Average values of L-series for real characters in function fields

Abstract

We establish asymptotic formulae for the first and second moments of quadratic Dirichlet L-functions, at the center of the critical strip, associated to the real quadratic function field k( √ P) and inert imaginary quadratic function field k( γ P) with P being a monic irreducible polynomial over a fixed finite field F q of odd cardinality q and γ a generator of F × q . We also study mean values for the class number and for the cardinality of the second K -group of maximal order of the associated fields for ramified imaginary, real, and inert imaginary quadratic function fields over F q . One of the main novelties of this paper is that we compute the second moment of quadratic Dirichlet L-functions associated to monic irreducible polynomials. It is worth noting that the similar second moment over number fields is unknown. The second innovation of this paper comes from the fact that, if the cardinality of the ground field is even then the task of average L-functions in function fields is much harder and, in this paper, we are able to handle this strenuous case and establish several mean values results of L-functions over function fields.

Conjectures for the Integral Moments and Ratios of L-functions in Even characteristic

“…When the denominator of u is a monic irreducible polynomial, Andrade, Bae and Jung [1] obtained asymptotic formulas for the first and second moments of L 1 2 , χ u when the sum is over all u ∈ I g+1 , u ∈ F g+1 and u ∈ F ′ g+1 .…”

“…for a positive constant c k . Conrey and Ghosh [14] presented the constant c k in a more explicit form and Keating and Snaith [34] conjectured a precise formula for c k for R(s) > − 1 2 based on the analogy with the characteristic polynomials of random matrices. Conrey, Farmer, Keating, Rubinstein and Snaith [12] described an algorithm for obtaining explicit expressions for lower terms for the conjectured full asymptotics of the moments of the Riemann zeta function.…”

Nonvanishing of hyperelliptic zeta functions over finite fields

“…Then, as in [11, Section 7] (more or less immediate from a theorem of Romagny and Wewers [19]), there is a scheme Hn c G,n over Z[1/|G|] whose k-points (as long as k has characteristic prime to |G|) are in bijection with the isomorphism classes of tame G-covers of P 1 which have n branch points on A 1 with monodromy type c. (We do not specify whether or how the cover is branched at ∞.) 2 In fact ([19, Theorem 2.1]), for a scheme S, the set Hn c G,n (S) corresponds to isomorphism classes of tame G-covers over S, suitably defined; we will not need to spell out that definition here. Once the n branch points are chosen on A 1 there are finitely many choices for f and φ.…”

“…By the proof of Corollary 2.6, we may take C ′ ℓ to be C ℓ /(ℓ − 1), where C ℓ is the constant appearing in the statement of Proposition 2.1. Moreover, we may take C ℓ to be 2K ℓ B ℓ and Q ℓ to be 4B 2 ℓ , where B ℓ , K ℓ are the constants appearing in the proof of Proposition 2.1 controlling the exponential growth of the Betti numbers of the relevant Hurwitz space. We now explain how to bound B ℓ explicitly.…”

“…Let x 1 , ..., x 2g+2 be the set of Weierstrass points of C. The 2-torsion subgroup J(C) [2] is spanned by the degree-0 2-torsion divisors x i − x j . That is, the group of divisors of the form ∑ a i x i with ∑ a i = 0 surjects onto J(C) [2]. Note also that x 1 + .…”