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21 Multidimensional Theta FunctionsProperties

§21.5 Modular Transformations

  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.


21.5.3 det𝚪=1,


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).