Atkin and Swinnerton-Dyer congruences and noncongruence modular forms

Abstract

Atkin and Swinnerton-Dyer congruences are special congruence recursions satisfied by coefficients of noncongruence modular forms. These are in some sense $p$-adic analogues of Hecke recursion satisfied by classic Hecke eigenforms. They actually appeared in different context and sometimes can be obtained using the theory of formal groups. In this survey paper, we introduce the Atkin and Swinnerton-Dyer congruences, and discuss some recent progress on this topic.

References in corpus (2)

• Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms
• Modular forms, de Rham cohomology and congruences

Cited by in corpus (5)

• Supercongruences for sporadic sequences
• Interpolated sequences and critical $L$-values of modular forms
• Hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions
• Some Numeric Hypergeometric Supercongruences
• On the Atkin and Swinnerton-Dyer type congruences for some truncated hypergeometric ${}_1{F}_0$ series