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We generalize some results of Greither and Popescu to a geometric Galois cover X → Y which appears naturally for example in extensions generated by p n -torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz–Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in [1]) with Goss ζ -function. [ABSTRACT FROM AUTHOR]

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]