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, $\Bbb{Q}_\infty/\Bbb{Q}$ 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 $\textrm{Sel}(\Bbb{Q}_{\infty},\textrm{Sym}^{2n}\otimes\textrm{det}^{-n} \rho_{\mu})$ 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 $\scr{L}$-invariant and a congruence number. We conjecture the non-vanishing of this $\scr{L}$-invariant; this implies therefore that these Selmer groups are cotorsion. We also show that our $\scr{L}$-invariants coincide with Greenberg's $\scr{L}$-invariants calculated by R. Harron and A. Jorza.