## Abstract

We give a general version of cancellation in exponential sums that arise as sums of products of trace functions satisfying a suitable independence condition related to the Goursat–Kolchin–Ribet criterion, in a form that is easily applicable in analytic number theory.

## 1. Introduction

In many (perhaps surprisingly many) applications to number theory, exponential sums over finite fields of the type
1.1
arise naturally, for some positive integer *k*≥1, where

— the function

*K*is a ‘trace function’ over**F**_{p}, of weight 0, for instance for some fixed polynomial*f*∈**Z**[*X*], a Kloo- sterman sum or its generalization to hyper-Kloosterman sums for some*r*≥2;— for 1≤

*i*≤*k*,*γ*_{i}∈PGL_{2}(**F**_{p}) acts on**F**_{p}by fractional linear transformation for instance*γ*_{i}⋅*x*=*a*_{i}*x*+*b*_{i}for some and*b*_{i}∈**F**_{p}(and the sum (1.1) is restricted to those*x*∈**F**_{p}which are not poles of any of the*γ*_{i}); and— finally,

*h*∈**F**_{p}.

The goal is usually to prove, except in special ‘diagonal’ cases, an estimate of the type

where the implied constant is independent of *p* and *h*, when *K* has suitably bounded ‘complexity’.

Note that if *K*(*x*) is a Kloosterman sum, or another similar normalized exponential sum in one variable, then opening the sums expresses (1.1) as a (*k*+1)-variable character sum, and (because of the normalization) the goal becomes to have square-root cancellation with respect to all variables.

We emphasize that we do not assume that the *γ*_{i} are distinct. Furthermore, such sums also arise with some factors *K*(*γ*_{i}⋅*x*) replaced with their conjugate , or indeed with factors *K*_{i}(*x*) which are not directly related. Such cases will be also handled in this paper.

As a sample of situations where such sums have arisen, we note:

— in many proofs of the Burgess estimate for short character sums, one has to deal with cases where

*h*=0 and*K*_{i}(*x*)=*χ*(*x*+*a*_{i}) or for some multiplicative character*χ*(e.g. [1, Corollary 11.24], Lemma 12.8);— cases where

*k*=2 and*γ*_{1},*γ*_{2}are diagonal are found in the thesis of Michel and his subsequent papers (e.g. [2]);— for

*k*=2,*γ*_{1}=1 and*h*=0, we obtain the general ‘correlation sums’ (for the Fourier transform of*K*) defined in [3]; these are crucial to our works [3,4,5];— special cases of this situation of correlation sums can be found (sometimes implicitly) in earlier works of Iwaniec [6], Pitt [7] and Munshi [8];

— the case

*k*=2,*γ*_{1}and*γ*_{2}diagonal,*h*arbitrary and*K*a Kloosterman sum in two variables (or a variant with*K*a Kloosterman sum in one variable and*γ*_{1},*γ*_{2}not upper triangular) occurs in the work of Friedlander & Iwaniec [9], and it is also used in the work of Zhang [10] on gaps between primes;— cases where

*k*is arbitrary, the*γ*_{i}are upper triangular and distinct, and*h*may be non-zero appear in the work of Fouvry*et al.*[11, Lemma 2.1], indeed in a form involving different trace functions*K*_{i}(*γ*_{i}⋅*x*) related to symmetric powers of Kloosterman sums;— the sums for

*k*arbitrary and*h*=0, with*K*a hyper-Kloosterman sum appear in the works of Fouvry*et al.*[12] and Kowalski & Ricotta [13] (with*γ*_{i}diagonal);— this last case, but with arbitrary

*h*and the*γ*_{i}being translations also appears in the work of Irving [14], and (for very different reasons) in work of Kowalski & Sawin [15]; and— another instance, with

*k*=4,*h*arbitrary and*γ*_{i}upper triangular, occurs in the work of Blomer & Milićević [16, §11].

The principles arising from algebraic geometry and algebraic group theory (in particular the so-called Goursat–Kolchin–Ribet criterion, as developed by Katz), together with the general form of the Riemann hypothesis over finite fields of Deligne lead to square-root cancellation in such sums in (also possibly surprisingly) many circumstances. However, this principle is not fully stated in a self-contained manner in any reference. Thus, this paper is devoted to a review of these principles. We have aimed to give statements that can be quoted easily in applications similar to the ones above, possibly with some additional algebraic leg-work.

All estimates will be derived ultimately from the following form of the Riemann hypothesis over finite fields (see §4 for a sketch). Given two constructible ℓ-adic sheaves and on , lisse on some dense open set *U*, we will denote
where we denote by the middle-extension dual of (see the notation section for details). We have:

### Proposition 1.1

*Let k*≥1 *and let* *be any k-tuple of* ℓ-*adic middle-extension sheaves on* *such that the* *are of weight 0, and let* *be an ℓ-adic middle-extension sheaf of weight 0. Let K*_{i} *be the trace function of* *and M that of* . *If*
1.2
*then we have*
*where C*≥0 *depends only on k and on the conductors*^{1} *of* *and of* .

Thus, we will concentrate below on finding and explaining criteria that ensure that the vanishing property (1.2) holds, deriving bounds for the corresponding sums from this proposition. However, for convenience, we will state formally a number of special cases of the resulting estimates. Moreover, it can happen in some applications that the precise main term in diagonal situations is important (e.g. [15]), and we give some statements that contain such estimates.

We begin by defining a special class of trace functions for which we can give a general estimate for (1.1).

### Definition 1.2 (Bountiful sheaves)

We say that an ℓ-adic sheaf on is *bountiful* provided the following conditions hold:

— the sheaf is a middle extension, pointwise pure of weight 0, of rank

*r*≥2;— the geometric monodromy group of is equal to either SL

_{r}or Sp_{r}(we will say that is of SL_{r}-type or Sp_{r}-type, respectively); and— the projective automorphism group 1.3 of is trivial.

If is of SL_{r}-type, we will also need to understand the set
which we define for any middle-extension ℓ-adic sheaf .

This definition implies that acts on by left-multiplication: for elements and , we have . This action is simply transitive (if *γ*_{1}, , we get with *γ*_{2}=*γγ*_{1}). This means that is either empty or is a right coset of .

There is another extra property: if , the fact that implies that .

In particular,^{2} for a sheaf with (e.g. a bountiful sheaf), there are only two possibilities: either is empty, or it contains a single element , and the latter is an involution: . If this second case holds, we say that is the *special involution* of . (For instance, we will see that for hyper-Kloosterman sums with *r* odd, there is a special involution which is *x*↦−*x*.)

The diagonal cases, where there is no cancellation in (1.1), will be classified by means of the following combinatorial definitions.

### Definition 1.3 (Normal tuples)

Let *p* be a prime, *k*≥1 an integer, ** γ** a

*k*-tuple of and

**a**

*σ**k*-tuple of

*Gal*(

**C**/

**R**)={1,

*c*}, where

*c*is complex conjugation.

(1) We say that ** γ** is

*normal*if there exists some such that is odd.

