Abstract
We shall start this chapter by proving “theta relations,” i.e., relations between theta functions. More precisely, we shall be interested in polynomial relations between θm(τ, z), θm(τ, 0) with constant coefficients. From the “labyrinth” of theta relations, we shall select just two, which are themselves not unrelated: The first one is called “Riemann’s theta formula” and the second one the “addition formula.” Their shortest proofs depend on the following lemma: Lemma 1. Let L denote a discrete commutative group and L1, L2 two subgroups of finite indices; let Φ denote an L1-function on L, i.e., a C-valued function on L such that the sum of ∣ Φ(ξ)∣ over L is convergent. Then we have $[L:{L_1}]\cdot \mathop\Sigma\limits_{\xi\epsilon{L_1}}\Phi(\xi) = \mathop\Sigma\limits_{\chi,\zeta} (\mathop\Sigma\limits_{\eta\varepsilon{L_2}} \chi(\eta + \zeta)\Phi((\eta + \zeta)),$ in which χ runs over the dual of L/L1 and ζ over a complete set of representatives of L/L2.
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© 1972 Springer-Verlag Berlin Heidelberg
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Igusa, Ji. (1972). Equations Defining Abelian Varieties. In: Theta Functions. Die Grundlehren der mathematischen Wissenschaften, vol 194. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-65315-5_4
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DOI: https://doi.org/10.1007/978-3-642-65315-5_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-65317-9
Online ISBN: 978-3-642-65315-5
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