Arithmetic of characteristic p special L-values

Abstract

Abstract. Recently the second author has associated a finite Fq[T ]-module

Arithmetic of function field units

“…We still denote by τ the continuous morphism of F (z)-algebras: τ ⊗ 1 : [7], proof of Theorem A and [9], Proposition 3.7, we deduce that…”

“…Let M be the projective limit for the norm map of the p-Sylow subgroups of the ideal class groups along the cyclotomic Z p -extension of Q(µ p ) : Q(µ p ∞ ). Let Γ = Gal(Q(µ p ∞ )/Q(µ p )), and let γ ∈ Γ such that: L. Taelman has recently introduced new arithmetic objects associated to Drinfeld modules ( [26]): class modules and modules of units; in the case of the Carlitz module, these latter objects have similar properties to that of the ideal class groups and units of number fields (see for example [7]). While Greenberg's Conjectures are "vertical" problems (one fixes a cyclotomic field and study the structures of the p-class groups along the cyclotomic Z p -extension), in this article we consider analogues of these problems for Taelman's class modules but in the "horizontal" case.…”

Twisting eigensystems of Drinfeld Hecke eigenforms by characters

“…[1, 1], [1,5], [2,2], [2,6], [2,10], [3,3], [3,7], [3,11], [3,15], [4,4], [4,8], [4,12], [4,16], [4,20], [5,1], [5,5], [6,2], [6,6], [6,10], [7,3], [7,7], [7,11], [7,15], [8,4], [8,8], [8,12], [8,16], [8,…”

Twisted characteristic p zeta functions

“…In particular, ∀n ∈ Z, the sum L P (n; t; z) := m≥0 ( a∈A+,deg θ a=m,a ≡0 (mod P ) a(t 1 ) · · · a(t s ) a n )z m ∈ T P defines an entire function on C s+1 P . We refer the reader to [7], [10], for various arithmetic properties of "special values" of the series L P (n; t; z), n ∈ Z.…”