(2) If *r*≥3 is an integer, we say that (** γ**,

**) is**

*σ**if there exists some such that and*

*r*-normal(3) If *r*≥3 is an integer, and is a given involution, we say that (** γ**,

**) is**

*σ**if there exists some such that and 1.4*

*r*-normal with respect to*ξ*

### Example 1.4

(1) The basic example of a pair (** γ**,

**) which is not**

*σ**r*-normal arises when

*k*is even and it is of the form since we then have for any

*γ*∈{

*γ*

_{1},…,

*γ*

_{k/2}}.

(2) Let be an involution. Some basic examples of pairs (** γ**,

**) which are not**

*σ**r*-normal with respect to

*ξ*are the following:

— If

*k*is even, pairs (for instance, if the*γ*_{i}are distinct, the left-hand side of (1.4) is then for each*γ*∈{*γ*_{1},…,*γ*_{k/2}}).— For

*r*=3,*k*=7, pairs where the left-hand side of (1.4) for*γ*(resp.*ξγ*) is

After these definitions, we have first an abstract statement, from which estimates follow immediately from proposition 1.1. In this statement, for a sheaf and *σ*∈Aut(**C**/**R**), we denote if *σ* is the identity, and if *σ*=*c* is complex conjugation.

### Theorem 1.5 (Abstract sums of products)

*Let p be a prime and let* *be a bountiful ℓ-adic sheaf on* .

*(1) Assume that* *is of Sp*_{r}*-type. For every k≥1, every k-tuple* *γ**of elements in* *and every h∈**F*_{p}, we have
*provided that either* *γ**is normal or that h≠0.*

*(2) Assume that* *is of SL*_{r}*-type. For every k≥1, for all k-tuples* *γ**of elements of* *and* *σ**of elements of Aut(**C**/*** R**), and for all h∈

*F*_{p}, we have provided that either h≠0, or that h=0 and either has no special involution, and (

*γ**,*

**) is r-normal;**

*σ**has a special involution ξ, p>r, and (*

*γ**,*

**) is r-normal with respect to ξ.**

*σ*To be concrete, we get:

### Corollary 1.6 (Bountiful sums of products)

*Let* *p* *be a prime and let* *K* *be the trace function modulo* *p* *of a bountiful sheaf* *with conductor* *c*. *Then, for any* *k*≥1, *there exists a constant* *C*=*C*(*k*,*c*) *depending only on* *c* *and* *k* *such that*:

(*1*) *If* *is self-dual, so that* *K* *is real-valued, then for any* *k*-*tuple* *γ**of elements of* PGL_{2}(*F*_{p}) *and for any* *h*∈*F*_{p}, *provided that* *either* *γ**is normal*, *or* *h*≠0, *we have*

(*2*) *If* *is of* SL_{r}-*type with* *r*≥3, *and* *p*>*r*, *then for* *k*-tuples ** γ** of elements of PGL

_{2}(

*F*_{p}) and

**of Aut(**

*σ***/**

*C***),**

*R**and for any*

*h*∈

*F*_{p},

*provided either that*(

**,**

*γ***)**

*σ**is*

*r*-

*normal, or*

*r*-

*normal with respect to the special involution of*

*if it exists, or that*

*h*≠0,

*we have*

*where we put*

*K*(

*γ*

_{i}⋅

*x*)

^{σi}=

*σ*

_{i}(

*K*(

*γ*

_{i}⋅

*x*)).

This is intuitively best possible, because if is self-dual and ** γ** is not normal, so that the distinct elements

*γ*

_{j}in

*γ*appear each with even multiplicity 2

*n*

_{j}, we get for

*h*=0 the sum in which there is no cancellation to be expected. The corresponding optimality holds for sheaves of

*SL*

_{r}-type, but this is less obvious.

It is sometimes important to determine even in this case what is the main term that may arise. Examples are given by the following statement.

### Corollary 1.7

*Let* *p* *be a prime and let* *K* *be the trace function modulo* *p* *of a bountiful sheaf* *with conductor* *c*. *Assume furthermore*:

—

*That the arithmetic monodromy group of**is equal to the geometric monodromy group*.—

*If**is of*SL_{r}-*type and has a special involution**ξ*,*that*.

*Then, for any* *k*≥1, *there exists a constant* *C* *depending only on* *c* *and* *k* *such that*:

(1) *If* *is of* Sp_{2g}-*type, then for any* *k*-*tuple* *γ**of elements of* PGL_{2}(*F*_{p}) *which is not normal and for any* *h*∈*F*_{p}, *there exists an integer* *m*(** γ**)≥1

*such that*

*If* *k* *is even and* *γ**consists of pairs of* *k*/2 *distinct elements, then* *m*(** γ**)=1.

*In general*,

*where*

*γ*

*runs over all elements occurring in the tuple*

**,**

*γ**n*

_{γ}

*is the multiplicity of*

*γ*

*in the tuple and*

*A*(

*n*)

*is the multiplicity of the trivial representation of*Sp

_{2g}

*in the*

*n*th

*tensor power of the standard representation of*Sp

_{2g}.

(2) *If* *is of* SL_{r}-*type with* *r*≥3, *then for* *k*-*tuples* *γ**of elements of* PGL_{2}(*F*_{p}) *and* *σ**of* Aut(** C**/

**),**

*R**such that*(

**,**

*γ***)**

*σ**is not*

*r*-

*normal, or not*

*r*-

*normal with respect to the special involution of*

*if it exists, there exists an integer*

*m*(

**,**

*γ***)≥1**

*σ**such that*

*If* *k* *is even*, *γ**consists of* *k*/2 *pairs of elements which are distinct or distinct modulo the special involution if it exists, and for each such pair* (*γ*_{i},*γ*_{j}), *one of* *σ*_{i} *is the identity and the other is* *c*, *then* *m*(** γ**,

**)=1.**

*σ**Otherwise*,

*m*(

**,**

*γ***)**

*σ**is bounded in terms of*

*k*

*and*

*r*

*only*.

The proofs of theorem 1.5, corollaries 1.6 and 1.7 will be found in §4, after we develop a more general framework in §2. Many examples of (trace functions of) bountiful sheaves, and also of the more general situation of the next section, together with more statements of the resulting estimates, are found in §3. Readers may wish to first read through this last section in order to see more examples of the estimates we obtain.

There is a certain inevitable tension in this paper between the fact that, on the one hand, we deal with rather general phenomena, and on the other hand most applications involve extremely concrete special cases. We hope to write a fuller book-length account of trace functions over finite fields that will resolve this conflict by providing much more detailed explanations and examples, but in the meantime, the current text should provide precise references for many applications. Any reader in a state of doubt concerning sums of the type considered here is welcome to contact the authors. We also mention that the arXiv version of this text contains some more details and examples.

*Notation and conventions*. (1) An ℓ-adic sheaf over an algebraic variety *X* defined over **F**_{p} will always mean a constructible -sheaf for some ℓ≠*p*; whenever the trace function of such sheaves are mentioned, it is assumed that an isomorphism has been chosen once and for all, and that the trace function is seen as complex-valued through this isomorphism.

(2) For a lisse sheaf (resp. a middle-extension sheaf on **A**^{1}), we denote by the dual lisse sheaf (resp. the middle-extension dual where *j* : *U*↪**A**^{1} is the open immersion of a dense open set where is lisse). If *ϱ* is a finite-dimensional representation of a group *G*, we denote by *D*(*ϱ*) the contragredient representation.

