Theta Functions pp 136-172

# Equations Defining Abelian Varieties

• Jun-ichi Igusa
Chapter
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 194)

## 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.