On the congruence ideal associated to $p$-adic families of Yoshida lifts
Ming-Lun Hsieh, Bharathwaj Palvannan
TL;DR
The paper develops a comprehensive framework linking p-adic families of Yoshida lifts to Iwasawa-theoretic main conjectures for Rankin–Selberg products. By constructing and exploiting a refined GSp4-Hida theory, generalized matrix algebras, and pseudocharacters, it establishes rigidity and existence/uniqueness results for Yoshida lifts within p-adic families and relates congruence ideals to the divisors of non-primitive Selmer groups. Under suitable hypotheses, it proves lower bounds (and in favorable cases equalities) between Selmer-divisors and p-adic L-function divisors, and demonstrates pseudo-cyclicity of dual primitive Selmer groups in a cyclotomic Iwasawa-theory setting. These results illuminate the interplay between automorphic congruences, Higman-type deformation theory, and higher-codimension phenomena in Iwasawa theory, contributing to a broader understanding of the Iwasawa–Greenberg main conjectures in the Rankin–Selberg context.
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.