(3) The conductor of an ℓ-adic middle-extension sheaf on is defined as
where is the set of singularities of in and the Swan conductor at *x*.

(4) We denote by *Z*(*G*) the centre of a group *G*, and by *G*^{0} the connected component of the identity in a topological or algebraic group *G*.

## 2. A general framework

We provide in this section, and the next, a very general statement concerning sheaves with trace functions of the type appearing in (1.1). This will be presented in a purely algebraic manner, and later sections will provide the diophantine interpretation that leads to the results of the first section, as well as to more general statements, which will be explained in the later sections.

We first make a definition that encapsulates some of the content of the Goursat–Kolchin–Ribet criterion of Katz (see [17, §1.8]).

### Definition 2.1 (Generous tuple)

Let *k*≥1 be an integer and *p* a prime. Let be a dense open set. Let be a tuple of ℓ-adic middle-extension sheaves on , all lisse on *U*. Denote by
the ℓ-adic representations corresponding to , and

We say that is * U-generous* if:

(1) the sheaves are geometrically irreducible and pointwise pure of weight 0 on

*U*;(2) for all

*i*, the normalizer of the connected component of the identity of the geometric monodromy group*G*_{i}of is contained in and its Lie algebra is simple (in particular, acts irreducibly on*V*_{i});(3) for all

*i*≠*j*, the pairs and are Goursat-adapted in the sense of [17, p. 24], where Std_{i}denotes the tautological representations*G*_{i}⊂*GL*(*V*_{i}); and(4) let

*G*be the Zariski closure of the image of*ϱ*and let be the representation such that is the composition ; then for all*i*≠*j*, and all one-dimensional characters*χ*of*G*, there is no isomorphism 2.1 as representations of*G*.

We say that is *strictly U-generous* if it is generous and the monodromy groups

*G*

_{i}are connected.

### Remark 2.2

The last condition holds in particular if, for *i*≠*j*, there is no rank 1 sheaf such that
and we will usually check it in this form.

### Example 2.3

We just give quick examples here, leaving more detailed discussions to §3.

(1) Let *U*=**G**_{m}. Given *n*≥1 even (resp. odd) and a *k*-tuple (*a*_{i}) of distinct elements of (resp. elements distinct modulo ±1), we take , where is the *n*-variable Kloosterman sheaf with trace function *Kl*_{n}(*x*;*p*) (see §3).

Then is strictly *U*-generous. This follows from the theory of Kloosterman sheaves, in particular the computation of the geometric monodromy groups by Katz [18], and the fact that there does not exist a rank 1 sheaf and a geometric isomorphism
for *a*≠1 if *n* is even, and for *a*∉{±1} if *n* is odd. (In other words, we have , and for *r*≥3 odd, contains the unique special involution *x*↦−*x*; see §3 for details.)

(2) Given self-dual and lisse on **G**_{m}, with geometric monodromy group equal to Sp_{r}, such that the projective automorphism group of is trivial, and a *k*-tuple (*a*_{i}) of distinct elements of , we may take on *U*=**G**_{m}, and is then strictly **G**_{m}-generous.

(3) Given lisse on **G**_{m} with geometric monodromy *containing* SL_{r} for some *r*≥3, such that
where **T**⊂PGL_{2} is the diagonal torus, and a *k*-tuple ** a**=(

*a*

_{i}) of elements of distinct modulo ±1, then the tuple is

**G**

_{m}-generous.

Indeed, all conditions of the definition are clearly met, except maybe for the non-existence of isomorphisms
for *i*≠*j*. But such an isomorphism would imply that there exists a rank 1 lisse sheaf on **G**_{m} and a geometric isomorphism
and this implies that
so that *a*_{i}=±*a*_{j}, contradicting our assumption on the tuple **a**.

(4) Given a *U*-generous tuple (resp. strictly *U*-generous tuple), any subtuple is still *U*-generous (resp. strictly *U*-generous). Similarly, if *V* ⊂*U* is another dense open set, the restrictions to *V* of a *U*-generous tuple is *V* -generous (and similarly for strictly generous tuples).

We now come back to the development of the general theory. The crucial point is the following lemma.

### Lemma 2.4 (Katz)

*Let* *be U-generous. Then the connected component of the identity of the geometric monodromy group G of the sheaf*
*on U is equal to the product*
*of the connected components of the geometric monodromy groups G*_{i} *of* . *If* *is strictly generous, then G*=*G*^{0}.

*Let* *be the finite Abelian étale covering corresponding to the surjective homomorphism*
*so that V* =*U and π is the identity on* *if* *is strictly U-generous. Then the geometric monodromy group of*
*is equal to G*^{0}. *Furthermore, the restriction to G*^{0} *of any irreducible representation of G is irreducible*.

### Proof.

In view of the definition, the computation of the monodromy groups is a special case of the Goursat–Kolchin–Ribet proposition of Katz [17, Prop. 1.8.2] (noting that, with the notation there, if the normalizer of is contained in , then acts irreducibly on *V* _{i}, because any sub-representation is stable under the action of ).

For the last part, let *τ* be an irreducible representation of *G*. Note that
by the second condition in the definition of a generous tuple, and the fact that any *g*∈*G* is of the form
for some *ξ*_{i}∈**G**_{m}∩*G*_{i}⊂*Z*(*G*_{i}) and , so that *g*=*zh* with *z*=(*ξ*_{i})∈*Z*(*G*) and *h*=(*g*_{i})∈*G*^{0}. It follows that for any *g*=*zh*∈*G*, we have

As *τ*(*z*) is a scalar (because *τ* if *G*-irreducible and *z* is central), we see that any *G*^{0}-invariant subspace is also *G*-invariant. ▪

### Remark 2.5

This result would not extend if we allow *G*_{i} not contained in : for instance, if *G*=O_{2r}, so that *G*^{0}=SO_{2r}, there exist irreducible representations of *G* which split in two irreducible sub-representations when restricted to *G*^{0}.

We then state a preliminary result, which for convenience we express in the language of Tannakian categories. For a *U*-generous tuple , we denote by the Tannakian category of sheaves on generated by the sheaves .

### Proposition 2.6

*Let* *be U-generous, and let* *be the finite Abelian étale covering corresponding to the surjective homomorphism*

*(1) The category* *is equivalent as a Tannakian category to the category of representations of the linear algebraic group G, a functor from the latter to* *giving this equivalence is*
*where* *is the representation of* *corresponding to the lisse sheaf*

*Furthermore, the restriction to G*^{0} *of a representation of G corresponds to the functor π**.

*(2) If* *is an irreducible object of* *then we have a geometric isomorphism*
*where Λ*_{i} *is an irreducible representation of* *for each i*. *Two such sheaves have isomorphic restriction to* *if and only if the respective Λ*_{i} *are the same*.

### Proof.

The first part is a standard fact. To deduce (2), we simply note that from the last part of lemma 2.4, the pullback is geometrically irreducible if is geometrically irreducible. We then obtain the stated formula from the classification of irreducible representations of a direct product. ▪

We now present a first classification theorem that is well suited to cases where all sheaves involved are self-dual.

