On the level raising of cuspidal eigenforms modulo prime powers
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On the level raising of cuspidal eigenforms modulo prime powers Emiliano Torti1 Received: 9 January 2019 / Accepted: 25 April 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this article, we prove level raising for cuspidal eigenforms modulo prime powers (for odd primes) of weight k ≥ 2 and arbitrary character, extending the result in weight two established by the work of Tsaknias and Wiese and generalizing (partially) Diamond– Ribet’s celebrated level-raising theorems. Keywords Congruences of modular forms · Modular Galois representations · Level raising Mathematics Subject Classification 11F33 · 11F80
1 Introduction The problem of raising the level of newforms modulo some power of a prime p is part of the study of congruences between modular forms of different levels. By congruences between modular forms we mean congruences between the lth coefficient in the qexpansion where l runs over all rational primes except a finite number. The level-raising phenomenon (modulo a prime p) was extensively studied in the past thirty years and it was definitely understood in the classical case thanks to the work of Ribet (see [25]) in weight two and trivial character, Diamond (see [12]) for weight k ≥ 2 and general characters, and Diamond–Taylor (see [13]). In particular, we are interested in the following: Theorem 1.1 (Ribet, Diamond) Let f be a newform of weight k ≥ 2, level N , character χ and let p be a prime not dividing N . If l is a prime not dividing pN and p 1 2 ϕ(N )Nl(k − 2)!, then the following are equivalent: (a) there exists a l-newform of level l N , say g, such that f ≡ g mod p
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Emiliano Torti [email protected]; [email protected] University of Luxembourg, Esch-sur-Alzette, Luxembourg
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E. Torti
(b) al2 ≡ χ (l)l k−2 (l + 1)2 mod p. Here, the symbol p denotes a prime ideal of the coefficient field of f lying above the rational prime p, the symbol ϕ denotes Euler’s totient function and al denotes the lth coefficient of the q-expansion of f . The fundamental idea of the existence of a non-trivial congruence module introduced and developed by Ribet in a geometric context (considering Jacobians attached to modular curves of the form X 0 (N )) was refined in cohomological terms by Diamond (see [11]). Proving the existence of a congruence module, whose non-triviality is granted by the level-raising condition (i.e. the condition (b) in the above theorem), is the heart of both the proofs of Ribet and Diamond, and its cohomological construction (together with a cohomological version of Ihara’s lemma) will allow us to apply Ribet’s analysis in a slightly more general context. As a natural extension of studying level-raising modulo p, one could ask if the same holds modulo p n for some positive integer n and if the natural generalization of the level-raising condition is the suitable one in the prime powers setting. More specifically, given a newform of level N whose lth coefficient (for a prime l not dividing pN ) satisfies a certain level of raising condition
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