Newton polygons of Hecke operators
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Newton polygons of Hecke operators Liubomir Chiriac1 · Andrei Jorza2 Received: 2 January 2020 / Accepted: 15 September 2020 © Fondation Carl-Herz and Springer Nature Switzerland AG 2020
Abstract In this computational paper we verify a truncated version of the Buzzard–Calegari conjecture on the Newton polygon of the Hecke operator T2 for all large enough weights. We first develop a formula for computing p-adic valuations of exponential sums, which we then implement to compute 2-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard–Calegari polynomial has a vertex at n ≤ 15, then it agrees with the Newton polygon of T2 up to n. Keywords Traces of Hecke operators · Slopes of modular forms Mathematics Subject Classification Primary: 11F33; Secondary: 11F30 · 11F85 Résumé Dans cet article, nous vérifions une version tronquée de la conjecture de Buzzard–Calegari concernant le polygone de Newton de l’opérateur Hecke T2 pour tous les poids suffisamment grands. Nous développons d’abord une formule pour les valuations p-adiques de sommes exponentielles, que nous utilisons ensuite pour calculer les valuations 2-adiques des traces d’opérateurs de Hecke agissant sur des espaces de formes cuspidales. Enfin, nous vérifions que si le polygone de Newton du polynôme de Buzzard–Calegari a un sommet en n ≤ 15, alors il coïncide avec le polygone de Newton de T2 jusqu’à n.
1 Introduction Let 2k ≥ 12 be an even number and let S2k be the finite-dimensional C-vector space of cusp forms of weight 2k on SL2 (Z). For a prime number p and f ∈ S2k , the action of the Hecke
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Liubomir Chiriac [email protected] Andrei Jorza [email protected]
1
Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR 97201, USA
2
Department of Mathematics, University of Notre Dame, 275 Hurley Hall, Notre Dame 46556, IN, USA
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L. Chiriac, A. Jorza
operator T p on f is given by (T p f )(z) =
p−1 1 z +r + p 2k−1 f ( pz). f p p r =0
Motivated by a question of Serre, Hatada [10] obtained several congruences modulo powers of 2 satisfied by the eigenvalues a2 of T2 , later improved by Emerton [8]. More precisely, among the normalized eigenforms in S2k the lowest 2-adic valuation of a2 is 3 (with multiplicity 1) if k ≡ 0 (mod 2), 4 (with multiplicity 1) if k ≡ 1 (mod 4), 5 (with multiplicity 1) if k ≡ 3 (mod 8), and 6 (with multiplicity 2) if k ≡ 7 (mod 8). Hatada’s congruences represent some of the first results concerning the 2-adic valuations of the eigenvalues of T2 , refered to as T2 -slopes. The list of slopes is determined by the 2-adic Newton polygon of the characteristic polynomial PT2 (X ) := det(1 − T2 X | S2k ) ∈ Z[X ], which is defined as the convex hull of the set of points (i, v2 (bi )), where v2 denotes the 2-adic valuation, and bi is the coefficient of X i in PT2 (X ). In [6] Buzzard and Calegari conjectured: Conjecture 1 (Buzzard–Calegari). If m = dim S2k then the polynomial PBC (X ) := 1 +
m
Xn
n=1
n 22 j (2k − 8
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