We propose a new heuristic approach to integral moments of
-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
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
-function and they
match the predictions of the Conry, Farmer, Keating, Rubinstein, and Snaith recipe (Proc. Lond.Math. Soc. 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
-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
-functions
in the large degree limit over function fields.
Keywords
L-functions, moments, function fields, polynomials, finite
fields, integral moments
