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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.

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11F11 Holomorphic modular forms of integral weight
11F25 Hecke-Petersson operators, differential operators (one variable)
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[1] DOI: 10.1016/j.jnt.2006.11.012 · Zbl 1144.11036 · doi:10.1016/j.jnt.2006.11.012
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