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. Chudnovsky and 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: \(0, 1, \infty \).
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This work was supported by an NSERC Discovery Grant and SFU VPR Bridging Grant.
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Chen, I., Glebov, G. On Chudnovsky–Ramanujan type formulae. Ramanujan J 46, 677–712 (2018). https://doi.org/10.1007/s11139-017-9948-8
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DOI: https://doi.org/10.1007/s11139-017-9948-8
Keywords
- Elliptic curves
- Elliptic functions
- Dedekind eta function
- Eisenstein series
- Hypergeometric function
- Picard–Fuchs differential equation