Frobenius’ theta formula

Abstract

In this section we want to combine Riemann’s theta formula (II.6) with the Vanishing Property (6.7) of the last section. An amazing cancellation takes place and we can prove that for hyperelliptic Ω, ϑ($\vec z$, Ω) satisfies a much simpler identity discovered in essence by Frobenius. We shall make many applications of Frobenius’ formula. The first of these is to make more explicit the link between the analytic and algebraic theory of the Jacobian by evaluating the constants c_k of Theorem 5.3. The second will be to give explicitly via thetas the solutions of Neumann’s dynamical system discussed in §4. Other applications will be given in later sections. Because one of these is to the Theorem characterizing hyperelliptic Ω by the Vanishing Property (6.7), we want to derive Frobenius’ theta formula using only this Vanishing and no further aspects of the hyperelliptic situation. Therefore, we assume we are working in the following situation:

• 1. B = fixed set with 2g+2 elements

• 2. U ⊂ B, a fixed subset with g+1 elements

• 3. ∞; ∈ B-U a fixed element

• 4. T → ŋ_T an isomorphism:

  $$ \left( {\begin{array}{*{20}c} {even subsets of B} \\ {modulo S \sim CS} \\ \end{array} } \right)\xrightarrow{ \approx }\frac{1} {2}{{\mathbb{Z}^{2g} } \mathord{\left/ {\vphantom {{\mathbb{Z}^{2g} } {\mathbb{Z}^{2g} }}} \right. \kern-\nulldelimiterspace} {\mathbb{Z}^{2g} }} $$

  such that

  $$ a)\eta _{S_1 \circ S_2 } = \eta _{S_1 } + \eta _{S_2 } $$
  $$ b)e_2 \left( {\eta _{S_1 } ,\eta _{S_2 } } \right) = \left( { - 1} \right)^{\# S_1 \cap S_2 } $$
  $$ c)e_ \star \left( {\eta _{\rm T} } \right) = \left( { - 1} \right)^{\frac{{\# \left( {{\rm T} \circ U} \right) - g - 1}} {2}} $$

• 5. satisfies ϑ-[η_T] (O,Ω) = 0 if #ToU ≠ g+1.

• 6. We fix ŋ_i∈ 1/2 ℤ^2g for all i ∈ B-∞ such that ŋ_i mod ℤ^2g equals η_{{i, ∞}} and also let η_∞ = 0. (This choice affects nothing essentially.) We shall use the notation

  $$ \begin{gathered} \varepsilon _S \left( k \right) = + 1if k \in S \hfill \\ - 1 if k \notin S \hfill \\ \end{gathered} $$

  for all k ∈ B, subsets S ⊂ B.

Uber die constanten Factoren der Thetareihen, Crelle, 98 (1885); see top formula, p. 249, Collected Works, vol. II.