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

Comments

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