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

Abstract

In this article, we describe how to compute slopes of p-adic ℒ-invariants of Hecke eigenforms of arbitrary weight and level by means of the Greenberg–Stevens formula. Our method is based on the 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 series, we construct a polynomial whose roots are precisely the 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 ℒ-invariants of different levels and weights for small primes p.