Neha Patil (Editor)

Theta constant

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In mathematics, a theta constant or Thetanullwert (German for theta zero value; plural Thetanullwerte) is the restriction θm(τ) = θm(τ,0) of a theta function θm(τ,z) with rational characteristic m to z = 0. The variable τ may be a complex number in the upper half-plane in which case the theta constants are modular forms, or more generally may be an element of a Siegel upper half plane in which case the theta constants are Siegel modular forms. The theta function of a lattice is essentially a special case of a theta constant.

Contents

Definition

The theta function θm(τ,z) = θa,b(τ,z)is defined by

θ a , b ( τ , z ) = ξ Z n exp [ π i ( ξ + a ) τ ( ξ + a ) t + 2 π i ( ξ + a ) ( z + b ) t ]

where

  • n is a positive integer, called the genus or rank.
  • m = (a,b) is called the characteristic
  • a,b are in Rn
  • τ is a complex n by n matrix with positive definite imaginary part
  • z is in Cn
  • t means the transpose of a row vector.
  • If a,b are in Qn then θa,b(τ,0) is called a theta constant.

    Examples

    If n = 1 and a and b are both 0 or 1/2, then the functions θa,b(τ,z) are the four Jacobi theta functions, and the functions θa,b(τ,0) are the classical Jacobi theta constants. The theta constant θ1/2,1/2(τ,0) is identically zero, but the other three can be nonzero.

    References

    Theta constant Wikipedia