Computing $\mathcal{L}$-invariants via the Greenberg-Stevens formula

TL;DR

This paper presents a method to compute $p$-adic $ ext{L}$-invariants of modular forms using the Greenberg-Stevens formula and overconvergent modular forms, supported by computational evidence.

Contribution

It introduces a novel computational approach to determine $ ext{L}$-invariants for arbitrary weight and level, utilizing derivatives of the characteristic series of the $U_p$ operator.

Findings

Successfully computes $ ext{L}$-invariants for various weights and levels.

Provides computational evidence for relations between slopes of $ ext{L}$-invariants.

Develops an efficient algorithm for constructing the $ ext{L}$-invariant polynomial.

Abstract

In this article, we describe how to compute slopes of $p$-adic $\mathcal{L}$-invariants of arbitrary weight and level by means of the Greenberg-Stevens formula. Our method is based on work of Lauder and Vonk on computing the reverse characteristic series of the $U_p$ operator on overconvergent modular forms. Using higher derivatives of this characteristic series, we construct a polynomial whose zeros are precisely the $\mathcal{L}$-invariants appearing in the corresponding space of modular forms with fixed sign of the Atkin-Lehner involution at $p$. In addition, we describe how to compute this polynomial efficiently. In the final section, we give computational evidence for relations between slopes of $\mathcal{L}$-invariants for small primes.