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

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.