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| Atkin–Swinnerton-Dyer congruences | |
|---|---|
| Name | Atkin–Swinnerton-Dyer congruences |
| Field | Number theory |
| Introduced | 1971 |
| Key people | A. O. L. Atkin, Peter Swinnerton-Dyer, Jean-Pierre Serre, Robert Langlands |
Atkin–Swinnerton-Dyer congruences are congruence relations discovered by A. O. L. Atkin and Peter Swinnerton-Dyer that connect Fourier coefficients of noncongruence modular forms with arithmetic data arising from algebraic varieties, Galois representations, and L-functions. The congruences were motivated by computational experiments in the early 1970s and led to links between the work of Jean-Pierre Serre, Robert Langlands, Nicholas Katz, and Barry Mazur on modularity, p-adic properties, and arithmetic geometry. These relations have influenced research around the Langlands program, Fontaine–Mazur conjecture, and the study of congruence subgroups in the theory of modular curves.
Atkin and Swinnerton-Dyer formulated congruences for Fourier coefficients in the setting of noncongruence subgroups of the modular group and modular curves, contrasting with classical results for congruence subgroups related to Hecke operators and Eichler–Shimura. Their observations involved comparisons with coefficients coming from newforms studied by Atkin–Lehner and Hecke eigenforms, and prompted interactions with the work of Deligne, Grothendieck, and Iwasawa on p-adic properties and Galois actions. The congruences concern primes p and often involve weight, level, and Nebentypus characters as in the theory developed by Shimura and Taniyama–Shimura.
The genesis of the congruences lies in computational studies by Atkin and Swinnerton-Dyer motivated by algorithms from Leech and numerical explorations connected to Ramanujan’s tau-function and investigations inspired by Petersson questions. Early numerical evidence paralleled conceptual advances by Serre on modular forms mod p and by Mazur on congruences between modular forms, while Deligne provided conceptual underpinnings via l-adic representations. Connections to the modularity theorem and later work by Wiles and Taylor–Wiles put these congruences into a broader context involving Galois representations and arithmetic geometry of elliptic curves studied at Cambridge and Princeton research groups.
Roughly, for a noncongruence subgroup Γ and a holomorphic form f on Γ with q-expansion coefficients a(n), Atkin and Swinnerton-Dyer observed that for many primes p and integers n there are congruences of the shape a(pn) ≡ A_p a(n) + B_p a(n/p) (mod p^m) where A_p and B_p relate to coefficients of classical newforms or to Frobenius traces on l-adic cohomology, as in examples studied by Atkin, Swinnerton-Dyer, and later by Katz and Scholl. Concrete instances appeared for noncongruence forms on genus-zero modular curves associated to sporadic groups like investigations reminiscent of Monstrous moonshine computations and examples tied to arithmetic of K3 surfaces and elliptic surfaces examined by Shioda and Livné.
In the congruence-subgroup context, Hecke operators give linear relations among Fourier coefficients for forms on groups like SL(2,Z), Γ0(N), and Γ1(N). For noncongruence subgroups, genuine Hecke operators do not act, so the Atkin–Swinnerton-Dyer congruences mimic Hecke relations via comparisons with eigenvalues from newforms studied by Atkin, Lehner, and Newman. Scholl constructed l-adic representations attached to noncongruence forms that produce Frobenius eigenvalues playing the role of Hecke eigenvalues, building on techniques from Deligne and Grothendieck cohomology, and relating to results of Serre on modular forms modulo p.
Katz and Scholl showed that p-adic properties of q-expansions link to l-adic Galois representations of absolute Galois groups such as Gal(ℚ̄/ℚ), echoing themes from the Fontaine–Mazur and Langlands correspondence. Work by Coleman and Iovita explored p-adic families and overconvergent phenomena, while Fontaine, Mazur, and Wiles contributed techniques to control deformation rings and modularity lifting relevant to these congruences. The resulting Galois representations often factor through motives studied by Grothendieck and appear in the cohomology of varieties like Kuga–Sato and elliptic surfaces analyzed by Shioda.
Proof approaches combine computational experiments from Atkin and Swinnerton-Dyer with theoretical input from Katz’s p-adic analysis, Scholl’s construction of l-adic representations, and cohomological methods from Deligne and Grothendieck. Techniques include rigid cohomology used by Berthelot, deformation theory from Mazur, and modularity lifting techniques pioneered by Wiles and Taylor. Some proofs in special cases leverage comparison theorems of Fontaine and Faltings and congruence criteria analogous to those in the work of Sturm and Atkin–Lehner.
Atkin–Swinnerton-Dyer congruences spurred research in the arithmetic of noncongruence forms, influenced the formulation of conjectures by Serre and Katz, and informed computational investigations related to Monstrous moonshine and the Ogg study of supersingular points. They influenced the classification of motives by Deligne and Scholl and connected to studies of rational points on modular curves undertaken by Mazur and Faltings. Later work by Li, Long, Zagier, and Hida explored broader classes of congruences, p-adic families, and connections to automorphic forms within the Langlands framework advanced by Langlands.
Major open questions include a general reciprocity linking noncongruence q-expansions to automorphic representations in the sense of Langlands, a full modularity statement analogous to the Modularity theorem for motives arising from noncongruence forms, and precise bounds in the style of the Ramanujan–Petersson conjecture for coefficients appearing in these congruences. Researchers such as Scholl, Katz, Coleman, and Darmon continue to investigate whether all Scholl motives are automorphic and how deformation theoretic methods inspired by Taylor and Wiles can resolve outstanding conjectures.