### Theorem 2.7 (Diagonal classification)

*Let* *be U-generous and let* *be the finite Abelian étale covering corresponding to the surjective homomorphism*

*Let* *be an ℓ-adic sheaf which is geometrically irreducible and lisse on U. Let*
*be a k-tuple of positive integers. Denote*

*We have*
*only if there exists a geometric isomorphism*
2.2
*on* *where, for all i, Λ*_{i} *is an irreducible representation of the group* *which is also a sub-representation of the representation* *of* *with Std*_{i} *denoting the natural faithful representation of* *corresponding to* .

*In fact, for* *given as above, we have*
*where* *denotes the multiplicity of Λ*_{i} *in* .

*If* *is strictly U-generous, then equality holds in this formula, and in particular the left-hand side is non-zero if and only if* *is of the form* *with Λ*_{i} *as above.*

*In general, if* *is of the form (*2.2*), then there exists a character χ of G/G*^{0} *such that*

*If all n*_{i} *are equal to 1, we denote* *. Then*
*unless* *and*
*in that case.*

The crucial point in the proof is the following very simple fact.

### Lemma 2.8

*With the notation of the theorem, assume that*

*Then* *is geometrically isomorphic to an object of* .

### Proof.

The co-invariant formula for lisse sheaves states that

The irreducibility of , and the semi-simplicity of the representations involved, shows that if this dimension is non-zero, then is geometrically isomorphic to a sub-sheaf of . But clearly this sheaf is itself an object of , hence the result by transitivity. ▪

### Proof of the theorem

By the lemma, is geometrically isomorphic to an object of . As it is also geometrically irreducible, lemma 2.4 shows that is also geometrically irreducible. Thus, by the proposition, it follows that
where the *Λ*_{i} are some irreducible representations of the group . We have then
where we can use invariants instead of coinvariants because the representations are semi-simple. But the *G*^{0}-invariants of the generic fibre of
are isomorphic (under the equivalence of the proposition) to the invariants of *G*^{0} on
hence to the tensor product over *i* of the -invariants of

Thus, we get the inequality for the dimension, and in particular the *G*^{0}-invariant space is non-zero if and only if *Λ*_{i} is a sub-representation of for all 1≤*i*≤*k*, and this gives a necessary condition for the *G*-invariant space to be non-zero.

In the opposite direction, if is given by (2.2) with *Λ*_{i} an irreducible sub-representation of , then we have

This invariant space is naturally a representation of *G*/*G*^{0}; since it is non-zero, it contains at least one character *χ*; one then checks easily that

Finally, if *n*_{i}=1 and , then since is irreducible in this case (e.g. because its restriction to *G*^{0} is irreducible as ), Schur's lemma gives the result. ▪

We state separately a more general version of theorem 2.7 which is useful when some sheaves are not self-dual.

### Theorem 2.9 (Diagonal classification, 2)

*Let* *be U-generous and let* *be the finite Abelian étale covering corresponding to the surjective homomorphism*

*Let* *be an ℓ-adic sheaf which is geometrically irreducible and lisse on U. Let*
*be k-tuples of integers such that n*_{i}*+m*_{i}*≥1 for all i. Denote*

*We have*

*only if there exists a geometric isomorphism*
2.3
*on* *where, for all i, Λ*_{i} *is an irreducible representation of the group* *which is also a sub-representation of the representation* *of* *with Std*_{i} *denoting the natural faithful representation of* *corresponding to* .

*In fact, for* *given as above, we have*
*where* *denotes the multiplicity of Λ*_{i} *in* *. If* *is strictly U-generous, then there is equality, and the converse also holds.*

*In general, if* *is given by (*2.3*), then there exists a character χ of G/G*^{0} *such that*

Clearly, the case ** n**=(0,…,0) recovers theorem 2.7.

### Proof.

This is the same as that of theorem 2.7, mutatis mutandis. ▪

Here is a simple corollary that can be very helpful.

### Corollary 2.10

*Let* *be U-generous. Let* *be an ℓ-adic sheaf. Let* *σ**be a k-tuple of elements of Aut*(** C**/

**).**

*R**If*

*then we have*

### Proof.

Note that this corresponds to the previous situation, with ** m** and

**such that**

*n**m*

_{i}+

*n*

_{i}=1 for all

*i*.

By considering a geometrically irreducible sub-sheaf of , we may assume that it is geometrically irreducible (as a sub-sheaf still satisfies the dimension bound and 〈⋅,⋅〉 is ‘bilinear’ with respect to direct sums). By the previous arguments, if
then we would have
where *Λ*_{i} is irreducible and occurs in *Std*_{i}. But this implies that *Λ*_{i}≃*Std*_{i}, and in particular that . ▪

## 3. Examples

We collect here examples of trace functions for which the results stated in the introduction or in the previous section apply and state some of the resulting bounds for convenience. These examples are taken for the most part from the many results of Katz, who has computed the monodromy groups of many classes of sheaves over **A**^{1} using a variety of techniques.

### (a) General construction

Quite generally, let be any finite tuple of middle-extension sheaves of weight 0 on such that the geometric monodromy groups *G*_{i} of the restriction of to a dense open set *U*_{i} where it is lisse, is such that is any of the groups

Then we can always extract a convenient generous subtuple as follows: let *U* be the intersection of the *U*_{i}, and let *J*⊂*I* be any set of representatives of *I* for the equivalence relation defined by *i*∼*j* if and only if
on *U* for some rank 1 sheaf lisse on *U*. Then is *U*-generous.

Indeed, condition (1) is clear, and (2) holds by the restrictions on (see also [19, 9.3.6] for the normalizer condition, and note that in the exceptional cases indicated, all automorphisms of the groups are inner, which implies the normalizer condition). Also, by Katz [17, Examples 1.8.1], the representations corresponding to *i*≠*j* in *J* are always Goursat-adapted, and finally the restriction to the representatives of the equivalence relation ensures the last condition.

Note that for any multiplicities *n*_{i}, *m*_{i}≥0 for *i*∈*I*, we have then geometric isomorphisms
for some rank 1 sheaf (depending on (*n*_{i},*m*_{i})) and
and it is therefore possible to use many of the results for the generous tuple to derive corresponding statements that apply to the original one. For an example of applying this principle, see the discussion of the Bombieri–Bourgain sums in §5.

In applications of this strategy, especially in the *SL*_{r} case, the following lemma will be useful.

### Lemma 3.1

*Let* *be an ℓ-adic sheaf modulo p*. *Then* *is either empty or is of the form* *for some* *such that* .

For , we recover the fact that is either empty or contains only an involution; if is equal to its normalizer, e.g. if it is a maximal and non-normal subgroup, then it shows that is either empty or equal to , which means that , or in other words that for some rank 1 sheaf . This means that, in some sense, is ‘almost’ self-dual.

### Proof.

More generally, consider a subgroup *H* of a group *G*, and a coset *T*⊂*G* of the form *T*=*ξH* that satisfies *g*^{2}∈*G* for all *g*∈*T* (as is the case of for the subgroup ).

We claim first that this situation occurs if and only if *T*=*ξH* for some *ξ*∈*G* such that *ξHξ*=*H*.

