Selmer groups of symmetric powers of ordinary modular Galois representations

TL;DR

This paper investigates the structure of Selmer groups associated with symmetric powers of ordinary modular Galois representations, relating their characteristic series to congruence ideals and $ ext{L}$-invariants, and extends previous results for $n=1$ to $n\, ext{up to}\,4$.

Contribution

It compares characteristic power series of Selmer groups to congruence ideals for symmetric powers, extending Hida's work and relating $ ext{L}$-invariants to Greenberg's invariants.

Findings

Expresses the first term of the Taylor expansion of $L(S)$ at $S=0$ in terms of $ ext{L}$-invariant and congruence number.

Conjectures the non-vanishing of the $ ext{L}$-invariant, implying cotorsion of Selmer groups.

Shows $ ext{L}$-invariants coincide with those of Greenberg, as computed by Harron and Jorza.

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.