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: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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