On the congruence ideal associated to $p$-adic families of Yoshida lifts

TL;DR

This paper investigates the relationship between $p$-adic families of Yoshida lifts and Selmer groups, establishing divisibility results for characteristic ideals and exploring implications for Iwasawa theory under certain hypotheses.

Contribution

It demonstrates that classical Yoshida lifts persist across Hida families and links the congruence ideal to Selmer group characteristic ideals, advancing understanding in $p$-adic automorphic forms and Iwasawa theory.

Findings

All classical specializations are Yoshida lifts under certain conditions.

The characteristic ideal of a Selmer group divides the Yoshida lift's congruence ideal.

Under additional assumptions, the dual Selmer group is pseudo-cyclic over the cyclotomic extension.

Abstract

We study congruences involving $p$-adic families of Hecke eigensystems of Yoshida lifts associated with two Hida families (say $\mathcal{F},\mathcal{G}$) of elliptic cusp forms. With appropriate hypotheses, we show that if a Hida family of genus two Siegel cusp forms admits a Yoshida lift at an appropriately chosen classical specialization, then all classical specializations are Yoshida lifts. Moreover, we prove that the characteristic ideal of the non-primitive Selmer group of (a self-dual twist of) the Rankin--Selberg convolution of $\mathcal{F}$ and $\mathcal{G}$ is divisible by the congruence ideal of the Yoshida lift associated with $\mathcal{F}$ and $\mathcal{G}$. Under an additional assumption inspired by pseudo-nullity conjectures in higher codimension Iwasawa theory, we establish the pseudo-cyclicity of the dual of the primitive Selmer group over the cyclotomic $\mathbb{Z}_p$-extension.