Equidistribution and independence of Gauss sums

We prove a general independent equidistribution result for Gauss sums associated to $n$ monomials in $r$ variable multiplicative characters over a finite field, which generalizes several previous equidistribution results for Gauss and Jacobi sums. As an application, we show that any relation satisfied by these Gauss sums must be a combination of the conjugation relation $G(\chi)G(\overline\chi)=\pm q$, Galois conjugation invariance and the Hasse-Davenport product formula.


Introduction
Let k = F q be a finite field of characteristic p.For every multiplicative character χ : k × → C × the associated Gauss sum is defined as where ψ : k → C × is the additive character given by ψ(t) = exp(2πiTr Fq/Fp (t)/p).
Fix an algebraic closure k of k, and for any m ≥ 1 let k m be the unique degree m extension of k in k.For any character χ : k × m → C × we denote by G m (χ) the corresponding Gauss sum over k m .Every character χ : k × → C × can be pulled back to a character of k m by composing with the norm map; we will also denote this character by χ if there is no ambiguity.The Hasse-Davenport relation (eg.[BEW98,Theorem 11.5.2])states that G m (χ) = G(χ) m (the lack of sign in this formula is due to the negative sign used in our definition of Gauss sums).
We say that a non-zero r-tuple b ∈ Z r is primitive if its coordinates are relatively prime and its first non-zero coordinate is positive.Any non-zero r-tuple a ∈ Z r can be writen uniquely as µb, where µ ∈ Z\{0} and b is primitive.We write all a i = µ i b i in that way.
Let Char k be the injective limit of the character groups k × m for m ≥ 1 via the maps k × m → k × md given by composition with the norm maps k × md → k × m (which can be identified with the set of finite order characters of the tame fundamental group of G m, k).We denote by V k the Q-vector space with basis the set Char k .For every i = 1, . . ., n, let v i ∈ V k be the element ξ µ i =ηi ξ, that is, the sum of all characters ξ ∈ Char k whose µ i -th power is η i .There are exactly ν i such characters, where ν i is the prime to p part of µ i .
Theorem 1. Suppose that, for every b ∈ Z r , the set of v i for the i = 1, . . ., n such that b i = b is linearly independent in V k .Then the sets {Φ m (χ)|χ ∈ S m } become equidistributed in (S 1 ) n with respect to the Haar measure as m → ∞.
Note that the condition holds, in particular, if all b i are distinct, that is, if no two a i 's are proportional.
Corollary 1.Let η 1 , . . ., η n : k × → C × be distinct characters and t 1 , . . ., t n ∈ k × , then the elements Proof.Here v i = η i for all i, which are clearly linearly independent in V k as they are distinct characters.
Another interesting case is when r = 1 and all η i are trivial: Corollary 2. Let 0 < d 1 < . . .< d n be prime to p integers and t 1 , . . ., t n ∈ k × , then the elements Proof.Now v i = ξ d i =1 ξ are linearly independent, since for all i, v i contains a character of order d i with non-zero coefficient, so it can not be a linear combination of the v j for j < i, which are themselves linear combinations of characters of order < d i .
From the main theorem we can easily deduce a more general version where we allow using different additive characters in each coordinate: Corollary 6.Under the hypothesis of Theorem 1, suppose given also elements α 1 , . . ., α n ∈ k × , and let Then the sets {Φ m (χ)|χ ∈ S m } become equidistributed in (S 1 ) n with respect to the Haar measure as m → ∞.
Proof.We have which are themselves equidistributed by the theorem.
From Katz' result [Kat88, Theorem 9.5] and the Hasse-Davenport product formula it can be easily deduced that all monomial relations among Gauss sums of the form G(ηχ n ) for different η and n ∈ Z which hold for "almost all" characters χ are a combination of the identities (1), (2) and (3) above.The second main result of the article uses Theorem 1 to show that the same holds true in the multi-variable case.The following statement will be made precise in section 4: Theorem 2. Suppose given η i , a i as in Theorem 1 and integers ǫ i for i = 1, . . ., n, an element t ∈ (k × ) r , a non-zero integer N , a subset U m ⊆ T m for every m ≥ 1 and a sequence of complex numbers {D m } m≥1 such that for every m ≥ 0 and every χ ∈ U m .Then the expression for some η, a and d|q − 1.
