Tamagawa defect of Euler systems

Abstract

As remarked by Mazur and Rubin [B. Mazur, K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (799) (2004)] one does not expect the Kolyvagin system obtained from an Euler system for a p-adic Galois representation T to be primitive (in the sense of the above mentioned reference) if p divides a Tamagawa number at a prime ℓ ≠ p ; thus fails to compute the correct size of the relevant Selmer module. In this paper we obtain a lower bound for the size of the cokernel of the Euler system to Kolyvagin system map in terms of the local Tamagawa numbers of T, refining a result of [B. Mazur, K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (799) (2004)]. We show how this partially accounts for the missing Tamagawa factors in Kato's calculations with his Euler system.