On the asymptotic growth of Bloch-Kato-Shafarevich-Tate groups of modular forms over cyclotomic extensions

TL;DR

This paper investigates the growth of Bloch-Kato-Shafarevich-Tate groups associated with non-ordinary modular forms over cyclotomic extensions, providing bounds linked to Iwasawa invariants and extending known results from elliptic curves.

Contribution

It establishes upper bounds for these groups in the non-ordinary case, generalizing formulas known for supersingular elliptic curves using p-adic Hodge theory.

Findings

Upper bounds for Shafarevich-Tate groups in non-ordinary case

Extension of elliptic curve results to modular forms

Connection with Iwasawa invariants and p-adic Hodge theory

Abstract

We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.