The results in this article can be seen as a case of the ℓ-adic Mellin transform theory developed by Katz in [Kat12] for the one-dimensional torus and generalized to higher-dimensional commutative algebraic groups by Forey, Fresán and Kowalski in [FFK].More precisely, let G be any connected commutative algebraic group over k, and P(G) the Tannakian perverse convolution category of Gk as defined in [FFK, Chapter 3], modulo isomorphism.The subcategory P 1 (G) of P(G) consisting of tannakian rank 1 objects is an abelian group under convolution.For objects The (arithmetic or geometric) tannakian group H i of each L i is a closed subgroup of GL(1) (so either GL(1) itself or the group of r i -th roots of unity for some r i ≥ 1), so the tannakian group H of M (that is, the group whose category of representations is the subcategory M of P(G) tensor-generated by M) can be viewed as a closed subgroup of GL(1) n which is mapped onto H i via the i-th projection ρ i : GL(1) n → GL(1) for every i = 1, . . ., n.Such subgroup is completely determined by the subgroup X of the group of characters of GL(1) n (which can be naturally identified with Z n where a ∈ Z n corresponds to the character ρ a (t) := t a = t a1 1 • • • t an n ) consisting of the characters that act trivially on H, more precisely, H = ρ∈X ker(ρ) (see eg. [Spr98, 3.2.10]).Now the object of the category M corresponding to (the restriction to (since convolution "is" the tensor operation in the category P(G)).So this restriction is trivial if and only if L * a1 is trivial in P(G), and we get the following result, which was kindly suggested to us by an anonymous referee and gives the proper high-level picture of the results in this article: In other words, the tannakian group of M reflects the multiplicative relations among the L i (including a particular L i being of finite order).In particular: This is precisely what we prove in this article for the geometric tannakian group of the direct sum of the objects [t i ] * α i * (L ψ ⊗ L ηi ) for i = 1, . . ., n (where m is the map t → t ai ) under the hypothesis of Theorem 1.We will not make use of this general theory here, and instead follow a higher dimensional analogue of Katz' approach in [Kat88], replacing the (implicit, as they were formally defined in a later work) use of hypergeometric sheaves by the higher dimensional hypergeometric objects and the explicit description of the abelian group P 1 (G r m ) given by Gabber and Loeser in [GL96].
In section 2 we review the main results of this theory that we will make use of.In section 3 we give the proof of Theorem 1, and in section 4 we use it to deduce Theorem 2. Finally, in the last section we prove a version on the main theorem in which the fields are allowed to have different characteristics.
We will choose a prime ℓ = p and work with ℓ-adic cohomology, and will assume a choice of embedding ι : Qℓ → C that we will use to identify elements of Qℓ and C without making any further mention to it.When speaking about purity of ℓ-adic objects, we will mean it with respect to the chosen embedding ι.
The author would like to thank the anonymous referees for their valuable comments on earlier versions of the article.

Hypergeometric perverse sheaves on the torus
The main reference for this section is [GL96].Let G r m,k be the r-dimensional split torus over k, and G r m, k its extension of scalars to k. Denote by D b c (G r m,k , Qℓ ) the derived category of ℓ-adic sheaves on G r m,k .We have a (!-)convolution operation are the projections and the multiplication map respectively.
Let P erv denote the subcategory of D b c (G r m,k , Qℓ ) consisting of the perverse objects.Any object K ∈ P erv has Euler characteristic χ(K) ≥ 0 [GL96, Corollaire 3.4.4],and K is said to be negligible if χ(K) = 0.The negligible objects form a thick subcategory P erv 0 of P erv; denote the quotient category by P erv.They also form an ideal for the convolution, and if K and L are perverse, the i-th perverse cohomology objects of K * L are negligible for i = 0 [GL96, Proposition 3.6.4],so the convolution gives a well defined operation P erv × P erv → P erv.With this operation, P erv becomes a Tannakian category, in which the "dimension" of an object is its Euler characteristic.
