Iwasawa Invariants for Symmetric Square Representations

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 L-functions 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 L-function, 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.