Algebra & Number Theory

A representation theory approach to integral moments of $L$-functions over function fields

Will Sawin

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We propose a new heuristic approach to integral moments of L-functions over function fields, which we demonstrate in the case of Dirichlet characters ramified at one place (the function field analogue of the moments of the Riemann zeta function, where we think of the character nit as ramified at the infinite place). We represent the moment as a sum of traces of Frobenius on cohomology groups associated to irreducible representations. Conditional on a hypothesis on the vanishing of some of these cohomology groups, we calculate the moments of the L-function and they match the predictions of the Conry, Farmer, Keating, Rubinstein, and Snaith recipe (Proc. Lond. Math. Soc. (3) 91 (2005), 33–104).

In this case, the decomposition into irreducible representations seems to separate the main term and error term, which are mixed together in the long sums obtained from the approximate functional equation, even when it is dyadically decomposed. This makes our heuristic statement relatively simple, once the geometric background is set up. We hope that this will clarify the situation in more difficult cases like the L-functions of quadratic Dirichlet characters to squarefree modulus. There is also some hope for a geometric proof of this cohomological hypothesis, which would resolve the moment problem for these L-functions in the large degree limit over function fields.

Article information

Algebra Number Theory, Volume 14, Number 4 (2020), 867-906.

Received: 17 October 2018
Revised: 12 December 2019
Accepted: 6 February 2020
First available in Project Euclid: 30 June 2020

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Mathematical Reviews number (MathSciNet)

Primary: 11M50: Relations with random matrices
Secondary: 11M38: Zeta and $L$-functions in characteristic $p$ 11T55: Arithmetic theory of polynomial rings over finite fields 14F20: Étale and other Grothendieck topologies and (co)homologies 20G05: Representation theory

L-functions moments function fields polynomials finite fields integral moments


Sawin, Will. A representation theory approach to integral moments of $L$-functions over function fields. Algebra Number Theory 14 (2020), no. 4, 867--906. doi:10.2140/ant.2020.14.867.

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