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Roots of L-Functions of Characters Over Function Fields, Generic

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Roots of L-Functions of Characters Over Function Fields, Generic Roots of L-functions of characters over function fields, generic linear independence and biases Corentin Perret-Gentil Abstract. We first show joint uniform distribution of values of Kloost- erman sums or Birch sums among all extensions of a finite field Fq, for ˆ almost all couples of arguments in Fq , as well as lower bounds on dif- ferences. Using similar ideas, we then study the biases in the distribu- tion of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick–Waxman and Keating–Rudnick, building on cohomological in- terpretations and determinations of monodromy groups by Katz. Our results are based on generic linear independence of Frobenius eigenval- ues of ℓ-adic representations, that we obtain from integral monodromy information via the strategy of Kowalski, which combines his large sieve for Frobenius with a method of Girstmair. An extension of the large sieve is given to handle wild ramification of sheaves on varieties. Contents 1. Introduction and statement of the results 1 2. Kloosterman sums and Birch sums 12 3. Angles of Gaussian primes 14 4. Prime polynomials in short intervals 20 5. An extension of the large sieve for Frobenius 22 6. Generic maximality of splitting fields and linear independence 33 7. Proof of the generic linear independence theorems 36 References 37 1. Introduction and statement of the results arXiv:1903.05491v2 [math.NT] 17 Dec 2019 Throughout, p will denote a prime larger than 5 and q a power of p. ˆ 1.1. Kloosterman and Birch sums. For an integer n 1, and a Fqn , we consider the Kloosterman sums ě P 1 tr x1 xr Kl n a e (1) r,q npr´1q{2 p `¨¨¨` q p q“ q ˆ p x ,...,xrPF ˆ ˙ 1 ÿ qn x1...xr“a Date: May 2019. Revised: September 2019. 2010 Mathematics Subject Classification. 14G10, 11T23, 11R58, 11J72, 11N36. 1 2 Roots of L-functions over function fields of integer rank r 2, as well as the Birch sums ě 1 tr ax x3 Biqn a e p ` q . (2) p q“ qn{2 p xPFˆ ˆ ˙ ÿqn Here, we adopt the usual notation e z exp 2πiz for any z C, and p q “ p q P tr : Fqn Fp is the field trace. Ñ For convenience, let us define the rank of Biqn to be r 2, and for r 2, we let “ ě fqn Klr,qn r 2 or Biqn r 2 (3) “ p ě q p “ q for every integer n 1. By the Deligne–Katz equidistribution theorem [Kat88] for Kloostermaně sums and Livné’s work [Liv87] for Birch sums (see also [Kat90]), as qn the values Ñ8 ˆ r, r R : r even fqn a : a F n equidistribute in Ωr r´ sĂ t p q P q u “ z C : z r : r odd, #t P | |ď u with respect to the pushforward tr˚ µr of the Haar measure µr on the com- pact group SUr : r odd Gr C , where Gr : p q “ #USpr : r even (e.g. the Sato-Tate measure when r 2). These statements encompass “ bounds on fqn (e.g. Deligne’s bound for hyper-Kloosterman sums), and the fact that fqn is real-valued whenever r is even. Moreover, they can alternatively be phrased as properties of the “angles” of Kloosterman and Birch sums, i.e. the θ1,f,q x ,...,θr,f,q x 0, 1 , such that r p q p q P r s ˆ fqn x e nθi,f,q x for all n 1, x F (4) p q“ p p qq ě P q i“1 ÿ (whose existence follows from deep work of Grothendieck, Deligne, Katz and others, and will be recalled in due time): they are distributed like the eigen- values of a Haar-random matrix in Gr C . p q Our first main result is the following generic linear independence state- ment: Theorem 1.1 (Generic pairwise linear independence). For r 2 fixed, let f be as in (3), and let ě 2r2`r´3 rank Gr 2 : r odd Er : dim Gr 2r2`3r “ ` 2 “ # 4 : r even. For almost all a, b Fˆ, that is for P q 2 log q 2 q 1 1 Or,p q 1 1 or,p 1 (5) p ´ q ` q1{p2Erq “ p ´ q p ` p qq „ ˆ ˙ Roots of L-functions over function fields 3 of them, the angles 1 j r 1 : r odd 1, θj,f,q a , θj,f,q b with