►Figure 21.4.1 provides surfaces of the scaled Riemann theta function , with
…This Riemann matrix originates from the Riemann surface represented by the algebraic curve ; compare §21.7(i).
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►For the scaled Riemann theta functions depicted in Figures 21.4.2–21.4.5
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►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 is determinate:
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§21.5(ii) Riemann Theta Functions with Characteristics
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►For explicit results in the case , see §20.7(viii).
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►For every Abelian function, there is a positive integer , such that the Abelian function can be expressed as a ratio of linear combinations of products with factors of Riemann theta functions with characteristics that share a common period lattice.
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