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.

We generalize Jacobi’s derivative formula for odd m by writing an m×m determinant of higher order derivatives at 0 of theta functions in 1 variable with characteristic vectors with entries in 1 2m 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 6m.

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 1 2m 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 6m.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1164
Classification: 14K25,  14H42
Keywords: Theta functions, elliptic curves
David Grant 1

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