Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On $λ$-invariants of congruent modular forms in the anticyclotomic, indefinite setting

Dac-Nhan-Tam Nguyen

TL;DR

The paper develops a precise, computable formula for comparing $λ$-invariants of congruent modular forms in the anticyclotomic indefinite setting where Selmer groups have positive rank, improving previous results that contained incomputable error terms. By leveraging residual Galois data and Greenberg-type local conditions, it derives the identity $λ(f_1) + 2 \sum_{ℓ|N_1N_2} λ_ℓ(f_1) = λ(f_2) + 2 \sum_{ℓ|N_1N_2} λ_ℓ(f_2)$ under the hypotheses $\bar{\rho}_{f_1} \simeq \bar{\rho}_{f_2}$ and $μ(f_1)=μ(f_2)=0$, provided $X(K, \mathbf{A}_{f_i})$ has no finite nonzero submodules. The key innovation is expressing the comparison in terms of residual data and local Euler factors $λ_ℓ(f_i)$, avoiding the problematic error terms noted in HL21, and the paper substantiates the theory with a concrete elliptic-curve example where the equality is numerically verified. This contributes to a more explicit understanding of Iwasawa invariants in the anticyclotomic setting and informs applications to $p$-adic $L$-functions and congruence families of modular forms.

Abstract

We give a precise, computable formula for comparing $λ$-invariants between modular forms in the anticyclotomic indefinite setting where the Selmer groups have positive rank. This is an improvement of Hatley-Lei \cite{HL19, HL21} where the authors give a formula with incomputable error terms.

On $λ$-invariants of congruent modular forms in the anticyclotomic, indefinite setting

TL;DR

The paper develops a precise, computable formula for comparing -invariants of congruent modular forms in the anticyclotomic indefinite setting where Selmer groups have positive rank, improving previous results that contained incomputable error terms. By leveraging residual Galois data and Greenberg-type local conditions, it derives the identity under the hypotheses and , provided has no finite nonzero submodules. The key innovation is expressing the comparison in terms of residual data and local Euler factors , avoiding the problematic error terms noted in HL21, and the paper substantiates the theory with a concrete elliptic-curve example where the equality is numerically verified. This contributes to a more explicit understanding of Iwasawa invariants in the anticyclotomic setting and informs applications to -adic -functions and congruence families of modular forms.

Abstract

We give a precise, computable formula for comparing -invariants between modular forms in the anticyclotomic indefinite setting where the Selmer groups have positive rank. This is an improvement of Hatley-Lei \cite{HL19, HL21} where the authors give a formula with incomputable error terms.
Paper Structure (5 sections, 6 theorems, 41 equations)

This paper contains 5 sections, 6 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Comparing $\lambda$-invariants of congruent modular forms
  4. Numerical example
  5. (by Jeffrey Hatley and Antonio Lei)

Key Result

Theorem 1.1

Let $f_1 \in S_{2r_1}(\Gamma_0(N_1)), f_2 \in S_{2r_2}(\Gamma_0(N_2))$ be modular forms that satisfy Heeg, adm and irr. Assume that $\bar{\rho}_{f_1} \simeq \bar{\rho}_{f_2}$ and $\mu(f_1) = \mu(f_2) = 0$. Moreover, assume that both $X(K, \mathbf{A}_{f_1}), X(K, \mathbf{A}_{f_2})$ do not have any f where $\lambda_\ell(f_i)$ are local constants defined in Definition defn:local-lambda.

Theorems & Definitions (21)

  • Theorem 1.1: Theorem \ref{['thm:lambda-inv']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 11 more