# Theta-series

$\theta$- series

A series of functions used in the representation of automorphic forms and functions (cf. Automorphic form; Automorphic function).

Let $D$ be a domain in the complex space $\mathbf C ^ {p}$, $p \geq 1$, and let $\Gamma$ be the discrete group of automorphisms of $D$. If $\Gamma$ is finite, then any function $H ( z)$, $z = ( z _ {1} \dots z _ {p} )$, meromorphic on $D$ gives rise to an automorphic function

$$\sum _ {\gamma \in \Gamma } H ( \gamma ( z)).$$

For infinite groups one needs convergence multipliers to obtain a theta-series. A Poincaré series, associated to a group $\Gamma$, is a series of the form

$$\tag{1 } \theta _ {m} ( z) = \ \sum _ {\gamma \in \Gamma } {} ^ {*} J _ \gamma ^ {m} ( z) H ( \gamma ( z)),$$

where $J _ \gamma ( z) = d \gamma ( z)/dz$ is the Jacobian of the function $z \mapsto \gamma ( z)$ and $m$ is an integer called the weight or the order. The asterisk means that summation is over those $\gamma \in \Gamma$ which yield distinct terms in the series. Under a mapping $z \mapsto \alpha ( z)$, $\alpha \in \Gamma$, the function $\theta _ {m} ( z)$ is transformed according to the law $\theta _ {m} ( \alpha ( z)) = J _ \alpha ^ {-} m ( z) \theta _ {m} ( z)$, and hence is an automorphic function of weight $m$, associated to $\Gamma$. The quotient of two theta-series of the same weight gives an automorphic function.

The theta-series

$$E _ {m} ( z) = \ \sum _ {\gamma \in \Gamma } {} ^ {*} J _ \gamma ^ {m} ( z)$$

is called an Eisenstein theta-series, or simply an Eisenstein series, associated with $\Gamma$.

H. Poincaré, in a series of articles in the 1880's, developed the theory of theta-series in connection with the study of automorphic functions of one complex variable. Let $\Gamma$ be a discrete Fuchsian group of fractional-linear transformations

$$\gamma ( z) = \ \frac{az + b }{cz + d } ,\ \ ad - bc = 1,$$

mapping the unit disc $D = \{ {z } : {| z | < 1 } \}$ onto itself. For this case the Poincaré series has the form

$$\tag{2 } \theta _ {m} ( z) = \ \sum _ {\gamma \in \Gamma } {} ^ {*} ( cz + d) ^ {-} 2m H \left ( \frac{az + b }{cz + d } \right ) ,$$

where $H$, for example, is a bounded holomorphic function on $D$. Under the hypothesis that $\Gamma$ acts freely on $D$ and that the quotient space $X = D/ \Gamma$ is compact, it has been shown that the series (2) converges absolutely and uniformly on $D$ for $m \geq 2$. With the stated conditions on $H$ and $\Gamma$, this assertion holds also for the series (1) in the case where $D$ is a bounded domain in $\mathbf C ^ {p}$. For certain Fuchsian groups the series (2) converges also for $m = 1$.

The term "theta-series" is also applied to series expansions of theta-functions, which are used in the representation of elliptic functions (cf. Jacobi elliptic functions) and Abelian functions (cf. Theta-function; Abelian function).

#### References

 [1] L.R. Ford, "Automorphic functions" , Chelsea, reprint (1951) MR1522111 Zbl 55.0810.04 Zbl 46.0621.01 Zbl 45.0693.07 [2] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 [3] R. Fricke, F. Klein, "Vorlesungen über die Theorie der automorphen Funktionen" , 1–2 , Teubner (1926) MR0183872 Zbl 32.0430.01 Zbl 43.0529.08 Zbl 42.0452.01

Let $\Lambda \subset \mathbf R ^ {n}$ be a lattice. The theta-series of the lattice $\Lambda$ is defined by

$$\theta _ \Lambda ( z ) = \ \sum _ {x \in \Lambda } q ^ {( x,x) } = \ \sum _ { m= } 1 ^ \infty N _ {m} q ^ {m} ,\ \ q = e ^ {\pi i z } ,$$

where $N _ {m}$ is the number of points in $\Lambda$ of squared length $m$. For instance, if $\Lambda$ is the lattice $\mathbf Z ^ {4} \subset \mathbf R ^ {4}$, then $N _ {m}$ is the number of ways of representing $m$ as a sum of four integral squares.

For the lattice $\mathbf Z \subset \mathbf R$ the theta-series is

$$\theta _ {\mathbf Z } ( z) = \ \sum _ {m=- \infty } ^ { {+ } \infty } q ^ {m ^ {2} } = \ 1 + 2q + 2q ^ {4} + 2q ^ {9} + 2q ^ {16} + \dots ,$$

which is the Jacobi theta-function $\theta _ {3} ( z )$.

For more details on theta-series of lattices, including formulas and tables for many (series of) important lattices such as root lattices and the Leech lattice, and applications, cf. [a2].

#### References

 [a1] A. Weil, "Elliptic functions according to Eisenstein and Kronecker" , Springer (1976) MR0562289 MR0562290 Zbl 0318.33004 [a2] J.H. Conway, N.J.A. Sloane, "Sphere packing, lattices and groups" , Springer (1988) MR0920369
How to Cite This Entry:
Theta-series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Theta-series&oldid=48964
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article