Jacobians in abelian-threefold isogeny classes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an abelian variety $A/\mathbb F_q$, its Weil polynomial is $f_A(T)=\det(T-\operatorname{Frob}_q\mid V_\ell A)$, where $\ell$ is a prime different from the characteristic, $V_\ell A=(\varprojlim_r A[\ell^r](\overline{\mathbb F}_q))\otimes_{\mathbb Z_\ell}\mathbb Q_\ell$, and $\operatorname{Frob}_q$ is the $q$-power Frobenius. It has degree twice $\dim A$. An isogeny is a surjective homomorphism of abelian varieties with finite kernel. The Jacobian $\operatorname{Jac}(C)$ of a smooth projective geometrically connected curve $C$ is its abelian variety of degree-zero divisor classes.

Statement recorded in the supplied source.

Given a prime power $q$ and a polynomial $f\in\mathbb Z[T]$ promised to be the Weil polynomial of an isogeny class of abelian threefolds over $\mathbb F_q$, determine whether there is a smooth projective geometrically connected genus-three curve $C/\mathbb F_q$ such that $f_{\operatorname{Jac}(C)}=f$. The Jacobian must be defined over the stated base field: geometric existence, or existence only of a quadratic twist of a Jacobian, does not settle this question. [S2]

Short English statement

Decide which finite-field isogeny classes of abelian threefolds contain an actual genus-three Jacobian over that field.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8800046 (AMR-087-0046). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Everett W. Howe, Low-genus curves over finite fields: Problems and variations, IPAM Number Theory and Cryptography: Open Problems workshop (9--13 October 2006), PDF pages 29--32, "The genus-3 existence problem. https://helper.ipam.ucla.edu/publications/SCWS1/SCWS1_6237.pdf.