an:05225254
Zbl 1144.11036
Ahlgren, Scott; Barcau, Mugurel
Congruences for modular forms of weights two and four
EN
J. Number Theory 126, No. 2, 193-199 (2007).
00211956
2007
j
11F33 11F11
modular forms mod \(p\); congruences; Atkin-Lehner operator
\textit{F. Calegari} and \textit{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 \textit{C. Breul} and \textit{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 \(p\).
Bernhard Heim (Bonn)
Zbl 1125.11320; Zbl 1042.11030