Indeed, (*ξg*)(*ξg*)∈*H* for all *g*∈*H* is equivalent to *ξgξ*∈*H* for all *g*∈*H*, i.e. to *ξHξ*⊂*H*. But then the converse inclusion also holds by taking the inverse:

Now from *ξHξ*=*H*, we get first in particular *ξ*^{2}∈*H*, and then
implies that *ξ*∈*N*(*H*). This gives the result in our case, and we may also note that the converse holds, namely if *ξ*∈*N*(*H*) satisfies *ξ*^{2}∈*H*, then
▪

Looking at the list of simple groups at the beginning of this section, it is clear that the only significant omission is that of for *r*≥2; in that case, it is indeed not true that the normalizer *O*_{2r} is contained in (see also remark 2.5). This complication may be problematic in some applications, as geometric monodromy groups *O*_{2r} do occur naturally (e.g. for certain hypergeometric sheaves and for elliptic curves over function fields, see §3). However, we have not (yet) encountered such cases in analytic number theory, and one can expect that some analogues of our statements could be proved using the classification of representations of *O*_{2r} and their restrictions to SO_{2r}.

### (b) Even rank Kloosterman sums

For *r*≥2 even, the normalized Kloosterman sums
are the trace functions of a self-dual bountiful sheaf on . Indeed, the geometric monodromy group is then Sp_{r} by [18, Theorem 11.1], and the projective automorphism group is trivial by proposition 3.6 below. In addition, one knows that the arithmetic monodromy group of is equal to its geometric monodromy group, so that corollary 1.7 applies to this sheaf. The conductor is given by (the rank is *r*, there are two singularities, one with zero Swan conductor, the other with Swan conductor 1).

Hence, from corollary 1.6, we get:

### Corollary 3.2

*Let r*≥2 *be an even integer. Let k*≥1 *be an integer. There exists a constant C*≥1, *depending only on k and r such that for any prime p, any h*∈*F*_{p} *and any* ** γ**=(

*γ*

_{1},…,

*γ*

_{k})∈

*PGL*

_{2}(

*F*_{p})

*and h*∈

*F*_{p},

*such that either*

—

*we have h*≠0,*or*—

*some component of**γ**occurs with odd multiplicity, i.e.**γ**is normal, as in definition 1.3*.

*Then we have*
*where the sum runs over x such that all γ*_{i}⋅*x are defined*.

### (c) Odd rank Kloosterman sums

For *r*≥2 odd, the normalized Kloosterman sums
are the trace functions of a non-self-dual bountiful sheaf on of SL_{r}-type, with conductor uniformly bounded over *p*, with special involution *x*↦−*x*. Indeed, the geometric monodromy group is SL_{r} by [18, Theorem 11.1], and the projective automorphism group is trivial by proposition 3.6 below, and we also have a geometric isomorphism

In addition, one knows that the arithmetic monodromy group of is equal to its geometric monodromy group, and hence corollary 1.7 also applies to this sheaf of SL_{r}-type. The conductor is *r*+3 as for even-rank Kloosterman sums.

Hence, from corollary 1.6, we get:

### Corollary 3.3

*Let r*≥2 *be an odd integer. Let k*≥1 *be an integer. There exists a constant C*≥1, *depending only on k and r such that for any prime p, any h*∈*F*_{p} *and any* ** γ**=(

*γ*

_{1},…,

*γ*

_{k})∈

*PGL*

_{2}(

*F*_{p})

^{k}

*and*

**=(**

*σ**σ*

_{1},…,

*σ*

_{k})∈Aut(

**/**

*C***)**

*R*^{k},

*such that either*

—

*we have h*≠0,*or*—

*the pair*(,*γ*)*σ**is**r-normal with respect to x*↦−*x*.

*Then we have*
*where the sum runs over x such that all γ*_{i}⋅*x are defined*.

Concretely, recall (see (1.4) and the examples following) that to say that the pair (** γ**,

**) is**

*σ**r*-normal with respect to

*x*↦−

*x*means that for

*some*component

*γ*of

**, we have where**

*γ*—

*a*_{1}is the number of*i*with*γ*=*γ*_{i}and*σ*_{i}=1,—

*a*_{2}is the number of*i*with and*σ*_{i}=*c*,—

*b*_{1}is the number of*i*with*γ*=*γ*_{i}and*σ*_{i}=*c*, and—

*b*_{2}is the number of*i*with and*σ*_{i}=1.

### (d) Hypergeometric sums

Hyper-Kloosterman sums have been generalized by Katz [17, ch. 8] to hypergeometric sums, which are analogues of general hypergeometric functions. Some give rise to bountiful sheaves, and many to generous tuples. We recall the definition: given a prime number *p*, integers *m*, *n*≥1, with *m*+*n*≥1, and tuples ** χ**=(

*χ*

_{i})

_{1≤i≤n}and

**=(**

*ϱ**ϱ*

_{j})

_{1≤j≤m}of multiplicative characters of , the hypergeometric sum

*Hyp*(

**,**

*χ***,**

*ϱ**t*;

*p*) is defined (see [17, 8.2.7]) for

*t*∈

**F**

_{p}by

where
so that the sum is over all (*n*+*m*)-tuples (** x**,

**)∈**

*y***F**

_{p}

^{n+m}such that

If *n*=*r*, *m*=0 and *χ*_{i}=1 for all *i*, then we recover the Kloosterman sums *Kl*_{r}(*t*;*p*). If *n*=2, *m*=0 and *χ*_{2}=1 but *χ*_{1} is non-trivial, we obtain Salié-type sums. This indicates that such sums should arise naturally in formulae like the Voronoi summation formula for automorphic forms with non-trivial nebentypus.

Katz shows (see [17, Theorem 8.4.2]) that if no character *χ*_{i} coincides with a character *ϱ*_{j} (in which case one says that ** χ** and

**are**

*ϱ**disjoint*), then for any ℓ≠

*p*, there exists an irreducible ℓ-adic middle-extension sheaf on , of weight 0, with trace function given by

*Hyp*(

**,**

*χ***,**

*ϱ**t*;

*p*). This sheaf is lisse on

**G**

_{m}, except if

*m*=

*n*, in which case it is lisse on

**G**

_{m}−{1}. It has rank . Moreover, these results of Katz show that the conductor of is bounded in terms of

*m*and

*n*only.

The basic results of Katz concerning the geometric monodromy group *G* of the hypergeometric sheaf depend on the following definitions of exceptional tuples of characters (see [17, Corollary 8.9.2, 8.10.1]).

### Definition 3.4

Let *k* be a finite field and let ** χ** and

**be an**

*ϱ**n*-tuple and an

*m*-tuple of characters of

*k*

^{×}.

(1) For

*d*≥1, the pair (,*χ*) is*ϱ**d*-Kummer-induced if*d*|(*n*,*m*) and if there exist*n*/*d*and*m*/*d*-tuples* and*χ** such that*ϱ*consists of all characters*χ**χ*such that*χ*^{d}is a component of*, and*χ*consists of all characters*ϱ**ϱ*such that*ϱ*^{d}is a component of*.*ϱ*(2) Assume

*n*=*m*. For integers*a*,*b*≥1 such that*a*+*b*=*n*, the pair (,*χ*) is (*ϱ**a*,*b*)-Belyi-induced if there exist characters*α*and*β*with*β*≠1 such thatconsists of all characters*χ**χ*such that either*χ*^{a}=*α*or*χ*^{b}=*β*, and ifconsists of all characters*ϱ**ϱ*such that*ϱ*^{n}=*αβ*.(3) Assume

*n*=*m*. For integers*a*,*b*≥1 such that*a*+*b*=*n*, the pair (,*χ*) is (*ϱ**a*,*b*)-inverse-Belyi-induced if and only if is (*a*,*b*)-Belyi-induced.

