About the Project
21 Multidimensional Theta FunctionsProperties

§21.5 Modular Transformations

Contents
  1. §21.5(i) Riemann Theta Functions
  2. §21.5(ii) Riemann Theta Functions with Characteristics

§21.5(i) Riemann Theta Functions

Let 𝐀, 𝐁, 𝐂, and 𝐃 be g×g matrices with integer elements such that

21.5.1 𝚪=[𝐀𝐁𝐂𝐃]

is a symplectic matrix, that is,

21.5.2 𝚪𝐉2g𝚪T=𝐉2g.

Then

21.5.3 det𝚪=1,

and

21.5.4 θ([[𝐂𝛀+𝐃]1]T𝐳|[𝐀𝛀+𝐁][𝐂𝛀+𝐃]1)=ξ(𝚪)det[𝐂𝛀+𝐃]eπi𝐳[[𝐂𝛀+𝐃]1𝐂]𝐳θ(𝐳|𝛀).

Here ξ(𝚪) is an eighth root of unity, that is, (ξ(𝚪))8=1. For general 𝚪, it is difficult to decide which root needs to be used. The choice depends on 𝚪, but is independent of 𝐳 and 𝛀. Equation (21.5.4) is the modular transformation property for Riemann theta functions.

The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which ξ(𝚪) is determinate:

21.5.5 𝚪=[𝐀𝟎g𝟎g[𝐀1]T]θ(𝐀𝐳|𝐀𝛀𝐀T)=θ(𝐳|𝛀).

(𝐀 invertible with integer elements.)

21.5.6 𝚪=[𝐈g𝐁𝟎g𝐈g]θ(𝐳|𝛀+𝐁)=θ(𝐳|𝛀).

(𝐁 symmetric with integer elements and even diagonal elements.)

21.5.7 𝚪=[𝐈g𝐁𝟎g𝐈g]θ(𝐳|𝛀+𝐁)=θ(𝐳+12diag𝐁|𝛀).

(𝐁 symmetric with integer elements.) See Heil (1995, p. 24). For a g×g matrix 𝐀 we define diag𝐀, as a column vector with the diagonal entries as elements.

21.5.8 𝚪 =[𝟎g𝐈g𝐈g𝟎g]  
θ(𝛀1𝐳|𝛀1) =det[i𝛀]eπi𝐳𝛀1𝐳θ(𝐳|𝛀),

where the square root assumes its principal value.

§21.5(ii) Riemann Theta Functions with Characteristics

21.5.9 θ[𝐃𝜶𝐂𝜷+12diag[𝐂𝐃T]𝐁𝜶+𝐀𝜷+12diag[𝐀𝐁T]]([[𝐂𝛀+𝐃]1]T𝐳|[𝐀𝛀+𝐁][𝐂𝛀+𝐃]1)=κ(𝜶,𝜷,𝚪)det[𝐂𝛀+𝐃]eπi𝐳[[𝐂𝛀+𝐃]1𝐂]𝐳θ[𝜶𝜷](𝐳|𝛀),

where κ(𝜶,𝜷,𝚪) is a complex number that depends on 𝜶, 𝜷, and 𝚪. However, κ(𝜶,𝜷,𝚪) is independent of 𝐳 and 𝛀. For explicit results in the case g=1, see §20.7(viii).