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 ck 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 → ŋT 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-[ηT] (O,Ω) = 0 if #ToU ≠ g+1.
-
6.
We fix ŋi∈ 1/2 ℤ2g for all i ∈ B-∞ such that ŋi mod ℤ2g 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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2007 Birkhäuser Boston
About this chapter
Cite this chapter
Mumford, D. (2007). Frobenius’ theta formula. In: Tata Lectures on Theta II. Modern Birkhäuser Classics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4578-6_8
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4578-6_8
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-4569-4
Online ISBN: 978-0-8176-4578-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)