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Selmer groups of symmetric powers of ordinary modular Galois representations

Xiaoyu Zhang

Abstract

Let $p$ be a fixed odd prime number, $μ$ be a Hida family over the Iwasawa algebra of one variable, $ρ_μ$ 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 $\mathrm{Sel}(\mathbb{Q}_{\infty},\mathrm{Sym}^{2n}\otimes\mathrm{det}^{-n}ρ_μ)$ 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.

Selmer groups of symmetric powers of ordinary modular Galois representations

Abstract

Let be a fixed odd prime number, be a Hida family over the Iwasawa algebra of one variable, its Galois representation, the -cyclotomic tower and the variable of the cyclotomic Iwasawa algebra. We compare, for and under certain assumptions, the characteristic power series of the dual of Selmer groups to certain congruence ideals. The case has been treated by H.Hida. In particular, we express the first term of the Taylor expansion at the trivial zero of in terms of an -invariant and a congruence number. We conjecture the non-vanishing of this -invariant; this implies therefore that these Selmer groups are cotorsion. We also show that our -invariants coincide with Greenberg's -invariants calculated by R.Harron and A.Jorza.

Paper Structure

This paper contains 11 sections, 40 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. $R=T$ theorems for definite unitary groups
  3. Some base changes
  4. Cyclic base change of automorphic representations
  5. Base change of deformation functors
  6. Selmer groups and congruence ideals
  7. Preliminary on commutative algebra
  8. Cotorsionness of Selmer groups
  9. Case $\mathrm{Sel} (\mathbb{Q}_{\infty},\mathcal{A}^1(\rho_\mu))$
  10. Case $\mathrm{Sel} (\mathbb{Q}_{\infty},\mathcal{A}^{n-1}(\rho_\mu))$ ($n>2$)
  11. Comparison of $\mathcal{L}$-invariants

Key Result

Theorem \oldthetheorem

Assume $\textbf{Big}(\overline{\rho}^{G_n}_\pi)$, $\textbf{Dist}(\overline{\rho}^{G_n}_\pi)$, $\textbf{RegU}(\overline{\rho}^{G_n}_\pi)$. Then the (fractional) congruence ideal $\tilde{\lambda}^\circ_0(\mathfrak{c}_{\tilde{\theta}^{G_n}_0})/\tilde{\lambda}^\circ_0(\mathfrak{c}_{\tilde{\theta}^{G_{n-

Theorems & Definitions (74)

  • Theorem \oldthetheorem: HidaTilouine
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: Proposition 2.7.2, Corollary 2.7.8 of Geraghty
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem: Proposition 2.7.4 of Geraghty, Proposition 2.8 of HidaTilouine
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • ...and 64 more