ρ — Adic Analogues of Ramanujan Type Formulas for 1/π

Following Ramanujan's work on modular equations and approximations of $\pi$, there are formulas for $1/\pi$ of the form Following Ramanujan's work on modular equations and approximations of $\pi$, there are formulas for $1/\pi$ of the form $\sum _{k=0}^{\infty }\frac{{\left(\frac{1}{2}\right)}_k{\left(\frac{1}{d}\right)}_k{\left(\frac{d-1}{d}\right)}_k}{k{!}^3}(ak+1){\left({\lambda }_d\right)}^k=\frac{\delta }{\pi }$ for $d=2,3,4,6,$ where ${ł}_d$ are singular values that correspond to elliptic curves with complex multiplication, and $a,\delta$ 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.