Reported Height Theorem for Abelian Isogeny Classes

Definitions and mathematical statement

An isogeny of abelian varieties over a field $K$ is a surjective homomorphism over $K$ with finite kernel. The isogeny class of an abelian variety depends on the field over which isogenies are required to be defined. The supplied extract does not specify that field, the height function, or the equivalence relation used to count members of an isogeny class.

The extract attributes to Mark Kisin, in joint work with Lucia Mocz, a theorem discussed under the title Heights in the isogeny class of an abelian variety. According to the supplied statement-verification note, this is a bounded-height finiteness theorem conditional on the Mumford--Tate conjecture. The extract records that the theorem is unconditional in cases where that conjecture is already known.

For clarity, let $\mathsf{MT}(A)$ denote the Mumford--Tate conjecture relevant to an abelian variety $A$, and let $\mathsf{T}(A)$ stand for the omitted theorem's conclusion. The reported logical relationship is schematically \[ \mathsf{MT}(A)\Longrightarrow\mathsf{T}(A), \] subject to all hypotheses of the omitted theorem. This is only a description of conditional dependence: $\mathsf{T}(A)$ is a placeholder, and the display is not a recovered mathematical theorem or a universally quantified conjecture about arbitrary abelian varieties.

Unresolved formulation. The supplied material does not give the theorem's complete hypotheses, height normalization, finiteness objects, boundary conditions, or quantifiers. Consequently, a precise bounded-height finiteness statement cannot be recovered from this record. The supplied verification expressly says that the report does not pose the unconditional extension as an open problem. No such extension is asserted or proposed here.

Short English statement

The record reports that a height theorem for abelian isogeny classes becomes unconditional whenever its Mumford--Tate hypothesis is known. It omits the theorem itself, so it does not determine a precise mathematical problem. Its supplied open label must be read together with the supplied unrecoverable-statement designation and the warning that the report does not pose an unconditional extension as an open problem.

Sources

• Local problem registry, supplied complete source record: permanent UnsolvedMath ID $30006014$, source code catalogue entry, title Reported Height Theorem for Abelian Isogeny Classes; collection: Oberwolfach Reports Open Problems. The supplied record has status open, statement status unrecoverable, and no clean statement.
• Supplied report citation: Arithmetic Geometry (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/33, pp. 1855--1912. Supplied source URL and literature-source link labeled "OWR report": https://doi.org/10.4171/owr/2024/33. The record separately gives proposed year 2024; the year discrepancy is retained without resolution.
• Supplied contribution attribution: Mark Kisin, joint work with Lucia Mocz, Heights in the isogeny class of an abelian variety. This attribution does not identify a proposer of a separate open problem; the proposer field is empty.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified here. It labels the general issue open and describes the extract as a conditionality observation, not a resolution of the Mumford--Tate conjecture or the associated height problem. The accompanying curation requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification note says that the official report presents a bounded-height finiteness theorem conditional on the Mumford--Tate conjecture, identifies already unconditional cases, and does not pose the unconditional extension as an open problem. It states that no separate clean statement is promoted and that the original record is retained publicly for provenance. These are supplied verification claims; the report has not been independently consulted here.