In
mathematics, the theta function of a lattice is a function whose coefficients give the number of vectors of a given norm.
Definition
One can associate to any (positive-definite)
lattice Λ a
theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
given by
:
The theta function of a
lattice is then a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
on the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
. Furthermore, the theta function of an even
unimodular lattice
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundam ...
of rank ''n'' is actually a
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory ...
of weight ''n''/2. The theta function of an integral lattice is often written as a power series in
so that the coefficient of ''q''
''n'' gives the number of lattice vectors of norm 2''n''.
References
*
Theta functions
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