Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras
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Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras Emmanuel Lecouturier1
Received: 18 May 2018 / Accepted: 20 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020 −1 Abstract Let N and p be primes such that p divides the numerator of N12 . In this paper, we study the rank g p of the completion of the Hecke algebra acting on cuspidal modular forms of weight 2 and level 0 (N ) at the p-maximal Eisenstein ideal. We give in particular an explicit criterion to know if g p ≥ 3, thus answering partially a question of Mazur. In order to study g p , we develop the theory of higher Eisenstein elements, and compute the first few such elements in four different Hecke modules. This has applications such as generalizations of the Eichler mass formula in characteristic p.
Contents 1 Introduction and results . . . . . . . . . . . . . 1.1 Modular forms . . . . . . . . . . . . . . . . 1.2 The supersingular module . . . . . . . . . . 1.3 Odd modular symbols . . . . . . . . . . . . 1.4 Even modular symbols . . . . . . . . . . . 1.5 Comparison . . . . . . . . . . . . . . . . . 2 The formalism of higher Eisenstein elements . . 2.1 Algebraic setting . . . . . . . . . . . . . . 2.2 The Newton polygon of TP . . . . . . . . 2.3 Pairing between higher Eisenstein elements 2.4 The special case of weight 2 and prime level 3 The supersingular module . . . . . . . . . . . .
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B Emmanuel Lecouturier
[email protected]
1
Yau Mathematical Sciences Center and Tsinghua University, Beijing, China
123
E. Lecouturier 3.1 Preliminary results and notation . . . . . . . . . . . . . . . . . . . . . . . 3.2 Overview of Sect. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Hasse polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The discriminant of the Hasse polynomial . . . . . . . . . . . . . . . 3.3.2 Relation between the Hasse polynomial and the modular polynomials 3.4 The supersingular module of Legendre elliptic curves . . . . . . . . . . . . 3.5 The case p ≥ 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 The case p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 The case p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Conjectural identities satisfied by the supersingular lambda invariants . . . 3.9 Eisenstein ideals of level 0 (N ) ∩ (2) . . . . . . . . . . . . . . . . . . . 4 Odd modular symbols . . . . . . . . . . . . . . . . . . .
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