On Chudnovsky-Ramanujan Type Formulae

In his well-known 1914 paper, Ramanujan gave a number of rapidly converging series for $1/\pi$ which involve modular functions of higher level. D. V. and G. V. Chudnovsky have 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 2013, $12.1 \times 10^{12}$ digits of $\pi$ were calculated by A. J. Yee and S. Kondo using the Chudnovsky series for $\pi$. The purpose of this work is to explain the method of D. V. and G. V. Chudnovsky and show how it can be generalised to derive formulae for transcendental constants starting from a one-parameter family of elliptic curves. We find all such closed-form Chudnovsky-Ramanujan type formulae for the family of elliptic curves parameterised by $J$ of level 1, where $J$ is the absolute $j$-invariant.