Iwasawa invariants for symmetric square representations

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