We derive the Atkin–Swinnerton–Dyer partition congruences for the modulus 11 by use of Winquist’s identity. Classification: 11P83

Classically, congruence subgroups of the modular group, which can be described by congruence relations, play important roles in group theory and modular forms. In reality, the majority of finite index subgroups of the modular group are noncongruence. These groups as well as their modular forms are central players of this survey article. Differences between congruence and noncongruence subgroups...

Let N(r,m, n) (resp. M(r,m, n)) denote the number of partitions of n whose ranks (resp. cranks) are congruent to r modulo m. Atkin and Swinnerton-Dyer gave the relations between the numbers N(r,m,mn+k) when m = 5, 7 and 0 ≤ r, k < m. Garvan gave the relations between the numbers M(r,m,mn+k) when m = 5, 7, and 11, 0 ≤ r, k < m. Here, we show that the methods of Atkin and Swinnerton-Dyer can be e...

Arithmetic subgroups are finite index subgroups of the modular group. Classically, congruence arithmetic subgroups, which can be described by congruence relations, are playing important roles in group theory and modular forms. In reality, the majority of arithmetic subgroups are noncongruence. These groups as well as their modular forms are central players of this survey article. Differences be...

and he conjectured further such congruences modulo arbitrary powers of 5, 7, and 11. Although the work of A. O. L. Atkin and G. N. Watson settled these conjectures many years ago, the congruences have continued to attract much attention. For example, subsequent works by G. Andrews, A. O. L. Atkin, F. Garvan, D. Kim, D. Stanton, and H. P. F. Swinnerton-Dyer ([An-G], [G], [G-K-S], [At-Sw2]), in t...

The main result of this paper is an instance of the conjecture made by Gouvêa and Mazur in [GM95], which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k should be spanned by the finite slope Hecke eigenforms. For N = 1, p = 2 and k = 0 we show that this follows from the combinatorial approach initiated by Emerton [Eme98] and S...