Editing Pi (section)

=== Modular forms and theta functions ===
[[File:Lattice with tau.svg|thumb|right|Theta functions transform under the [[lattice (group)|lattice]] of periods of an elliptic curve.]]

The constant {{pi}} is connected in a deep way with the theory of [[modular form]]s and [[theta function]]s.  For example, the [[Chudnovsky algorithm]] involves in an essential way the [[j-invariant]] of an [[elliptic curve]].

[[Modular form]]s are [[holomorphic function]]s in the [[upper half plane]] characterized by their transformation properties under the [[modular group]] <math>\mathrm{SL}_2(\mathbb Z)</math> (or its various subgroups), a lattice in the group <math>\mathrm{SL}_2(\mathbb R)</math>.  An example is the [[Jacobi theta function]]

<math display=block>\theta(z,\tau) = \sum_{n=-\infty}^\infty e^{2\pi i nz \ +\  \pi i n^2\tau}</math>

which is a kind of modular form called a [[Jacobi form]].<ref name="Mumford 1983 1–117">{{cite book |first=David |last=Mumford |author-link=David Mumford |title=Tata Lectures on Theta I |year=1983 |publisher=Birkhauser |location=Boston |isbn=978-3-7643-3109-2 |pages=1–117}}</ref>  This is sometimes written in terms of the [[nome (mathematics)|nome]] <math>q=e^{\pi i \tau}</math>.

The constant {{pi}} 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

<math display=block>\theta(z+\tau,\tau) = e^{-\pi i\tau -2\pi i z}\theta(z,\tau),</math>

which implies that {{math|θ}} transforms as a representation under the discrete [[Heisenberg group]].  General modular forms and other [[theta function]]s also involve {{pi}}, once again because of the [[Stone–von Neumann theorem]].{{r|Mumford 1983 1–117}}