ď ď ´ (6) p q p q 1 j r 2 : r even # ď ď { are Q-linearly independent. The implied constants depend only on r, p, and only on r in the case of Kloosterman sums. r Remark 1.2. The restriction on j in (6) is necessary since θj,f,q x 0, j“1 p q“ and if r is even, the angles come by pairs: θ x θj,f,q x (1 j r{2`j,f,qp q“´ř p q ď ď r 2). { Remark 1.3. Actually, we will more generally prove Theorem 1.1 for almost all tuples of t 1 arguments, when t o ?log q (e.g. t fixed), with (5) replaced by1 ě “ p q δr odd t t r C log q q 1 1 Or,p p q (7) p ´ q ` q1{ptErq ˆ ˆ ˙˙ for an absolute constant C 1. The implied constants depends again only on r in the case of Kloostermaně sums. This has several interesting consequences. First, we obtain the joint dis- tribution of almost all pairs of values of f in extensions of a fixed base field: Corollary 1.4. For r 2, let f be as in (3), a Kloosterman or Birch ě 2 ´1{p2Erq ˆ sum. For all but Or,p q 1 log q q couples a, b Fq , the random vector p ´ q p q P ` ˘ Xa,b fqn a ,fqn b “ p p q p qq 1ďnďN (with the uniform measure on´ 1,N N) converges¯ in law as N to r sX Ñ8 tr g1 , tr g2 , p q p q with g1, g2 independent uniformly´ distributed¯ in a maximal torus of Gr C . p q Explicitly, tr gi is distributed like p q r{2 j“1 2 cos 2πθj : r even r´1 p q r´1 (8) # j 1 e θj e j 1 θj : r odd, ř “ p q` p´ “ q with θj independent uniformř in 0, 1 . Equivalently,ř the distribution of tr gi m r s p q is that of tr hi for any m r and hi uniform in Gr C with respect to the Haar measure.p q The impliedě constant in Landau’s notationp q depends only on r in the case of Kloosterman sums. Remark 1.5. Applying Deligne’s equidistribution theorem and [Kat88, Kat90] would show that fqn a b1 ,...,fqn a bt converges in law p ` q p ` q aPFqn , a`bi‰0 n t (with respect to the´ uniform measure), as q ¯ , to a random vector in Ωr Ñ8 bt distributed with respect to the product measure tr µr , when bi Fqn p ˚ q P 1 Here and from now on, δB will denote the Kronecker symbol with respect to a binary δr odd variable B, i.e. δB “ 1 if B is true, 0 otherwise. In particular, r is equal to r if the latter is odd, and to 1 otherwise. 4 Roots of L-functions over function fields ˆ / ˆ / ˆ / ˆ ˆ F (fixed) F F n F m F p q q mě1 q “ q P a, b Ť Figure 1. The asymptotic setting for Section 1.1. are t o log qn distinct shifts (see e.g. [PG17], where the dependencies of the“ errorsp p fromqq [FKM15] with respect to t are made explicit). However, this only gives information among values that are explicitly related, by fixed shifts. Remark 1.6 (Discrepancy). For the distribution of a single Kloosterman sum of rank 2, conditionally on a linear independence hypothesis, Ahmadi and Shparlinski [AS10] also obtained bounds on the discrepancy, using lower bounds arising from Baker’s theorem. Their results are stated for curves, but the last paragraph of [AS10, Section 5.2] explains how they readily extend to Kloosterman sums. Our Theorem 1.1 shows that their discrepancy bounds hold for almost all arguments, and using the same technique, a bound on the discrepancy in Corollary 1.4 could as well be given. Another corollary is the following absence of bias among values of Birch sums and Kloosterman sums in extensions: Corollary 1.7. Let fqn be either Klr,qn (r 2 even), Biqn , or – if r 3 ě 2 ´1{p2Eěrq is odd – ReKlr,qn or ImKlr,qn . For all but Or,p q 1 log q q p ´ q p q couples a, b Fˆ, we have P q ` ˘ 1 n N : fqn a fqn b Pn N fqn a fqn b : |t ď ď p qă p qu| ď p qă p q “ N ´ ¯ 1 2 as N . Ñ { Ñ8 The implied constant in Landau’s notation depends only on r in the case of Kloosterman sums. Finally, Theorem 1.1 also yields the following lower bounds, through the method of Bombieri and Katz [BK10]. The first one is not explicit and the value of n is not effective, while the second is weaker but does not suffer from these issues.
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