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Modular forms and theta functions

The constant π is connected in a deep way with the theory of modular forms and theta functions. For example, the Chudnovsky algorithm involves in an essential way the j-invariant of an elliptic curve.

Modular forms are holomorphic functions in the upper half plane characterized by their transformation properties under the modular group S L 2 (Z) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z})} (or its various subgroups), a lattice in the group S L 2 (R) {\displaystyle \mathrm {SL} _{2}(\mathbb {R})}. An example is the Jacobi theta function θ (z, τ) = ∑ n = − ∞ ∞ e 2 π i n z + i π n 2 τ {\displaystyle \theta (z,\tau)=\sum _{n=-\infty }^{\infty }e^{2\pi inz+i\pi n^{2}\tau }} which is a kind of modular form called a Jacobi form. This is sometimes written in terms of the nome q = e π i τ {\displaystyle q=e^{\pi i\tau }}.

The constant π is the unique constant making the Jacobi theta function an automorphic form, which means that it transforms in a specific way. Certain identities hold for all automorphic forms. An example is θ (z + τ, τ) = e − π i τ − 2 π i z θ (z, τ), {\displaystyle \theta (z+\tau,\tau)=e^{-\pi i\tau -2\pi iz}\theta (z,\tau),} which implies that θ transforms as a representation under the discrete Heisenberg group. General modular forms and other theta functions also involve π, once again because of the Stone–von Neumann theorem.