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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 (truncated) Euler characteristic χ ( G , S e l A ( K ) p ) of the p -part of the Selmer group of A over K . In the special case G ≃ Z p d and A a constant ordinary variety, using Akashi series, we show how the Euler characteristic of the dual of S e l A ( K ) p is related to special values of a p -adic L -function. [ABSTRACT FROM AUTHOR]

Let F be a global function field of characteristic p > 0 and A/F an abelian variety. Let K/F be an ℓ-adic Lie extension ( ℓ ≠ p) unramified outside a finite set of primes S and such that Gal( K/F) has no elements of order ℓ. We shall prove that, under certain conditions, Sel( K) has no nontrivial pseudo-null submodule. [ABSTRACT FROM AUTHOR]