Tamagawa defect of Euler systems

TL;DR

This paper investigates how Tamagawa numbers affect the primitiveness of Euler systems derived from p-adic Galois representations, providing bounds on the cokernel size and refining prior results to account for Tamagawa factors.

Contribution

It establishes a lower bound for the cokernel of the Euler system to Kolyvagin system map in terms of Tamagawa numbers, refining previous theoretical results.

Findings

Provides a bound on the cokernel size related to Tamagawa numbers.

Refines Kolyvagin system theory to include Tamagawa factors.

Partially explains missing Tamagawa factors in Kato's Euler system calculations.

Abstract

As remarked in [Kolyvagin systems, by Barry Mazur and Karl Rubin] Proposition 6.2.6 and Buyukboduk[ arXiv:0706.0377v1 ] Remark 3.25 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 [Kolyvagin systems, by Barry Mazur and Karl Rubin] Definition 4.5.5) if p divides a Tamagawa number at a prime \ell different from 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 (see Theorem 3.2.4 of [Kolyvagin systems, by Barry Mazur and Karl Rubin] for a definition of this map) in terms of the Tamagawa numbers of T, refining [Kolyvagin systems, by Barry Mazur and Karl Rubin] Propostion 6.2.6. We show how this partially accounts for the missing Tamagawa factors in Kato's calculations with his Euler system.