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Congruences for modular forms of weights two and four. (English) Zbl 1144.11036
F. Calegari and W. Stein [in: Algorithmic number theory. 6th international symposium, ANTS-VI, Burlington, VT, USA, June 13–18, 2004. Proceedings. Berlin: Springer. Lecture Notes in Computer Science 3076, 140–152 (2004; Zbl 1125.11320)] stated several conjectures about discriminants of Hecke algebras of prime level. In the paper under review, the authors prove the following one:
Let \(p\) be a prime, \(p \geq 5\), and \(\Theta\) the Ramanujan operator. Let \(\mathcal{P}\) be the maximal ideal of \(\overline{\mathbb Z}_p\). Suppose that \(f \in S_2(\Gamma_0(p),\overline{\mathbb Z}_p)\) and \(g \in S_4 (\Gamma_0(p),\overline{\mathbb Z}_p)\) are Hecke eigenforms. Assume
\( \Theta \,\, f \equiv g \pmod{\mathcal{P}}.\)
Then the Atkin-Lehner eigenvalues of \(f\) and \(g\) have opposite signs.
They also mention another proof indicated by F. Calegari based on a theorem of C. Breul and A. Mezard [Duke Math. J. 115, No. 2, 205–310 (2002; Zbl 1042.11030)]. The proof given in the paper under consideration employs elementary, but very tricky arguments based on special properties of \(\Theta\), the quasi modular form \(E_2\), Eisenstein series of weight \(2\), and Serre’s and Swinnerton-Dyer’s theory of modular forms modulo \(p\).

11F33 Congruences for modular and \(p\)-adic modular forms
11F11 Holomorphic modular forms of integral weight
Full Text: DOI
[1] Breuil, C.; Mézard, A., Multiplicités modulaires et représentations de \(\operatorname{GL}_2(\mathbf{Z}_p)\) et de \(\operatorname{Gal}(\overline{\mathbf{Q}}_p / \mathbf{Q}_p)\) en \(l = p\), Duke math. J., 115, 2, 205-310, (2002), MR1944572 (2004i:11052)
[2] Calegari, F.; Stein, W.A., Conjectures about discriminants of Hecke algebras of prime level, (), 140-152, MR2137350 · Zbl 1125.11320
[3] Diamond, F.; Im, J., Modular forms and modular curves, (), 39-133, MR1357209 (97g:11044) · Zbl 0853.11032
[4] Lang, S., Introduction to modular forms, (1995), Springer Berlin, Corrected reprint of the 1976 original, MR1363488 (96g:11037)
[5] Serre, J.-P., Formes modulaires et fonctions zêta p-adiques, (), 191-268, MR0404145 (53 #7949a)
[6] Shimura, G., Introduction to the arithmetic theory of automorphic functions, (1994), Princeton Univ. Press Princeton, NJ, Reprint of the 1971 original, MR1291394 (95e:11048)
[7] Swinnerton-Dyer, H.P.F., On l-adic representations and congruences for coefficients of modular forms, (), 1-55, MR0406931 (53 #10717a) · Zbl 1017.34042
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