Implicit computation with testing characters and homogeneous spaces

Definitions and mathematical statement

A principal homogeneous space for a group object $G$ over a field $K$ is a $G$-torsor: after a suitable base extension its action is free and transitive. It need not have a $K$-rational point; specifying one would trivialize the torsor. This is standard terminology only, since the source does not identify $G$, $K$, or the torsor category.

The problem asks to work efficiently with the slide's testing characters and principal homogeneous spaces while avoiding explicit construction. The testing characters, their domains/targets, and the tests or pairings they perform are not defined in the record. Nor is the ambient descent, Selmer or cryptographic problem specified.

A precise implicit-computation result would supply a succinct or oracle representation, list the operations to be supported, prove that they reproduce the required explicit-object computations, and bound their resource use in the actual input sizes. Merely computing an explicit character or torsor and suppressing its display would not meet the stated avoidance goal. But without the objects, operations and encoding, no such algorithmic theorem can be formulated uniquely. No particular character map or cohomology-class representation is inferred from the generic names.

Short English statement

Represent and use the source's testing characters and torsors implicitly, with efficient required tests or operations. The objects, operations and input model are not provided.

Sources

• Local UnsolvedMath registry, permanent ID 8800095, exact source code catalogue entry, "Implicit computation with testing characters and homogeneous spaces"; attributed to Ming-Deh Huang, no proposed year supplied. Registry status open, classification OPEN-TRIAGE.
• IPAM Workshop on Number Theory and Cryptography---Open Problems (2006), attachment 6673, PDF page 30, https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule. Supplied pdf-and-legacy-ppt visual audit accessed July 29, 2026; provenance records 17 PDFs/two PPT decks inspected with repeated builds excluded. No PDF/PPT was accessed here.
• The supplied summary mentions implicit Selmer/character computation but says it located no verified application resolving this specific task. It provides no precise defining or algorithmic reference.
• Provenance flags current primary-literature status and redistribution terms NEEDS_REVIEW.