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A generalization of Ramanujan’s congruence to modular forms of prime level. (English) Zbl 1441.11100

Summary: We prove congruences between cuspidal newforms and Eisenstein series of prime level, which generalize Ramanujan’s congruence. Such congruences were recently found by N. Billerey and R. Menares [Math. Res. Lett. 23, No. 1, 15–41 (2016; Zbl 1417.11094)], and we refine them by specifying the Atkin-Lehner eigenvalue of the newform involved. We show that similar refinements hold for the level raising congruences between cuspidal newforms of different levels, due to K. A. Ribet [in: Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 503–514 (1984; Zbl 0575.10024)] and F. Diamond [Astérisque 196–197, 205–213 (1991; Zbl 0783.11022)]. The proof relies on studying the new subspace and the Eisenstein subspace of the space of period polynomials for the congruence subgroup \(\Gamma_0(N)\), and on a version of Ihara’s lemma.

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
11F03 Modular and automorphic functions

Software:

PARI/GP; Magma
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Full Text: DOI arXiv

References:

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