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Hodge theory of Kloosterman connections. (English) Zbl 1498.14019

This paper is concerned with a conjecture by D. Broadhurst [Commun. Number Theory Phys. 10, No. 3, 527–569 (2016; Zbl 1362.81072)]. In particular, the paper proves the conjecture about \(L\)-functions of symmetric power moments of Kloosterman sums.
Let \(p\) be a prime, and let \({\mathbb{F}}_p\) be the finite field with \(p\) elements, and \(\bar{\mathbb{F}}_p\) be its algebraic closure. If \(q\) is a power of \(p\), let \({\mathbb{F}}_q\) be the subfield of \(\bar{\mathbb{F}}_p\) with \(q\) elements, and let \({\mathrm{tr}}\) be the trace map from \(\mathbb{F}_q\to \mathbb{F}_p\). Fix a nontrivial additive character \(\psi: \mathbb{F}_p\to \mathbb{C}^{\times}\). For each \(a\in\mathbb{F}_q^{\times}\), the Kloosterman sum is the real number \(Kl_2(a;q)=\sum_{x\in\mathbb{F}^{\times}_q} \psi(\mathrm{tr}_{\mathbb{F}_q/\mathbb{F}_p}(x+a/x))\). By Weil, it is known that \(Kl_2(a;q)=-(\alpha_a+\beta_a)\) for some algebraic integers \(\alpha_a, \beta_a\) of absolute value \(\sqrt{q}\) with \(\alpha_a\beta_a=q\). For any integer \(k\geq 1\), the \(k\)-th symmetric power of Kloosterman sums is defined as \(Kl_2^{{\mathrm{Sym}}^k}(a;q)=\sum_{i=0}^k \alpha_a^i\beta_a^{k-i}\), and summing over over \(a\), the moments is defined as \(m_2^k(q)=\sum_{a\in{\mathbb{F}}_q^{\times}} Kl_2^{{\mathrm{Sym}}^k}(a;q)\). Form the zeta-function \(Z_k(p;T)=\exp(\sum_{n=1}^{\infty}m_2^k(p^n)\frac{T^n}{n})\). Then \(Z_k(p;T)\) is a polynomial which is always divisible by \(1-T\). Denote by \(M_k(p;T)\) the polynomial obtained from \(Z_k(p;T)\) removing the so-called trivial factors. Then all its roots have the absolute value \(p^{-(k+1)/2}\). The global \(L\)-function over \({\mathbb{Q}}\) is then defined by \(L_k(s)=\prod_{p\not\in S} M_k(p;s)\) where \(S\) is a set of bad primes appropriately defined depending on \(k\) is odd or even.
The main results are formulated in the following two theorems.
Theorem 1: Assume that \(k\) is odd. Let \(S\) be the set of odd primes smaller than or equal to \(k\). Then the function \(L_k(s)\) admits meromorphic continuation to the complex plane and satisfies the function equation \(\Lambda_k(s)=\Lambda_k(K-2-s)\).
Theorem 2: Assume that \(k\) is even (e.g., \(k=2m+4\) or \(2m+2\) with \(m\) even). Let \(S\) be the set of all primes smaller than or equal to \(k/2\). Then the function \(L_k(s)\) meromorphically extends to the complex plane. Morevoer, there exists a sign \(\varepsilon_k\in\{\pm 1\}\), an integer \(r_k\geq 0\), and a reciprocal of a polynomial with rational coefficients \(L_k(2;T)\) such that, setting \(\Lambda_k(s)=2^{r_ks/2}\Lambda_k(2;2^{-s})\Lambda_k^{\prime}(s)\) (where \(\Lambda_k(s)^{\prime}\) is an extension of \(L_k(s)\) outside the prime \(2\)), the following functional equation holds: \(\Lambda_k(s)=\varepsilon_k\Lambda_k(k+2-s)\).
Theorems 1 and 2 and the modulairty of \(L\)-functions are known for \(k\leq 8\).
The idea of proof is to give a cohomological interpretation for the \(L\)-functions. Let \(Kl_2\) denote the Kloosterman sheaf, which is a rank-\(2\) lisse sheaf on \(\mathbb{G}_{m,{\mathbb{F}}_p}\). Let \(\mathrm{Sym}^k Kl_2\) be the symmetric power of \(Kl_2\). Then \(Z_k(p,T)\) is given by the characteristic polynomial of the Frobenius \(F_p\) on \(H^1_{et,c}(\mathbb{G}_{m,\bar{\mathbb{F}}_p}, {\mathrm{Sym}}^k Kl_2)\). Removing the trivial factors from \(Z_p(p,T)\) amounts to replace the etale cohomology with compact support with middle extension cohomology \(H^1_{et,mid}(\mathbb{G}_{m, \bar{\mathbb{F}}_p}, {\mathrm{Sym}}^k Kl_2)\). Let \(M_k(p,T)\) be the characteristic polynomial of \(F_p\) on this middle extension cohomology.
Motives of symmetric powers of Kloosterman connections are constructed, and their Hodge numbers are computed in terms of the irregular Hodge filrtation on their relaizations as exponential mixed Hodge structures. It is shown that \(H^1(\mathbb{G}_m, \mathrm{Sym}^k Kl_2)\) has the mixed Hodge structure of weight at least \(k+1\), and that all Hodge numbers are either \(0\) or \(1\). This implies the potential automorphy of these motives.

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F80 Galois representations
11L05 Gauss and Kloosterman sums; generalizations
14F40 de Rham cohomology and algebraic geometry
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)

Citations:

Zbl 1362.81072
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References:

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