Selmer groups of symmetric powers of ordinary modular Galois representations

Abstract

Let $p$ be a fixed odd prime number, $\mu$ be a Hida family over the Iwasawa algebra of one variable, $\rho_{\mu}$ its Galois representation, $\mathbb{Q}_\infty/\mathbb{Q}$ the $p$-cyclotomic tower and $S$ the variable of the cyclotomic Iwasawa algebra. We compare, for $n\leq 4$ and under certain assumptions, the characteristic power series $L(S)$ of the dual of Selmer groups $\mathrm{Sel}(\mathbb{Q}_{\infty},\mathrm{Sym}^{2n}\otimes\mathrm{det}^{-n}\rho_{\mu})$ to certain congruence ideals. The case $n=1$ has been treated by H.Hida. In particular, we express the first term of the Taylor expansion at the trivial zero $S=0$ of $L(S)$ in terms of an $\mathcal{L}$-invariant and a congruence number. We conjecture the non-vanishing of this $\mathcal{L}$-invariant; this implies therefore that these Selmer groups are cotorsion. We also show that our $\mathcal{L}$-invariants coincide with Greenberg's $\mathcal{L}$-invariants calculated by R.Harron and A.Jorza.