On Chudnovsky-Ramanujan Type Formulae

TL;DR

This paper explains and generalizes Chudnovsky-Ramanujan type formulas for 1/π using elliptic curves, modular functions, and differential equations, providing a comprehensive framework for deriving such rapidly converging series around singular points.

Contribution

It introduces a unified elliptic curve and differential equation approach to derive and generalize Chudnovsky-Ramanujan formulas for 1/π at various singular points.

Findings

Derived formulas valid around singular points 0, 1, and ∞.

Unified framework using modular curves and Picard-Fuchs equations.

Complete classification of Chudnovsky-Ramanujan type series for level 1.

Abstract

In a well-known 1914 paper, Ramanujan gave a number of rapidly converging series for $1/\pi$ which are derived using modular functions of higher level. D. V. and G. V. Chudnovsky in their 1988 paper derived an analogous series representing $1/\pi$ using the modular function $J$ of level 1, which results in highly convergent series for $1/\pi$, often used in practice. In this paper, we explain the Chudnovsky method in the context of elliptic curves, modular curves, and the Picard-Fuchs differential equation. In doing so, we also generalize their method to produce formulae which are valid around any singular point of the Picard-Fuchs differential equation. Applying the method to the family of elliptic curves parameterized by the absolute Klein invariant $J$ of level 1, we determine all Chudnovsky-Ramanujan type formulae which are valid around one of the three singular points: $0, 1, \infty$.