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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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##### References:
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