New phenomena arising from L-invariants of modular forms

TL;DR

This paper introduces a practical method for computing L-invariants of p-new eigenforms using p-adic L-series, providing extensive data analysis and new insights into their distribution and relation to Galois representations.

Contribution

It presents a novel algorithm for computing L-invariants sensitive to Galois representations and offers a large dataset for analyzing their distribution and underlying structures.

Findings

Conjecture of a statistical law for L-invariant valuations.

Large dataset of over 150,000 L-invariants compiled.

New perspectives on L-invariants related to Galois representations.

Abstract

This article explains how to practically compute L-invariants of p-new eigenforms using p-adic L-series and exceptional zero phenomena. As proof of the utility, we compiled a data set consisting of over 150,000 L-invariants. We analyze qualitative and quantitative features found in the data. This includes conjecturing a statistical law for the distribution of the valuations of L-invariants in a fixed level as the weights of eigenforms approach infinity. One novel point of our investigation is that the algorithm is sensitive to compiling data for fixed Galois representations modulo p. Therefore, we explain new perspectives on L-invariants that are related to Galois representations. We propose understanding the structures in our data through the lens of deformation rings and moduli stacks of Galois representations.