Jacobians in the isogeny class of E^g

Abstract: Let $E$ be an ordinary elliptic curve over a finite field $\mathbb{F}_q$ such that $R=\mathrm{End}(E)$ is generated by the Frobenius endomorphism. There is an equivalence of categories which associates to each abelian variety $A$ in the isogeny class of $E^g$ an $R$-lattice $L$ of rank $g$. Given $L$ (with a hermitian form describing a polarization $a$ on $A$), we show how to make $(A,a)$ concrete, i.e. we give an embedding of $(A,a)$ into a projective space by computing its algebraic theta constants. Using these data and an algorithm to compute Siegel modular forms algebraically, we can decide when $(A,a)$ is a Jacobian over $\mathbb{F}_q$ when $g \leq 3$ (and over $\bar{\mathbb{F}}_q$ when $g=4$). We illustrate our algorithms with the problem of constructing curves over $\mathbb{F}_q$ with many rational points.

Joint work with Markus Kirschmer, Fabien Narbonne and Damien Robert

algebraic geometrynumber theory

Audience: researchers in the topic

SFU NT-AG seminar

Series comments: Description: Research/departmental seminar

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