%0 Journal Article
%T ¦Ñ ¡ª Adic Analogues of Ramanujan Type Formulas for 1/¦Ð
%A Sarah Chisholm
%A Alyson Deines
%A Ling Long
%A Gabriele Nebe
%A Holly Swisher
%J Mathematics
%D 2013
%I MDPI AG
%R 10.3390/math1010009
%X Following Ramanujan¡¯s work on modular equations and approximations of ¦Ð, there are formulas for 1/¦Ð of the form [PLEASE CHECK FORMULA IN THE PDF] for d = 2, 3, 4, 6, where ¦Ë d are singular values that correspond to elliptic curves with complex multiplication, and ¦Á, ¦Ä are explicit algebraic numbers. In this paper we prove a ¦Ñ-adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication.
%K Ramanujan type supercongruences
%K Atkin and Swinnerton-Dyer congruences
%K hypergeometric series
%K elliptic curves
%K complex multiplication
%K periods
%K modular forms
%K Picard¨CFuchs equation
%U http://www.mdpi.com/2227-7390/1/1/9