Equations Defining Abelian Varieties

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 θ_m(τ, z), θ_m(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.