We say that (** χ**,

**) is Kummer-induced (resp. Belyi-induced, inverse-Belyi-induced) if there exists some**

*ϱ**d*≥2 (resp. some

*a*,

*b*≥1) such that the pair is

*d*-Kummer-induced (resp. (

*a*,

*b*)-Belyi-induced, (

*a*,

*b*)-inverse-Belyi-induced).

We then have the following.

— If

*n*=*m*, let*Λ*denote the multiplicative characterAssume that (

,*χ*) is neither Kummer-induced, Belyi-induced, nor inverse-Belyi-induced. Then*ϱ**G*^{0}is either trivial, SL_{n}, SO_{n}or Sp_{n}; if*Λ*=1, it is either SL_{n}or Sp_{n}, if*Λ*≠1 but*Λ*^{2}=1, then*G*^{0}is either 1 or SO_{n}or SL_{n}, and if*Λ*^{2}≠1, then*G*^{0}is either 1 or*SL*_{n}(see [17, Theorem 8.11.2]). The problem of determining which case occurs is discussed by Katz; most intricate is the criterion for*G*^{0}to be trivial (see [17, §8.14–8.17]), which is however applicable in practice.— If

*n*≠*m*, let be the rank of the sheaf. Assume that (,*χ*) is not Kummer induced. Then, provided , and*ϱ**p*does not divide an explicit positive integer, we have:*G*^{0}=SL_{r}if*n*−*m*is odd (and*G*≠*G*^{0}if |*n*−*m*|=1);*G*^{0}=SL_{r}, SO_{r}or Sp_{r}if*n*−*m*is even and either*r*∉{7,8,9} or |*n*−*m*|≠6 (see [17, Theorem 8.11.3]). Here also, more precise criteria for which*G*^{0}arises exist, as well as a classification of the few exceptional possibilities when |*n*−*m*|=6 and*r*∈{6,7,8}.

### Example 3.5

If ** ϱ** is the empty tuple,

*n*≥2 and

**is an**

*χ**n*-tuple where all components are trivial, then it follows immediately from the definition that (

**,**

*χ***) is not Kummer-induced. Thus, the last result recovers, for**

*ϱ**p*large enough in terms of

*n*, the fact that the geometric monodromy group of contains SL

_{n}if

*n*is odd, and contains either

*SO*

_{n}or Sp

_{n}if

*n*is even.

In order to apply the results of the previous section, it is of course very useful to have some information concerning the projective automorphism groups of hypergeometric sheaves. Many cases are contained in the following result.

### Proposition 3.6

*(1) Let* *χ*_{1}, *ϱ*_{1} *and* *χ*_{2}, *ϱ*_{2} *be any n*_{1}-*tuple (resp. m*_{1}-*tuple, n*_{2}-*tuple, m*_{2}-*tuple) with* *χ*_{1} *disjoint from* *ϱ*_{1} *and* *χ*_{2} *disjoint from* *ϱ*_{2}, *and with* *m*_{1}+*n*_{1}≥1, *m*_{2}+*n*_{2}≥1. *Let* . *Then we have a geometric isomorphism*
3.1
*if and only if a*=1 *and* *χ*_{1}∼*χ*_{2} and *ϱ*_{1}∼*ϱ*_{2}.

(2) Let *m*≠*n* with *m*+*n*≥1 be integers with and (*m*,*n*)≠(1,2), (*m*,*n*)≠(2,1). Let ** χ** and

**be disjoint tuples of characters of . The projective automorphism group is then trivial.**

*ϱ*(3) With notation as in (*2*), the set is non-empty if and only if the integer *n*−*m* is odd, and the tuples ** χ** and

**are both invariant under inversion. In this case, the special involution is**

*ϱ**x*↦−

*x*, i.e. we have

(4) If *n*=*m*≥2, then for any disjoint *n*-tuples (** χ**,

**), the group is a subgroup of the finite group**

*ϱ*(5) With notation as in (*4*), the set is either empty or is a subset of *Γ*, which is of the form *T*=*ξH* for some subgroup *H*⊂*Γ* and some *ξ*∈*N*_{Γ}(*H*) such that *ξ*^{2}∈*H*.

### Proof.

(1) For all , the components of ** χ** (resp.

**) can be recovered from the sheaf as the tame characters occurring in the representation of the inertia group at 0 (resp. at ) corresponding to this sheaf, and the multiplicity appears as the size of the associated Jordan block (see [17, Theorem 8.4.2] (6), (7), (8)). Thus, (3.1) is only possible if**

*ϱ*

*χ*_{1}∼

*χ*_{2}and

*ϱ*_{1}∼

*ϱ*_{2}.

We assume this is the case now, i.e. that

We then obtain *a*=1 from [17, Lemma 8.5.4] and the fact that the Euler–Poincaré characteristic of a hypergeometric sheaf is −1.

(2) We may assume that *n*>*m*, using inversion otherwise. Assume that is such that
for a rank 1 sheaf . By comparing ramification behaviour, we see that *γ* must be diagonal (if *γ*^{−1}(0)≠0, then must be tamely ramified at 0 to have the tensor product tamely ramified at *γ*^{−1}(0), as is; but then the inertia invariants at *γ*^{−1}(0) are zero for the tensor product, a contradiction to [17, Theorem 8.4.2 (6)], and the case of gives a similar contradiction).

Thus, implies a geometric isomorphism
on some dense open set *j* : *U*↪**G**_{m}. By [17, Lemma 8.11.7.1], under the current assumption (*n*,*m*)≠(2,1), this implies that for some multiplicative character *Λ*.

But then we have
by Katz [17, 8.3.3] where *Λ*** χ**=(

*Λχ*

_{i})

_{i}and

*Λ*

**=(**

*ϱ**Λϱ*

_{j})

_{j}. We are therefore reduced to a geometric isomorphism and by (1), it follows that

*a*=1, i.e.

*γ*=1.

(3) As in the previous case, we see that any element must be diagonal, so that *γ*⋅*x*=*ax* for some . As *γ*, if it exists, is an involution, we obtain *a*^{2}=1, and therefore the only possibility for the special involution is *x*↦−*x*.

We now assume that for some rank 1 sheaf .

Again by [17, Lemma 8.11.7.1], the sheaf is a Kummer sheaf . We have by combining [17, 8.3.3] and [17, Lemma 8.7.2] (using also the fact that Kummer sheaves are geometrically multiplication invariant). Thus the assumption means that

If *n*−*m* is even, this cannot happen by (1); if *n*−*m* is odd, on the other hand, this happens if and only if and , as claimed.

(4) Let *n*=*m*≥2 and so that
for some rank 1 sheaf . The right-hand side is ramified at (because *n*≥2 and the description of local monodromy from [17, Theorem 8.4.2] (8) shows that the ramification of the hypergeometric sheaf cannot be eliminated by tensoring with a character), and hence *γ* must permute the points 0, 1, . This shows that *γ*∈*Γ*.

