p-adic analogues of Ramanujan type formulas for 1/π
Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form for d = 2, 3, 4, 6, where λd are singular values that correspond to elliptic curves with complex multiplication, and α, δ are explicit algebraic numbers. In this paper we prove a p-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.
Publication Source (Journal or Book title)
Chisholm, S., Deines, A., Long, L., Nebe, G., & Swisher, H. (2013). p-adic analogues of Ramanujan type formulas for 1/π. Mathematics, 1 (1), 9-30. https://doi.org/10.3390/math1010009