On Shafarevich-Tate groups and analytic ranks in families of modular forms, II. Coleman families

TL;DR

This paper investigates the algebraic and analytic ranks in $p$-adic families of modular forms, establishing conditions under which ranks and Shafarevich-Tate groups behave predictably across specializations, supporting Greenberg's conjecture.

Contribution

It proves new results relating ranks and Shafarevich-Tate groups in Coleman families of modular forms, providing evidence for Greenberg's conjecture on analytic ranks.

Findings

Finite $p$-primary Shafarevich-Tate groups for specializations under certain conditions.

Analytic ranks of specializations match those of the original form for almost all primes.

Results support the conjecture relating ranks in families of modular forms.

Abstract

This is the second article in a two-part project whose aim is to study algebraic and analytic ranks in $p$-adic families of modular forms. Let $f$ be a newform of weight $2$, square-free level $N$ and trivial character, let $A_f$ be the abelian variety attached to $f$, whose dimension will be denoted by $d_f$, and for every prime number $p\nmid N$ let $\boldsymbol f^{(p)}$ be a $p$-adic Coleman family through $f$ over a suitable open disc in the $p$-adic weight space. We prove that, for all but finitely many primes $p$ as above, if $A_f(\mathbb Q)$ has rank $r\in\{0,d_f\}$ and the $p$-primary part of the Shafarevich-Tate group of $A_f$ over $\mathbb Q$ is finite, then all classical specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have finite $p$-primary Shafarevich-Tate group and $r/d_f$-dimensional image of the relevant $p$-adic \'etale Abel-Jacobi map. As a second contribution, assuming the non-degeneracy of certain height pairings \`a la Gillet-Soul\'e between Heegner cycles, we show that, for all but finitely many $p$, if $f$ has analytic rank $r\in\{0,1\}$, then all classical specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have analytic rank $r$. This result provides some evidence for a conjecture of Greenberg on analytic ranks in families of modular forms.