2016/09/08 by Imin Chen, Chen, Imin, Gleb Glebov +1
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1609.05778
openalex publication_date 2016/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a well-known 1914 paper, Ramanujan gave a number of rapidly converging series for 1/π 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/π using the modular function J of level 1, which results in highly convergent series for 1/π, 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, ∞.