In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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. Theta functions are parametrized by points in a tube ...
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, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...
. Furthermore, the theta function of an even
unimodular lattice
In geometry and mathematical group theory, a unimodular lattice is an integral Lattice (group), lattice of Lattice (group)#Dividing space according to a lattice, determinant 1 or −1. For a lattice in ''n''-dimensional Euclidea ...
of rank ''n'' is actually a
modular form
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...
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''.
See also
*
Siegel theta series In mathematics, a Siegel theta series is a Siegel modular form associated to a positive definite lattice, generalizing the 1-variable theta function of a lattice.
Definition
Suppose that ''L'' is a positive definite lattice. The Siegel theta ser ...
*
Theta constant
References
*
Theta functions
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