_{4(b−1)/b}. This answers question (i) in Introduction. As for question (ii), we conclude from Theorems 4.2.2 and 4.7.1 that eigenvalues of type λ_{a,χ} (resp. of type λ_{a,ω}) are uniformly distributed with respect to the Sato-Tate measure, as the cardinality of the underlying finite field of Terras graph tends to infinity.

References

[1] J. Angel, N. Celniker, S. Poulos, A. Terras, C. Trimble, and E. Velasquez. Special functions on finite upper half planes. volume 138 of Contemp. Math., pages 1–26. Amer. Math. Soc., 1992.

[2] N. Celniker, S. Poulos, A. Terras, C. Trimble, and E. Velasquez. Is there life on finite upper half planes? volume 143 of Contemp. Math., pages 65–88. Amer. Math. Soc., 1993.

[3] P. Deligne. Le conjecture de Weil, II. Publ. Math. IHES, 52:138–252, 1980.

[4] P. Deligne. Cohomologie Étale (SGA 4\frac{1}{2}), volume 569 of Lecture Notes in Math. Springer-Verlag, 1977.

[5] R. Evans. Character sums as orthogonal eigenfunctions of adjancency operators for Cayley graphs. volume 168 of Contemporary Math., pages 33–50. Amer. Math. Soc., 1994.

[6] S. Gelbart and H. Jacquet. Forms of GL(2) from the analytic point of view. In Automorphic Forms, Representations, and L-functions, Part 1, volume 33 of Proc. Symp. Pure Math., pages 213–251. Amer. Math. Soc., 1979.

[7] N. Katz. Estimates for Soto-Andrade sums. J. reine angew. Math., 438:143–161, 1993.

[8] G. Laumon. Semi-coninuité du conducteur de Swan. Astérisque, 82–83:173–219, 1981.

[9] W.-C. W. Li. Number Theory with Applications. World Scientific, 1996.

[10] W.-C. W. Li. Eigenvalues of Ramanujan graphs. In Emerging Applications of Number Theory (Minneapolis, MN, 1996), volume 109 of IMA Vol. Math. Appl., pages 387-403, Springer, New York, 1999.

[11] M. Morgenstern. Existence and explicit constructions of q + 1 regular Ramanujan graphs for every prime power q. J. Comb. Theory, series B, 62:44–62, 1994.

[12] A. Terras. Fourier Analysis on Finite Groups and Applications. London Math. Soc. Student Texts, vol. 43, Cambridge University Press, 1999.

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