Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Constructing Galois representations with large Iwasawa $λ$-Invariant

Anwesh Ray

Abstract

Let $p\geq 5$ be a prime. We construct modular Galois representations for which the $\mathbb{Z}_p$-corank of the $p$-primary Selmer group (i.e., $λ$-invariant) over the cyclotomic $\mathbb{Z}_p$-extension is large. More precisely, for any natural number $n$, one constructs a modular Galois representation such that the associated $λ$-invariant is $\geq n$. The method is based on the study of congruences between modular forms, and leverages results of Greenberg and Vatsal. Given a modular form $f_1$ satisfying suitable conditions, one constructs a congruent modular form $f_2$ for which the $λ$-invariant of the Selmer group is large. A key ingredient in acheiving this is the Galois theoretic lifting result of Fakruddin-Khare-Patrikis, which extends previous work of Ramakrishna. The results are subject to certain additional hypotheses, and are illustrated by explicit examples.

Constructing Galois representations with large Iwasawa $λ$-Invariant

Abstract

Let be a prime. We construct modular Galois representations for which the -corank of the -primary Selmer group (i.e., -invariant) over the cyclotomic -extension is large. More precisely, for any natural number , one constructs a modular Galois representation such that the associated -invariant is . The method is based on the study of congruences between modular forms, and leverages results of Greenberg and Vatsal. Given a modular form satisfying suitable conditions, one constructs a congruent modular form for which the -invariant of the Selmer group is large. A key ingredient in acheiving this is the Galois theoretic lifting result of Fakruddin-Khare-Patrikis, which extends previous work of Ramakrishna. The results are subject to certain additional hypotheses, and are illustrated by explicit examples.

Paper Structure

This paper contains 5 sections, 10 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminary notions
  3. Congruent Galois Representations
  4. Lifting Galois Representations
  5. Modular deformations with large $\lambda$-invariant

Key Result

Lemma 3.3

Let $f_1$ and $f_2$ be as above. Then $f_1$ is $p$-ordinary if and only if $f_2$ is $p$-ordinary.

Theorems & Definitions (26)

  • Conjecture 2.4
  • Definition 3.1
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Corollary 3.5
  • proof
  • Theorem 3.6
  • Corollary 3.7
  • proof
  • ...and 16 more