(5) Arguing as in (4) with an isomorphism we see that . Then the statement is just the conclusion of lemma 3.1 in this special case. ▪

In view of these results, one can feel confident that sums of products of hypergeometric sums can be handled using the results of this paper, at least in many cases. The trickiest case would be when *G*^{0}=*O*_{r} with *r* even, which does occur (e.g. if *n*−*m*≥2 is even, *n* is even, the tuples ** χ** and

**are stable under inversion, and is non-trivial of order 2, see [17, Theorem 8.8.1, Lemma 8.11.6]).**

*ϱ*### (e) Fourier transforms of multiplicative characters

Many examples of sheaves with suitable monodromy groups are discussed in [17, 7.6–7.14], arising from Fourier transforms of other (rather simple) sheaves. We discuss one illustrative case, encouraging the reader to look at Katz's results if he or she encounters similar-looking constructions.

We consider a polynomial *g*∈**F**_{p}[*X*] and a non-trivial multiplicative character *χ* modulo *p*. We assume that no root of *g* is of order divisible by the order of *χ*. We then form the sheaf
i.e. the Fourier transform of the Kummer sheaf with trace function *χ*(*g*(*x*)), where *ψ* is the additive character *e*(⋅/*p*). The trace function of is

### Proposition 3.7

*With notation as above, let* *r* *be the number of distinct roots of* *g* *in* . *Assume that* *r*≥2 *and* *p*>2*r*+1. *Assume furthermore that the only solutions of the equations*
3.2
*where* (*x*_{1},…,*x*_{4}) *range over the roots of* *g* *in* *are given by* *x*_{3}=*x*_{1}, *x*_{4}=*x*_{2} or *x*_{2}=*x*_{1} *and* *x*_{3}=*x*_{4}. *Then* *is a middle-extension sheaf of weight 0, of rank* *r*, *lisse on* *G*_{m}, *and with geometric monodromy group containing* SL_{r}. *Furthermore, we have*
*and* *if* *r*≥3. *The conductor of* *is bounded in terms of* *only*.

We leave this proposition without proof (see the arXiv version of this paper for details); all the results follow from the work of Katz [17, 7.9.1–7.9.3] on such trace functions. Concerning the last statement, note that if *r*=2, the sheaf is of SP_{2}-type (since Sp_{2}=SL_{2}) so that is not relevant in that case.

## 4. Sums of products with fractional linear transformations

We can now quickly prove the results stated in §1 using the framework established previously.

### Proof of theorem 1.5

We begin with the easier Sp-type case. Let ** γ*** be the tuple of distinct elements of

**, and**

*γ**n*

_{γ}the multiplicity of any such element in

**. Let**

*γ**U*be the common open set in

**A**

^{1}where all

*γ*∈

*** are defined. Arguing as in example 2.3 (2), we see that the tuple is strictly**

*γ**U*-generous, simply because is bountiful of Sp

_{r}-type.

By theorems 2.7 and 2.9, we see that if
there must exist some geometric isomorphism
where *Λ*_{γ} are irreducible representations of the geometric monodromy group *G*=Sp_{r} of such that *Λ*_{γ} is a sub-representation of Std^{⊗nγ}. Just for dimension reasons, each *Λ*_{γ} must be a one-dimensional character. But definition 1.2 implies in particular that *G* has no non-trivial character, so that *Λ*_{γ}=1, which implies that must be geometrically trivial, i.e. that *h*=0.

This already proves the first part of theorem 1.5 when *h*≠0. Now assume *h*=0. Then the condition that the trivial representation be a sub-representation of Std^{⊗nγ} holds if and only if *n*_{γ} is even, and thus
if and only if all multiplicities *n*_{γ} are even, which means if and only if ** γ** is

*not*normal.

We now come to the SL_{r}-type case. If has a special involution *ξ*, let be a rank 1 sheaf such that
4.1
and we note that (as a character of the fundamental group of ) the sheaf has order dividing *r* (by taking the determinant on both sides). For convenience, we let *ξ*=1 and , if there is no special involution.

Let ** γ*** be a tuple of representatives of the elements of

**for the equivalence relation (which is indeed an equivalence relation because**

*γ**ξ*

^{2}=1).

Then, arguing as in example 2.3 (3), we see that the tuple is strictly *U*-generous, because is bountiful of SL_{r}-type and because
for some rank 1 sheaf , implies that
and thus either does not occur (if has no special involution) or happens only if *γ*_{i}=*ξγ*_{j}, so that *γ*_{i}∼*γ*_{j}, which is excluded for distinct components of ** γ***.

For *γ*∈** γ***, we denote
so that, by bringing together equivalent

*γ*

_{i}'s, we obtain a geometric isomorphism 4.2 for some rank 1 sheaf , which is a tensor product of sheaves of the form or . In particular, has order dividing

*r*since does.

We now get from theorem 2.9 that if
then
where *Λ*_{γ} is an irreducible representation of SL_{r} which is a sub-representation of the tensor product . As SL_{r} has no non-trivial one-dimensional characters, this shows that this condition cannot occur unless *Λ*_{γ} is trivial for all *γ*, which implies then that
4.3
is trivial.

If has no special involution, this immediately implies that *h*=0. If has a special involution, on the other hand, we recall that has order *r*, while has order *p* if *h*≠0. Hence (4.3) is impossible if *p*>*r* and *h*≠0, and moreover, in that case we also get from (4.3) that must be trivial.

Thus, in all cases of theorem 1.5, we reduce to understanding the case *h*=0. As *Λ*_{γ} is trivial, we have also the condition that the trivial representation is a sub-representation of the tensor product
for all *γ* in ** γ***.

But the trivial representation of SL_{r} is a sub-representation of Std^{⊗n}⊗*D*(Std)^{⊗m} if and only if *r*|*n*−*m* (e.g. [13, proof of Proposition 4.4]), and this means that if there is a main term, then for all *γ*∈** γ***, which means precisely that (

**,**

*γ***) is not**

*σ**r*-normal (if there is no special involution) or not

*r*-normal with respect to

*ξ*(if there is one). ▪

### Remark 4.1

We see from the proof that the condition *p*>*r* in theorem 1.5 (when has a special involution) can be relaxed: especially, it is not needed if we have
(i.e. if in (4.1) can be taken to be the trivial sheaf), as we only used *p*>*r* to deduce that in (4.2) is trivial, which is automatically true in this case.

For completeness, we explain the proof of proposition 1.1 (see [3, §8,§9] for similar arguments).

### Proof of proposition 1.1

Let *U*⊂**A**^{1} be the maximal open set where all sheaves and are lisse. We have

As the sheaves are all mixed of weights ≤0, we have
where *C*_{1} is the product of the ranks of the sheaves. This means that it is enough to deal with the sum over *x*∈*U*(**F**_{p}).

By the Grothendieck–Lefschetz trace formula, we have as the and terms vanish, by assumption for and because we have a tensor product of middle-extension sheaves for .

