Euler characteristic and Akashi series for Selmer groups over global function fields

Abstract

Let A be an abelian variety defined over a global function field F of positive characteristic p and let K/F be a p-adic Lie extension with Galois group G. We provide a formula for the Euler characteristic χ(G, Sel A (K)p) of the p-part of the Selmer group of A over K. In the special case G = Z d p and A a constant ordinary variety, using Akashi series, we show how the Euler characteristic of the dual of Sel A (K)p is related to special values of a p-adic L-function.

On Euler characteristics of Selmer groups for abelian varieties over global function fields

“…Proof. Since we are assuming that Sel A t (F ) ℓ is finite and ψ K is surjective, then equation (6) shows that Coker(ψ G K ) ≃ H 1 (G, Sel A (K) ℓ ). Therefore…”

“…Let F be a global function field of characteristic p > 0 and let K/F be an ℓ-adic Lie extension (ℓ = p) with Galois group G and unramified outside a finite and nonempty set S of primes of F . The case ℓ = p, which requires a few more technical tools related to flat cohomology, will be treated in a different paper [6].…”