$\mathcal L$-invariants and deep congruences between newforms

TL;DR

This paper investigates deep congruences between $p$-new modular forms of level $ ext{Gamma}_0(p)$, proposing a conjecture linking these congruences to the $ ext{L}$-invariant and Atkin--Lehner signs, supported by explicit computations and theoretical evidence.

Contribution

It introduces a novel conjecture relating congruences of modular forms to their $ ext{L}$-invariants and Atkin--Lehner signs, supported by explicit computational evidence.

Findings

Conjecture that eigenforms have 'twins' with high power congruences linked to $ ext{L}$-invariants.

Explicit computational evidence supporting the conjecture.

Formulation of a local conjecture on congruences between semistable Galois representations.

Abstract

We study congruences modulo powers of a prime $p$ between pairs of $p$-new modular Hecke eigenforms of level $\Gamma_0(p)$ and same weight $k$. Based on explicit computations, we conjecture that every such eigenform $f$ admits a twin to which it is congruent modulo a surprisingly high power of $p$, whose exponent is close to the opposite of the valuation of the $\mathcal L$-invariant of $f$, and whose Atkin--Lehner sign is opposite to that of $f$. This is a new phenomenon that is not explained by the known results on the $p$-adic variation of eigenforms. Inspired by the global picture, we formulate a local conjecture describing congruences between semistable representations of fixed weight, varying $\mathcal L$-invariant, and opposite Atkin--Lehner signs. We give some theoretical evidence towards our conjectures.