A classical theta function is a holomorphic function defined by a Fourier series over integer vectors,
with coefficients given by the exponential
function of a quadratic form. In the standard
several-variable form, take 
and a symmetric 
complex
matrix 
whose imaginary part is positive
definite. The Riemann theta function
is
 |
(1)
|
The series converges absolutely and uniformly on compact sets. For 
, the quasi-periodicity
law is
 |
(2)
|
The factor arises by shifting the summation index by 
(Mumford 1983, pp. 120-122).
For the one-variable Jacobi theta functions, the third function satisfies 
,
where the nome is 
and 
lies in the upper half-plane.
The other Jacobi theta functions come from
half-integer shifts in the summation variable and argument, encoded by theta characteristics
(NIST DLMF).
The theta series of a point lattice 
is a specialization at 
. If 
is the Gram matrix of a basis of 
, the squared vector
norm of the vector with integer coordinates 
in that basis is 
. Thus, for real 
,
 |
(3)
|
For an integral point lattice, the coefficient of 
in 
counts vectors of squared vector
norm 
.
The term also appears in metaplectic analogs. Kubota's construction recovers the classical theta function in degree two and gives Kubota's
cubic theta function in degree three over the imaginary
quadratic field 
(Yoshimoto 1987).
