Theta function of a lattice: Difference between revisions

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In [[mathematics]], the '''theta function of a lattice''' is a function whose coefficients give the number of vectors of a given norm.

#redirect[[E8 lattice#Theta function]]

==Definition==

One can associate to any (positive-definite) [[Lattice (discrete subgroup)|lattice]] Λ a [[theta function]] given by

:<math>\Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0.</math>

The theta function of a [[Lattice (discrete subgroup)|lattice]] is then a [[holomorphic function]] on the [[upper half-plane]]. Furthermore, the theta function of an even [[unimodular lattice]] of rank&nbsp;''n'' is actually a [[modular form]] of weight&nbsp;''n''/2. The theta function of an integral lattice is often written as a power series in <math>q = e^{2i\pi\tau}</math> so that the coefficient of ''q''^''n'' gives the number of lattice vectors of norm&nbsp;2''n''.

== See also ==

* [[Siegel theta series]]

* [[Theta constant]]

==References==

*{{dlmf|id=21|title=Multidimensional Theta Functions|first=Bernard |last=Deconinck}}

[[Category:Theta functions]]

{{numtheory-stub}}