A higher-order generalization of Jacobi’s derivative formula and its algebraic geometric analogue

David Grant

doi:10.5802/jtnb.1164

David Grant ¹

¹ Department of Mathematics University of Colorado Boulder Boulder, CO 80309-0395 USA

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 361-386.

Abstract
Résumé

We generalize Jacobi’s derivative formula for odd $m$ by writing an $m \times m$ determinant of higher order derivatives at 0 of theta functions in 1 variable with characteristic vectors with entries in $\frac{1}{2 m} ℤ$ as an explicit constant times a power of Dedekind’s $η$ -function. We do so by deriving it from an algebraic geometric version that holds in characteristic not dividing $6 m$ .

Nous généralisons la formule de la dérivée de Jacobi en écrivant, pour un $m$ impair, un déterminant de taille $m$ composé de dérivées d’ordre supérieur évaluées en $0$ des fonctions thêta d’une variable avec vecteurs caractéristiques à coordonnées dans $\frac{1}{2 m} ℤ$ comme une constante explicite multipliée par une puissance de la fonction $η$ de Dedekind. Nous déduisons ce résultat de sa version algébro-géométrique, qui est valable si la caractéristique ne divise pas $6 m$ .

Received: 2020-02-06
Revised: 2021-02-02
Accepted: 2021-05-18
Published online: 2021-09-10

DOI: 10.5802/jtnb.1164

Classification: 14K25, 14H42
Keywords: Theta functions, elliptic curves

Author's affiliations:

David Grant ¹

¹ Department of Mathematics University of Colorado Boulder Boulder, CO 80309-0395 USA

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     journal = {Journal de th\'eorie des nombres de Bordeaux},
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David Grant. A higher-order generalization of Jacobi’s derivative formula and its algebraic geometric analogue. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 361-386. doi : 10.5802/jtnb.1164. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1164/

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