Let ψ : k → C × be the additive character defined in the introduction, and χ : k × → C × any multiplicative character.Let L ψ and L χ be the corresponding (restriction of) Artin-Schreier and Kummer sheaves on G m,k [Del77, 1.7] and We will need the following lemma: Proof.Perverse objects have finite length, so by the long exact sequence of cohomology sheaves associated to an exact sequence of perverse objects it suffices to show this for the simple components of K. Since χ(K) = 1 and the Euler characteristic is additive, all but one of the simple components of K are negligible, and the other one is a simple hypergeometric object K 0 such that Ψ 2 (K 0 ) = Ψ 2 (K).

Proof of Theorem 1
This section is devoted to the proof of Theorem 1. Recall that we have fixed a finite field k = F q , n non-zero r-tuples a 1 , . . ., a n ∈ Z r , n multiplicative characters η 1 , . . ., η n : k × → C × and n elements t 1 , . . ., t n ∈ (k × ) r .For every χ ∈ S m the element Φ m (χ) ∈ (S 1 ) n is given by for every χ ∈ S m , we may assume without loss of generality that µ i > 0 for every i = 1, . . ., n.
In order to prove the equidistribution of Φ m (χ) as m → ∞ we need to show that, for every continuous function f : (S 1 ) n → C, we have lim where µ is the Haar measure on (S 1 ) n .Since (S 1 ) n is abelian, every such f is a class function, so it suffices to show this for the traces of irreducible representations of (S 1 ) n , which are dense in the space of class functions by the Peter-Weyl theorem.These irreducible representations are just the characters ) n dµ, so let us assume that c = 0.Then, since (S 1 ) n Λ c dµ = 0, we need to show that lim m→∞ Σ m (Λ c ) = 0.And this is clearly a consequence of the following Proposition 2. Let a = i min j:aij =0 |a ij |.There exists a constant A(c) such that, for every m > log q (1 + a), Proof.For the sake of notation simplicity, let us assume m = 1 and denote Σ 1 , S 1 , T 1 and Φ 1 by Σ, S, T and Φ respectively.Write We split this as where We will start by evaluating the first sum: where the character χ ∈ T is given by χ(t) = r l=1 χ l (t l ) for one-dimensional χ 1 , . . ., χ r ∈ k × .The inner sum vanishes unless n i=1 |ci| j=1 t ǫi il x ǫia il ij = 1, in which case it is equal to q − 1, so we get where X ⊆ (k × ) ||c|| is the subset consisting of the (x ij ) 1≤i≤n,1≤j≤|ci| such that = 1 for every l = 1, . . ., r.We can rewrite this sum as Let α i : G m → G r m be the morphism of tori given by t → t ai := (t ai1 , . . ., t air ).Then α i factors as β i • [µ i ], where β i : t → t bi is a closed embedding and [µ i ] : G m → G m is the µ i -th power map.For every i, the function is the trace function, on G r m , of the complex if ǫ i = −1, since α i * commutes with duality (being a finite map) and the Verdier dual of H(ψ, . Arithmetically, it is pure of weight −1, since H(ψ, η i ) is pure of weight 1.By the Lefschetz trace formula, we conclude that Σ 1 is (q − 1) r q −||c||/2 q c i <0 |ci| = (q − 1) r q −|c|/2 times the Frobenius trace at t = 1 of the !-convolution: ) which is a hypergeometric object as seen in the previous section.The class of its pull-back to G r m, by [Kat88,4.3](for the power of p part) and [Kat90, 8.9.1] (for the prime to p part), so by [Kat90, 8.1.10(2a)],and its class in the group of hypergeometric objects is where, as explained in the previous section, we identify the set S of one-dimensional subtori of G r m, k with the set of primitive b ∈ Z r .Therefore, we have For every b ∈ S and ξ ∈ Char k , the coefficient of (b, ξ) in this sum is for every non-zero b and every ξ ∈ Char k , so 0 = in the vector space V k for every non-zero b ∈ Z r , which contradicts the hypothesis that the elements ξ µ i =ηi ξ for i such that b i = b are linearly independent (since at least one c i is non-zero).Therefore Ψ 2 (K) = 0, and then lemma 1 implies that H 0 (K) = 0. We conclude that where A K := i dim H i (K)1, since K is mixed of weights ≤ |c|, being the convolution of ||c|| pure objects, i:ci>0 c i of them of weight 1 and i:ci<0 −c i of them of weight −1, and then H i (K) is mixed of weights ≤ |c| + i ≤ |c| − 1 for every i ≤ −1.
