Computing $ℒ$-invariants via the Greenberg–Stevens formula

Anni, Samuele; Böckle, Gebhard; Gräf, Peter; Troya, Álvaro

Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 727-746.

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 $p$.

Dans cet article, nous montrons comment calculer les pentes des invariants-$ℒ$ $p$-adiques de formes modulaires de niveaux et de poids arbitraires en appliquant la formule de Greenberg–Stevens. Notre méthode repose sur les travaux de Lauder et Vonk sur le calcul de la série caractéristique réciproque de l’opérateur $U_p$ sur les formes modulaires surconvergentes. En utilisant les dérivées supérieures de cette série, nous construisons un polynôme dont les racines sont exactement les invariants-$ℒ$ apparaissant dans l’espace correspondant des formes modulaires de signe fixé pour l’action de l’involution d’Atkin–Lehner en $p$. En outre, nous montrons comment calculer ce polynôme efficacement. Dans la dernière section, pour des petits nombres premiers $p$, nous donnons des évidences numériques en faveur de l’existence des relations entre les pentes des invariants-$ℒ$ de différents niveaux et poids.

Classification: 11F03,  11F85
Keywords: classical and $p$-adic modular forms

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