Iwasawa Invariants for Symmetric Square Representations
Abstract
Let $p\geq 5$ be a prime, and $\mathfrak{p}$ a prime of $\bar{\mathbb{Q}}$ above $p$. Let $g_1$ and $g_2$ be $\mathfrak{p}$ordinary, $\mathfrak{p}$distinguished and $p$stabilized cuspidal newforms of nebentype characters $\epsilon_1, \epsilon_2$ respectively, and weight $k\geq 2$, whose associated newforms have level prime to $p$. Assume that the residual representations at $\mathfrak{p}$ associated to $g_1$ and $g_2$ are absolutely irreducible and isomorphic. Then, the imprimitive $p$adic Lfunctions associated with the symmetric square representations are shown to exhibit a congruence modulo $\mathfrak{p}$. Furthermore, the analytic and algebraic Iwasawa invariants associated to these representations of the $g_i$ are shown to be related. Along the way, we give a complete proof of the integrality of the $\mathfrak{p}$adic Lfunction, normalized with Hida's canonical period. This fills a gap in the literature, since, despite the result being widely accepted, no complete proof seems to ever have been written down. On the algebraic side, we establish the corresponding congruence for Greenberg's Selmer groups, and verify that the Iwasawa main conjectures for the twisted symmetric square representations for $g_1$ and $g_2$ are compatible with the congruences.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 DOI:
 10.48550/arXiv.2111.14304
 arXiv:
 arXiv:2111.14304
 Bibcode:
 2021arXiv211114304R
 Keywords:

 Mathematics  Number Theory;
 11R23
 EPrint:
 Accepted for publication in Research in the Mathematical Sciences