Abstract
In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigenforms. We also show that there is an automorphic L-function over \({\mathbb{Q}}\) whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms.
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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). Am. Math. Soc., Providence, R.I., pp. 1–25 (1971)
Cartier, P.: Groupes formels, fonctions automorphes et fonctions zeta des courbes elliptiques. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, pp.291–299
Fang, L., Hoffman, J.W., Linowitz, B., Rupinski, A., Verrill, H.: Modular forms on noncongruence subgroups and Atkin–Swinnerton-Dyer relations (preprint, 2005)
Langlands R.P. (1980). Base Change for GL(2), Annals of Mathematics Studies, vol.96. Princeton University Press, Princeton
Li W.-C.W., Long L., Yang Z. (2005). On Atkin and Swinnerton-Dyer congruence relations. J Number Theory 113(1): 117–148
Livné R. (1987). Cubic exponential sums and Galois representations. Comtemp. Math. 67: 247–261
Scholl A.J. (1985). Modular forms and de Rham cohomology; Atkin–Swinnerton-Dyer congruences. Invent. Math. 79(1): 49–77
Scholl A.J. (1988). The l-adic representations attached to a certain noncongruence subgroup. J. Reine Angew. Math. 392: 1–15
Scholl, A.J.: On some l-adic representations of galois group attached to noncongruence subgroups. http://arXiv.org/abs/math/0402111(2004)
Serre, J.P.: Résumé de cours 1984-5. Collège de France
Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971, Kanô Memorial Lectures, No. 1
Shioda T. (1972). On elliptic modular surfaces. J. Math. Soc. Jpn. 24: 20–59
Stienstra J., Beukers F. (1985). On the Picard–Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces. Math. Ann. 271(2): 269–304
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The research of the second author was supported in part by an NSA grant #MDA904-03-1-0069 and an NSF grant #DMS-0457574. Part of the research was done when she was visiting the National Center for Theoretical Sciences in Hsinchu, Taiwan. She would like to thank the Center for its support and hospitality. The third author was supported in part by an NSF-AWM mentoring travel grant for women. She would further thank the Pennsylvania State University and the Institut des Hautes Études Scientifiques for their hospitality.
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Atkin, A.O.L., Li, WC.W. & Long, L. On Atkin and Swinnerton-Dyer congruence relations (2). Math. Ann. 340, 335–358 (2008). https://doi.org/10.1007/s00208-007-0154-7
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DOI: https://doi.org/10.1007/s00208-007-0154-7