Asymptotic formula for Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic extensions

TL;DR

This paper derives an asymptotic formula for the size of the $p$-primary Tate-Shafarevich groups of certain supersingular elliptic curves over anticyclotomic extensions, advancing understanding of their growth in this setting.

Contribution

It provides a new asymptotic formula for Tate-Shafarevich groups of supersingular elliptic curves over anticyclotomic extensions under specific hypotheses.

Findings

Asymptotic growth rate of Tate-Shafarevich groups established

Results apply to elliptic curves with supersingular reduction at $p$

Advances understanding of arithmetic invariants in anticyclotomic towers

Abstract

Let $p\ge 5$ be a prime number and $E/\mathbf{Q}$ an elliptic curve with good supersingular reduction at $p$. Under the generalized Heegner hypothesis, we investigate the $p$-primary subgroups of the Tate--Shafarevich groups of $E$ over number fields contained inside the anticyclotomic $\mathbf{Z}_p$-extension of an imaginary quadratic field where $p$ splits.