Effective representation of principally polarized abelian threefolds

Definitions and mathematical statement

A principally polarized abelian threefold over a field $k$ is a pair $(A,\lambda)$ where $A/k$ is an abelian variety of dimension three and $\lambda:A\to A^\vee$ is an ample-line-bundle polarization that is an isomorphism. It is a Jacobian over $k$ if there exists a smooth projective geometrically connected genus-three curve $C/k$ and an isomorphism of principally polarized varieties \[ (A,\lambda)\cong(\operatorname{Jac}(C),\lambda_C). \] This differs from becoming a Jacobian only over $\overline k$ or merely being isogenous to one.

Give an effective input representation of $(A,\lambda)$ suitable for deciding this Jacobian property. The representation should include the principal polarization, not just the underlying abelian variety, and provide finite accessible data for whatever arithmetic/geometric operations the intended decision procedure requires.

The source does not specify the base field, its computational presentation, characteristic restrictions, or admissible exact/approximate input model. Algebraic equations with group/polarization data, period matrices, and theta/level structures are possible but inequivalent frameworks; the summary mentions them without selecting one. It also does not state whether the target is the base-field or geometric Jacobian property. Consequently the task is to define a representation/decision framework for a specified category rather than to assume a uniform finite encoding of all complex data or an unprovided general Schottky test.

Short English statement

Specify effective data for a principally polarized abelian variety of dimension three that lets an algorithm decide whether it is a genus-three Jacobian. The base field, polarization encoding, and descent versus geometric question must be fixed.

Sources

• Everett Howe, IPAM Workshop on Number Theory and Cryptography --- Open Problems (2006), attachment 6237, PDF pages 33--34, schedule source https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule; accessed 2026-07-29. The supplied provenance records a PDF/legacy-PPT visual audit.
• The supplied summary names Serre's obstruction $\chi_{18}$ and period/theta/normalized representations without a full reference, computational hypotheses, or a universally applicable criterion.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $8800048$, source code catalogue entry, Effective representation of principally polarized abelian threefolds. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.