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

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

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

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.

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 $p$.

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

[1] Atkin, A. Oliver L.; Lehner, Joseph Hecke operators on ${\Gamma }_0\left(m\right)$, Math. Ann., Volume 185 (1970), pp. 134-160 | DOI | Zbl

[2] Barré-Sirieix, Katia; Diaz, Guy; Gramain, François; Philibert, Georges Une preuve de la conjecture de Mahler–Manin, Invent. Math., Volume 124 (1996) no. 1-3, pp. 1-9 | DOI | MR | Zbl

[3] Bellaïche, Joël Eigenvarieties, families of Galois representations, $p$-adic $L$-functions (2010) (unpublished course notes)

[4] Bellaïche, Joël Critical $p$-adic $L$-functions, Invent. Math., Volume 189 (2012), pp. 1-60 | DOI | MR | Zbl

[5] Bergdall, John Upper bounds for constant slope $p$-adic families of modular forms, Sel. Math., New Ser., Volume 25 (2019) no. 4, 59, 24 pages | MR | Zbl

[6] Buzzard, Kevin Questions about slopes of modular forms, Automorphic forms (I) (Astérisque), Volume 298, Société Mathématique de France, 2005, pp. 1-15 | Numdam | Zbl

[7] Buzzard, Kevin Eigenvarieties, $L$-functions and Galois representations (London Mathematical Society Lecture Note Series), Volume 320, Cambridge University Press, 2007, pp. 59-120 | MR | Zbl

[8] Chenevier, Gaëtan Quelques courbes de Hecke se plongent dans l’espace de Colmez, J. Number Theory, Volume 128 (2008) no. 8, pp. 2430-2449 | DOI | MR | Zbl

[9] Coleman, Robert F. Classical and overconvergent modular forms, Invent. Math., Volume 124 (1996) no. 1-3, pp. 215-241 | DOI | MR | Zbl

[10] Coleman, Robert F. $p$-adic Banach spaces and families of modular forms, Invent. Math., Volume 127 (1997) no. 3, pp. 417-479 | DOI | MR | Zbl

[11] Coleman, Robert F.; Stevens, Glenn; Teitelbaum, Jeremy T. Numerical experiments on families of $p$-adic modular forms, Computational perspectives on number theory (Chicago, IL, 1995) (AMS/IP Studies in Advanced Mathematics), Volume 7, American Mathematical Society, 1995, pp. 143-158 | DOI | Zbl

[12] Colmez, Pierre Invariants $ℒ$ et dérivées de valeurs propres de Frobenius, Représentations $p$-adiques de groupes $p$-adiques III: Méthodes globales et géométriques (Astérisque), Volume 331, Société Mathématique de France, 2010, pp. 13-28 | Numdam | Zbl

[13] Gouvêa, Fernando Quadros; Mazur, Barry On the characteristic power series of the $U$ operator, Ann. Inst. Fourier, Volume 43 (1993) no. 2, pp. 301-312 | DOI | Numdam | MR | Zbl

[14] Gräf, Peter M. A control theorem for $p$-adic automorphic forms and Teitelbaum’s $ℒ$-invariant, Ramanujan J., Volume 50 (2019) no. 1, pp. 13-43 | DOI | MR | Zbl

[15] Greenberg, Ralph; Stevens, Glenn $p$-adic $L$-functions and $p$-adic periods of modular forms, Invent. Math., Volume 111 (1993) no. 2, pp. 407-447 | DOI | MR | Zbl

[16] Guitart, Xavier; Masdeu, Marc Periods of modular ${\mathrm{GL}}_2$-type abelian varieties and $p$-adic integration, Exp. Math., Volume 27 (2018) no. 3, pp. 344-361 | DOI | Zbl

[17] Lauder, Alan G. B. Computations with classical and $p$-adic modular forms, LMS J. Comput. Math., Volume 14 (2011), pp. 214-231 | DOI | MR | Zbl

[18] Martin, Kimball Refined dimensions of cusp forms, and equidistribution and bias of signs, J. Number Theory, Volume 188 (2018), pp. 1-17 | DOI | MR | Zbl

[19] Mazur, Barry; Tate, John T.; Teitelbaum, Jeremy T. On $p$-adic analogues of the conjectures of Birch and Swinnerton–Dyer, Invent. Math., Volume 84 (1986), pp. 1-48 | DOI | MR | Zbl

[20] Teitelbaum, Jeremy T. Values of $p$-adic $L$-functions and a $p$-adic Poisson kernel, Invent. Math., Volume 101 (1990) no. 2, pp. 395-410 | DOI | MR | Zbl

[21] Vonk, Jan Computing overconvergent forms for small primes, LMS J. Comput. Math., Volume 18 (2015), pp. 250-257 | DOI | MR | Zbl