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Congruence primes for cusp forms of weight \(k\geq 2\). (English) Zbl 0783.11022

Courbes modulaires et courbes de Shimura, C. R. Sémin., Orsay/Fr. 1987-88, Astérisque 196-197, 205-213 (1991).
[For the entire collection see Zbl 0745.00052.]
This paper generalizes to forms of higher weight a result of K. A. Ribet [Proc. Int. Congr. Math. 1983, Vol. 1, 501-514 (1984; Zbl 0575.10024)] on congruence relations between newforms of different level: for a newform \(f=\sum a_ nq^ n\) of weight \(k\geq 2\), character \(\chi\) and level \(N\), and primes \(p\), \(l\) satisfying \(l\mid Np\) and \(p\mid{1\over 2}\varphi(N)Nl(k-2)!\), there exists a newform \(g\) of the same weight and character, but of level \(dl\) for some divisor \(d\) of \(N\), which is congruent to \(f\bmod p\), if and only if \(a_ l^ 2\equiv \chi(l)l^{k-2}(l+1)^ 2\bmod p\).
For the proof, the author works with certain parabolic cohomology groups instead of subgroups of the Jacobian of a modular curve. In the same way as Ribet’s result, this theorem can be used to lower the level of certain modular representations [cf. H. Carayol, Duke Math. J. 59, 785-801 (1989; Zbl 0703.11027)].

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
11F11 Holomorphic modular forms of integral weight
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