# Difference between revisions of "Theta-series"

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261031.png" />, for example, is a bounded holomorphic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261032.png" />. Under the hypothesis that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261033.png" /> acts freely on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261034.png" /> and that the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261035.png" /> is compact, it has been shown that the series (2) converges absolutely and uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261036.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261037.png" />. With the stated conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261039.png" />, this assertion holds also for the series (1) in the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261040.png" /> is a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261041.png" />. For certain Fuchsian groups the series (2) converges also for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261042.png" />. | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261031.png" />, for example, is a bounded holomorphic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261032.png" />. Under the hypothesis that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261033.png" /> acts freely on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261034.png" /> and that the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261035.png" /> is compact, it has been shown that the series (2) converges absolutely and uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261036.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261037.png" />. With the stated conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261039.png" />, this assertion holds also for the series (1) in the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261040.png" /> is a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261041.png" />. For certain Fuchsian groups the series (2) converges also for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092610/t09261042.png" />. | ||

− | The term | + | 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|Jacobi elliptic functions]]) and Abelian functions (cf. [[Theta-function|Theta-function]]; [[Abelian function|Abelian function]]). |

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.R. Ford, "Automorphic functions" , Chelsea, reprint (1951) {{MR|1522111}} {{ZBL|55.0810.04}} {{ZBL|46.0621.01}} {{ZBL|45.0693.07}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R. Fricke, F. Klein, "Vorlesungen über die Theorie der automorphen Funktionen" , '''1–2''' , Teubner (1926) {{MR|0183872}} {{ZBL|32.0430.01}} {{ZBL|43.0529.08}} {{ZBL|42.0452.01}} </TD></TR></table> |

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====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Weil, "Elliptic functions according to Eisenstein and Kronecker" , Springer (1976) {{MR|0562289}} {{MR|0562290}} {{ZBL|0318.33004}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.H. Conway, N.J.A. Sloane, "Sphere packing, lattices and groups" , Springer (1988) {{MR|0920369}} {{ZBL|}} </TD></TR></table> |

## Latest revision as of 22:57, 30 March 2012

*-series*

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

Let be a domain in the complex space , , and let be the discrete group of automorphisms of . If is finite, then any function , , meromorphic on gives rise to an automorphic function

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

(1) |

where is the Jacobian of the function and is an integer called the weight or the order. The asterisk means that summation is over those which yield distinct terms in the series. Under a mapping , , the function is transformed according to the law , and hence is an automorphic function of weight , associated to . The quotient of two theta-series of the same weight gives an automorphic function.

The theta-series

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

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 be a discrete Fuchsian group of fractional-linear transformations

mapping the unit disc onto itself. For this case the Poincaré series has the form

(2) |

where , for example, is a bounded holomorphic function on . Under the hypothesis that acts freely on and that the quotient space is compact, it has been shown that the series (2) converges absolutely and uniformly on for . With the stated conditions on and , this assertion holds also for the series (1) in the case where is a bounded domain in . For certain Fuchsian groups the series (2) converges also for .

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 be a lattice. The theta-series of the lattice is defined by

where is the number of points in of squared length . For instance, if is the lattice , then is the number of ways of representing as a sum of four integral squares.

For the lattice the theta-series is

which is the Jacobi theta-function .

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://www.encyclopediaofmath.org/index.php?title=Theta-series&oldid=23995