By Deligne's proof of the Riemann hypothesis [20], as the tensor product is of weight 0, all eigenvalues of Frobenius acting on the cohomology space have modulus , and hence

Finally, using the Euler–Poincaré formula, one sees that the dimension of this space is bounded in terms of the conductors of and of , and in terms of *k* (see [21, Lemma 3.3] for some details). ▪

As already mentioned, corollary 1.6 is an immediate consequence of theorem 1.5 and proposition 1.1. Corollary 1.7 is similar, except that in the argument of proposition 1.1, there is a main term in the trace formula which is (for the Sp-type case) given by

However, the extra assumption that the geometric monodromy group coincides with the arithmetic monodromy group means that all eigenvalues of the Frobenius acting on are equal to *p*. Hence this contribution is equal to
and for given (as in the proof of theorem 1.5) by
with *Λ*_{γ} an irreducible representation of *G* which is a sub-representation of Std^{⊗nγ}, we have
where each multiplicity is at most *k*, and is equal to 1 if *n*_{γ}=1. The result follows immediately.

The case of SL_{r}-type is similar and left to the reader; the extra condition that (without a twist by a non-trivial rank 1 sheaf) allows us to deduce (4.2) with trivial, from which the non-vanishing of follows when (** γ**,

**) is not**

*σ**r*-normal with respect to the special involution. (We already observed that under this condition we do not need to assume

*p*>

*r*in theorem 1.5.)

## 5. Applications

We present here some applications of the general case developed in §2, going beyond the results of the introduction and of the previous section. The first recovers an estimate of Katz used by Fouvry and Iwaniec in their study of the divisor function in arithmetic progressions [22], the second discusses briefly the sums of Bombieri & Bourgain [23].

### (a) The Fouvry–Iwaniec sum

In [22], for primes *p* and , the exponential sum
arises, where the sum is over , and we abbreviate *Kl*_{2}(*x*)=*Kl*_{2}(*x*;*p*). This is not of the type of §1, as the arguments of the Kloosterman sums are not simply of the form *γ*_{i}⋅*t*. However, it fits the general framework of §2 with the 4-tuple
where

Let *U*=**G**_{m}−{1,*β*/*α*}. We claim that this 4-tuple is *U*-generous if *α*≠*β* (which is certainly a necessary condition, since otherwise *f*_{1}=*f*_{2}). Indeed, since the geometric monodromy group of each is SL_{2}=Sp_{2} (because the geometric monodromy group of is SL_{2}, and SL_{2} has no finite index algebraic subgroup), we need to check that there is no geometric isomorphism
for *i*≠*j* and a rank 1 sheaf . But taking the dual and then tensoring, such an isomorphism implies
on the open set where the left-hand side of the original isomorphism (hence also the right-hand side) is lisse. As , this implies that
on *V* .

But since is ramified at 0 and , the ramification loci *S*_{i} of the sheaves are, respectively
and are therefore distinct, proving the desired property of *U*-generosity.

As the sum *S*(*α*,*β*;*p*) concerns the tensor product of
with the trivial sheaf , which is a tensor product of the trivial representations, which is not a sub-representation of Std, it follows therefore that
and hence by proposition 1.1 that
for all primes *p* and *α*≠*β* in , where the implied constant is absolute. In the appendix to [22], Katz gives a precise estimate of the implied constant.

### (b) The Bombieri–Bourgain sums

The Bombieri–Bourgain sums are defined by
(see [24, p. 513]), where
for some *b*∈**F**_{p} and (*a*_{1},…,*a*_{k})∈**F**_{p}^{k}, where

— (

*χ*,*χ*_{1},…,*χ*_{k}) are non-trivial multiplicative characters modulo*p*,—

*f*_{i}∈**F**_{p}[*X*],*g*∈**F**_{p}[*X*] are non-zero polynomials, and—

*g*_{i}∈**F**_{p}[*X*] and*G*∈**F**_{p}[*X*] may be zero.

This sum is of the type considered in §2, with (or rather those corresponding to the distinct parameters as this is not assumed to be the case).

Under (different) suitable conditions on these parameters, Bombieri & Bourgain [23, Lemma 33] and Katz [24, Theorem 1.1] give estimates for *S* of the type
where the implied constant depends only on *k* and the degrees of the polynomials involved. Both proofs actually avoid involving monodromy groups which illustrates that sometimes an estimate for a sum of products might be easier to obtain than those involved in the previous sections. We show how to recover quickly the desired square-root cancellation in the case that occurs for the application considered by Bombieri and Bourgain, by a hybrid of Katz's argument and those of the previous sections.

In [23], the conditions are: *p* is odd, *g*_{i}=*G*=0, , , the *f*_{i} and *g* have only simple roots, and all *χ*_{i} and *χ* are equal and are of order 2. We then first note that if some *f*_{i} has degree 1, the resulting Fourier transform
is geometrically isomorphic to a tensor product
(we use here that ), so that by combining these with we may assume that all *f*_{i} are of degree 2. Note that *g* is replaced by *X*^{k}*g*, where *k* is the number of *i* with . As *χ* has order 2, we have either *k* even and
so that the previous assumptions on *g* remain valid, or *k* odd and
where if *g*(0)=0, or otherwise; in the first case it may be that , but in that case the unique zero of is in **G**_{m} since *g* has simple roots. In particular, in all cases, we see that *g* is replaced by a polynomial with at least one (simple) root in **G**_{m}.

If all *f*_{i} were of degree 1, we are left with
with *g* non-constant, which satisfies the desired conditions. We therefore assume that some *f*_{i} are of degree 2.

For a polynomial *f*_{i} of degree 2, by completing squares, we see that the Fourier transform
is geometrically isomorphic to a tensor product of for some *h* and of the Fourier transform corresponding to a polynomial of the form *X*^{2}+*c*_{i}. We may therefore assume that all *f*_{i} are of this form.

Finally, it is easy to see that

In particular, such sheaves are of rank 2, lisse on **G**_{m} and have geometric monodromy group . We therefore obtain a strictly **G**_{m}-generous tuple by taking for the Fourier transforms corresponding to the *c*_{i}'s, modulo the equivalence relation *c*_{i}∼*c*_{j} if and only if

We can now conclude: as *g* has a simple zero in **G**_{m}, the sheaf has at least one ramification point inside **G**_{m}, and therefore the irreducible sheaf cannot be a sub-sheaf of the tensor product
which is lisse on **G**_{m}.

### Remark 5.1

Even if , *g*=*αX* and *α*≠0, we can obtain the square-root bounds provided we have at least one sheaf : by the results of §2, the condition
would imply that is geometrically isomorphic to
for some *m*_{i}≥0. By rank considerations, we have *m*_{i}=0, and this implies that is geometrically trivial, which is impossible as *g* is non-constant.

## Funding statement

P.M. was partially supported by the SNF (grant no. 200021-137488) and the ERC (Advanced Research grant no. 228304). É.F. thanks ETH Zürich, EPF Lausanne and the Institut Universitaire de France for financial support.

## Acknowledgements

Thanks to Z. Rudnick for feedback and suggestions concerning the paper, and to A. Irving for asking a question that led us to find a slip in a previous version. We also thank the referee for his or her careful reading of the text.

## Footnotes

One contribution of 8 to a Theo Murphy meeting issue ‘Number fields and function fields: coalescences, contrasts and emerging applications’.

↵1 See the notation section for a remainder of the definition.

↵2 See lemma 3.1 for a more general statement, based on these properties, that limits the possible structure of .

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.