The μ-Invariant of Isogenies

Abstract

Two abelian varieties A and B over a number field k which are isogeneous have the same Hasse-Weil L-function. This fact is an immediate consequence of the definition of these L-functions. But as was pointed out by B. Perrin-Riou by way of an example the (Iwasawa theoretic) p-adic L-functions of A and B, for a fixed p, may differ if the degree of the isogeny is divisible by p. A look at their definition shows that they differ at most by a power of p. In this paper we will give a complete and explicit determination of that constant in terms of the local behaviour of the kernel of the isogeny.