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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 $$p$$.

##### MSC:
 11F33 Congruences for modular and $$p$$-adic modular forms 11F11 Holomorphic modular forms of integral weight
##### Keywords:
modular forms mod $$p$$; congruences; Atkin-Lehner operator
Full Text:
##### References:
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