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

Abstract

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 $n^{it}$ 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. $\left(3\right)$ 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.