We now proceed to estimate the second summand of (4).We have and |G(η i χ ai )| ≤ √ q for every χ ∈ T and i = 1, . . ., n, so and |{χ|η i χ ai = 1}| ≤ a i (q − 1) r−1 , where a i = min j {|a ij | for j such that a ij = 0} (if a i = a ij0 , for every choice of χ j for j = j 0 there are at most a i choices for χ j0 such that η i χ ai = 1).Therefore In particular, we have |S| = |T |−|T \S| ≥ (q −1) r −a(q −1) r−1 .By (4) we conclude that, for q > 1 + a (which, in particular, implies S = ∅): which concludes the proof of the estimate and therefore of Theorem 1.

Independence of Gauss sums
In this section, we will apply the equidistribution theorem 1 to show that all (monomial) relations between Gauss sums that hold for "almost all" multiplicative characters are a combination of the Hasse-Davenport relation, the conjugation relation G(χ)G( χ) = χ(−1)q and the Galois invariance relation G(χ p ) = G(χ).
Let k = F q be a finite field of characteristic p as in the previous sections.Let r be a positive integer, and G the free abelian (multiplicative) group with basis the set {e η,a } indexed by the pairs (η, a) where η : k × → C × is a multiplicative character and a ∈ Z r a non-zero r-tuple.Every r-tuple χ = (χ 1 , . . ., χ r ) of multiplicative characters of k a group homomorphism ev χ : G → C × that maps e η,a to the Gauss sum G(ηχ a ).More generally, for every m ≥ 1 and every r-tuple χ of multiplicative characters of k m , we get a homomorphism ev m,χ : G → C × that maps e η,a to G m (ηχ a ).We define the following elements of G: (1) Given a character η : (where, in the last product, η is seen as a character on k) and and for every positive d|q − 1 and every r-tuple χ such that η d χ da = 1, we have by the Hasse-Davenport product formula (3).
Let H ⊆ G be the subgroup generated by the P (η, a), Q(η, a) and R(η, a, d) for every η : k × → C × , non-zero a ∈ Z r and d|q − 1.If x ∈ H, from the previous paragraph we deduce that there exists some constants D and n and some t ∈ (k × ) r such that for every m ≥ 1 and all χ except at most n of them, χ(t)ev m,χ (x) = D m : if x = P (η, a) we can take t = (−1) a and D = η(−1)q, if x = Q(η, a) we can take t = (1, . . ., 1) and D = 1, if x = R(η, a, d) we can take t = d da and D = η(d −d ) ξ d =1 G(ξ), and we conclude by multiplicativity.The main result of this section is a converse of this.Theorem 3. Let x ∈ G and assume that there exist an element t ∈ (k × ) r , a nonzero integer N , a subset U m ⊆ T m for every m ≥ 1 and a sequence of complex numbers {D m } m≥1 such that lim m→∞ |U m | q m = 1 and (χ(t)ev m,χ (x)) N = D m for every m ≥ 1 and every χ ∈ U m .Then x ∈ H.
The theorem says that any relation satisfied by Gauss sums associated to monomials for "almost all" r-tuples of multiplicative characters must be a combination of the identities (1), (2) and (3).

Proof. Let x =
n i=1 e ǫi ηi,ai with (η i , a i ) distinct and ǫ i ∈ Z\{0}, and let S m ⊆ T m , as in the introduction, be the set of χ such that η i χ ai := η i χ ai1 1 • • • χ air r = 1 for every i = 1, . . ., n.Then D m must have absolute value q mǫN/2 for sufficiently large m (large enough so that S m ∩ U m is non-empty), where ǫ = i ǫ i , since the Gauss sums associated to non-trivial characters of k m have absolute value q m/2 .Write a i = µ i b i for all i, where µ ∈ Z\{0} and b i ∈ Z r is primitive.
