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Modular forms, de Rham cohomology and congruences. (English) Zbl 1337.14021
Authors’ abstract: In this paper we show that Atkin and Swinnerton-Dyer type of congruences hold for weakly modular forms (modular forms that are permitted to have poles at cusps). Unlike the case of original congruences for cusp forms, these congruences are nontrivial even for congruence subgroups. On the way we provide an explicit interpretation of the de Rham cohomology groups associated to modular forms in terms of “differentials of the second kind”. As an example, we consider the space of cusp forms of weight 3 on a certain genus zero quotient of Fermat curve \(X^N+Y^N=Z^N\). We show that the Galois representation associated to this space is given by a Grössencharacter of the cyclotomic field \( \mathbb{Q}(\zeta _N)\). Moreover, for \(N=5\) the space does not admit a “\(p\)-adic Hecke eigenbasis” for (nonordinary) primes \(p\equiv 2,3 \pmod {5}\), which provides a counterexample to Atkin and Swinnerton-Dyer’s original speculation.
Independently, J. Kibelbek [Proc. Am. Math. Soc. 142, No. 12, 4029–4038 (2014; Zbl 1309.11039)] has given an example of a space of weight 2 modular forms that does not admit a \(p\)-adic Hecke eigenbasis (also for nonordinary primes \(p\)).

MSC:
14F40 de Rham cohomology and algebraic geometry
11F33 Congruences for modular and \(p\)-adic modular forms
11F80 Galois representations
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References:
[1] Atkin, A. O. L.; Li, Wen-Ching Winnie; Long, Ling, On Atkin and Swinnerton-Dyer congruence relations. II, Math. Ann., 340, 2, 335-358, (2008) · Zbl 1157.11015
[2] Atkin, A. O. L.; Swinnerton-Dyer, H. P. F., Modular forms on noncongruence subgroups. Combinatorics, Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968, 1-25, (1971), Amer. Math. Soc., Providence, R.I.
[3] \bibBEWbook author=Berndt, Bruce C., author=Evans, Ronald J., author=Williams, Kenneth S., title=Gauss and Jacobi sums, series=Canadian Mathematical Society Series of Monographs and Advanced Texts, pages=xii+583, publisher=John Wiley & Sons, Inc., New York, date=1998, isbn=0-471-12807-4, review=\MR 1625181 (99d:11092),
[4] Cartier, P., Groupes formels, fonctions automorphes et fonctions zeta des courbes elliptiques. Actes du Congr\`es International des Math\'ematiciens, Nice, 1970, 291-299, (1971), Gauthier-Villars, Paris
[5] Deligne, Pierre, \'Equations diff\'erentielles \`a points singuliers r\'eguliers, Lecture Notes in Mathematics, Vol. 163, iii+133 pp., (1970), Springer-Verlag, Berlin-New York · Zbl 0244.14004
[6] Ditters, Bert, Sur les congruences d’Atkin et de Swinnerton-Dyer, C. R. Acad. Sci. Paris S\'er. A-B, 282, 19, Ai, A1131-A1134, (1976) · Zbl 0342.14019
[7] Katz, Nicholas M., \(p\)-adic properties of modular schemes and modular forms. Modular functions of one variable, III, Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972, 69-190, (1973), Lecture Notes in Mathematics, Vol. 350, Springer, Berlin
[8] Kibelbek, Jonas, On Atkin and Swinnerton-Dyer congruences for noncongruence modular forms, Proc. Amer. Math. Soc., 142, 12, 4029-4038, (2014) · Zbl 1309.11039
[9] Li, Wen-Ching Winnie; Long, Ling; Yang, Zifeng, On Atkin-Swinnerton-Dyer congruence relations, J. Number Theory, 113, 1, 117-148, (2005) · Zbl 1083.11027
[10] Long, Ling, On Atkin and Swinnerton-Dyer congruence relations. III, J. Number Theory, 128, 8, 2413-2429, (2008) · Zbl 1175.11021
[11] Rohrlich, David E., Points at infinity on the Fermat curves, Invent. Math., 39, 2, 95-127, (1977) · Zbl 0357.14010
[12] Scholl, A. J., Modular forms and de Rham cohomology; Atkin-Swinnerton-Dyer congruences, Invent. Math., 79, 1, 49-77, (1985) · Zbl 0553.10023
[13] Scholl, A. J., Motives for modular forms, Invent. Math., 100, 2, 419-430, (1990) · Zbl 0760.14002
[14] Shimura, Goro, Introduction to the arithmetic theory of automorphic functions, xiv+267 pp., (1971), Kan\^o Memorial Lectures, No. 1, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J. · Zbl 0872.11023
[15] Weil, Andr\'e, Jacobi sums as “Gr\"ossencharaktere”, Trans. Amer. Math. Soc., 73, 487-495, (1952) · Zbl 0048.27001
[16] Yang, Tonghai, Cusp forms of weight \(1\) associated to Fermat curves, Duke Math. J., 83, 1, 141-156, (1996) · Zbl 0860.11021
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