Recognizing genus-three Jacobians over the base field

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. A principal polarization $\lambda:A\to A^\vee$ is the isomorphism to the dual abelian variety induced by an ample line bundle. A Jacobian is regarded with its canonical principal polarization $\lambda_C$.

Statement recorded in the supplied source.

For a field $k$ and a principally polarized abelian threefold $(A,\lambda)/k$, determine a criterion for the existence of a smooth projective geometrically connected genus-three curve $C/k$ and an isomorphism of polarized abelian varieties \[ (A,\lambda)\simeq(\operatorname{Jac}(C),\lambda_C) \] defined over $k$. Address the base-field descent obstruction as well as the geometric principal-polarization condition. [S2]

Short English statement

Recognize when a principally polarized abelian threefold is the Jacobian of a curve over its own base field.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8800047 (AMR-087-0047). 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 31--34, "A half-open problem. https://helper.ipam.ucla.edu/publications/SCWS1/SCWS1_6237.pdf.