We claim that there exists some b ∈ Z r \{0} such that the elements ξ µ i =ηi ξ ∈ V k for the i = 1, . . ., n such that b i = b are linearly dependent.Otherwise, let u ∈ (k × m0 ) r be an element with coordinates in some finite extension k m0 of k such that u ǫ1 = t.By theorem 1 the elements (q −m/2 χ(u)G m (η 1 χ a1 ), q −m/2 G m (η 2 χ a2 ), . . ., q −m/2 G m (η n χ an )) for χ ∈ S m would become equidistributed on (S 1 ) n as m → ∞ (m being a multiple of m 0 ).Since the homomorphism (S 1 ) n → S 1 given by (t 1 , . . ., t n ) → n i=1 t ǫi i maps the Haar measure of (S 1 ) n to the Haar measure of S 1 , we conclude that the elements become equidistributed on S 1 as m → ∞, and then so do their N -th powers (since t → t N is a surjective homomorphism).This clearly contradicts the hypothesis that (almost all of) these N -th powers coincide.
Multiplying by suitable powers of elements of H of the form e η,a e η,−a , we may assume that µ i > 0 for all i.We now proceed by induction on µ := i (µ i − 1) = i µ i − n.If µ = 0, then µ i = 1 for all i.By the claim, there must be some b such that the elements η i ∈ V k for i such that b i = b are linearly dependent.But that can only happen if two of them coincide, which contradicts the distinctness of the (η i , a i ).
Let µ > 0. If some µ i is a multiple of p, we can multiply x by an element of H of the form e −1 η p ,pa e η,a , which decreases µ by p − 1, and proceed by induction.So we may assume that all µ i are prime to p.By the claim, there is some b such that the elements ξ µ i =ηi ξ ∈ V k for the i = 1, . . ., n such that b i = b are linearly dependent.Pick some non-trivial dependency relation, and let i be such that b i = b, ξ µ i =ηi ξ appears with non-zero coefficient in it, and µ i is the largest among the i's with these properties.
Suppose that, for some d|µ i , there were two different d-th roots of η i defined over k.Then their ratio is a non-trivial character of order e for some e|d which is defined over k.We deduce that all e-th roots of η i are defined over k (one is obtained by raising a d-th root to the d/e-th power, and then all others by multiplying by powers of the character of order e).We can then multiply x by the element e −1 ηi,µibi ξ e =ηi e ξ,(µi/e)bi ∈ H or its inverse, which decreases i (µ i − 1) by e − 1, and proceed by induction.
So we may assume that, for every d|µ i , there is at most one d-th root of η i defined over k.If there is a d-th root and a d ′ -th root, then there is a lcm(d, d ′ )-th root by Bézout, so there is some maximal d|µ i such that η i has a (unique) d-th root θ defined over k and, for every e|µ i , η i has an e-th root defined over k if and only if e|d, in which case the e-th root in question is θ d/e .Pick a character ξ 0 ∈ Char k such that ξ µi/d 0 = θ (in particular, ξ µi 0 = η i ) and a character ǫ ∈ Char k of order µ i .Then (ξ 0 ǫ) µi = η i , so ξ 0 ǫ appears with nonzero coefficient in ξ µ i =ηi ξ.By the linear dependence relation, it must appear in ξ µ j =ηj ξ for some other j = i with b j = b.By the distinctness of the (η i , a i ) we can not have µ i = µ j , so by the maximality of µ i we must have µ j < µ i .Since (ξ 0 ǫ) µi = η i and (ξ 0 ǫ) µj = η j are defined over k, so is (ξ 0 ǫ) µ0 where µ 0 = gcd(µ i , µ j ) < µ i .Then (ξ 0 ǫ) µ0 is a (µ i /µ 0 )-th root of η i defined over k, so (µ i /µ 0 )|d and (ξ 0 ǫ) µ0 = θ d/(µi/µ0) = (ξ µi/d 0 ) d/(µi/µ0) = ξ µ0 0 .Therefore ǫ µ0 is trivial, which contradicts the fact that ǫ has order µ i .

Independence of p
In this final section we will prove a version of Theorem 1 where we allow the fields over which the characters χ are defined to have different characteristics.
