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mod \(p\) congruences for cusp forms of weight four for \(\Gamma_0(pN)\). (English) Zbl 1273.11076
For a prime \(p\geq 5\), let \(\overline{\mathbb Z}_p\) be the ring of integers in an algebraic closure of the field of \(p\)-adic numbers, with \({\mathfrak p}\) its maximal ideal. Suppose that \(f\in S_2(\Gamma_0(pN), \overline{\mathbb Z}_p)\) and \(g\in S_4(\Gamma_0(pN), \overline{\mathbb Z}_p)\) are cusp forms of level \(pN\) and weights 2 and 4, respectively, over the ring \(\overline{\mathbb Z}_p\). Let \(W= W_p\) be a Fricke involution acting on modular forms, and commuting with all Hecke operators \(T_n\) such that \(\gcd(n, pN)= 1\).
The authors consider the following question: Let \(f\) and \(g\) be newforms satisfying the congruence \(\Theta f\equiv g\pmod{\mathfrak p}\), where \(\Theta= q{d\over dq}\) is the usual derivation operator. Is it true that the eigenvalues of \(W\) for \(f\) and \(g\) have opposite signs ? For \(N=1\) the answer is “yes” as conjectured by S. Calegari and W. A. Stein [Algorithmic number theory. 6th international symposium, ANTS-VI, Lect. Notes Comput. Sci. 3076, 140–152 (2004; Zbl 1125.11320)] and proved by S. Ahlgren and M. Barcau [J. Number Theory 126, No. 2, 193–199 (2007; Zbl 1144.11036)].
The authors show that the answer “yes” remains valid for \(N>4\) if \(f\) satisfies a condition on what is called its filtration, and they give examples showing that the answer can be “yes” or “no” if this condition is not satisfied.

11F33 Congruences for modular and \(p\)-adic modular forms
11F11 Holomorphic modular forms of integral weight
11F25 Hecke-Petersson operators, differential operators (one variable)
Full Text: DOI
[1] DOI: 10.1016/j.jnt.2006.11.012 · Zbl 1144.11036 · doi:10.1016/j.jnt.2006.11.012
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[3] DOI: 10.1007/978-3-540-24847-7_10 · doi:10.1007/978-3-540-24847-7_10
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[6] DOI: 10.1215/S0012-7094-90-06119-8 · Zbl 0743.11030 · doi:10.1215/S0012-7094-90-06119-8
[7] DOI: 10.1007/BF01344466 · Zbl 0278.10026 · doi:10.1007/BF01344466
[8] DOI: 10.1007/BFb0072985 · doi:10.1007/BFb0072985
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