Let K ∈ D b c (P r k , Qℓ ).The complexity c(K) ∈ N of K is defined in [SFFK, Definition 3.2] as the maximum, for 0 ≤ s ≤ r, of the sum of the (geometric) Betti numbers of the restriction of K to a generic linear subspace of P r k of dimension s.More generally, for a quasi-projective variety X with an embedding u : X → P r k , the complexity of an object K ∈ D b c (X, Qℓ ) is c u (K) := c(u !K) [SFFK, Definition 6.3].We will consider X = G r m,k embedded in P r k in the natural way.The following lemma optimizes the upper bound on the complexity that follows from the general formalism in [SFFK].
Lemma 2. Let η : k × → C × be a character, a ∈ Z r a non-zero r-tuple and t ∈ (k × ) r .Let α : G m,k → G r m,k be the homomorphism of tori given by t → t a .Then the complexity of K t,a,η := δ t * α * H(ψ, η) ∈ D b c (G r m,k , Qℓ ) is bounded by 2 max i |a i |.
Proof.Since taking convolution with δ t is just appying a translation, which preserves the set of generic linear subspaces, we may assume t = 1.
In that case, K t,a,η is supported on a one-dimensional subtorus, so its restriction to a generic linear subspace of P r k of dimension s is empty except for s = r, r − 1.For s = r, since α is a finite map, we have H i (P r k, u !K t,a,η ) = H i c (G r m, k, K t,a,η ) = H i c (G m, k, H(ψ, χ)) which is one-dimensional for i = 0, and vanishes for all other i.

Corollary 7 .
The (arithmetic or geometric) tannakian group of M is the entire GL(1) n if and only if L * a1 1 * • • • * L * an n is (arithmetically or geometrically, repectively) non-trivial for every non-zero a ∈ Z n .
objects on G r m, k are of this form though, as they may arise from characters which are not defined over k).The isomorphism classes of hypergeometric objects of P erv on G r m, k form an abelian group under convolution [GL96, Corollaire 8.1.6].Let S be the set of one-dimensional subtori of G r m, k.We will identify it with the set of primitive r-tuples b ∈ Z r , the r-tuple b corresponding to the image of the embedding i b : G m, k → G r m, k given by t → t b .Let also C(G m, k) denote the set of continuous ℓ-adic characters of the tame fundamental group of G m, k.The subset of C(G m, k) consisting of finite order characters can be identified with the set Char k via the chosen embedding ι : Qℓ → C. Then by [GL96, Théorème 8.6.1]there is an isomorphism Ψ = (Ψ 1 , Ψ 2 ) between the group of isomorphism classes of hypergeometric objects on G r m, k and the product ( k× ) r

k
is the inclusion of an irreducible smooth subvariety of dimension d and F is an irreducible lisse sheaf on V [BBDG18, Théorème 4.3.1].If d > 0, then H 0 (j !* (F [d])) = 0 by [BBDG18, Corollaire 1.4.24].All negligible simple objects must have d > 0, since otherwise they would be punctual objects, which have Euler characteristic ≥ 1.
and a non-zero a ∈ Z r , let P (η, a) := e η,a e η,−a .(2) Given a character η : k × → C × and a non-zero a ∈ Z r , let Q(η, a) := e −1 η p ,pa e η,a .(3) Given a character η : k × → C × , a non-zero a ∈ Z r and a positive d|q − 1, let R(η, a, d) := e −1 η d ,da ξ d =1 e ηξ,a = e −1 η d ,da ξ d =η d e ξ,a .For every r-tuple χ of multiplicative characters of k m such that ηχ a = 1, we have ev m,χ (P (η, a)) = G m (ηχ a )G m (η χa ) = ηχ a (−1)q m = χ((−1) a )η(−1) m q m d ne be the distinct d i 's, each d nj appearing m j times.By Corollary 2, the elements Qℓ ) is perverse with Euler characteristic 1, and so is δ t * i * H(ψ, χ) = trans t * i * H(ψ, χ) for any t ∈ k × , where trans t : G r m,k → G r m,k is the translation map x → tx and δ t is the punctual object Qℓ supported on t [GL96, Proposition 8.1.3].By multiplicativity, any convolution of objects of this form is perverse with Euler characteristic 1.Such objects are called hypergeometric (not all hypergeometric