Unspecified Abelian Surface BSD Conjecture

Definitions and mathematical statement

The supplied record does not determine a complete open problem. Its extraction joins an incomplete discussion of a converse theorem to an abelian-surface application of the Birch--Swinnerton-Dyer conjecture. The following notation and conditional fragment record the recoverable mathematical content; they do not supply the missing hypotheses or identify an open conjecture.

An abelian surface over ${\mathbb{Q}}$ is an abelian variety of dimension $2$. For such a surface $A$, write $A({\mathbb{Q}})$ for its group of rational points and $\operatorname{End}_{{\mathbb{Q}}}(A)$ for its ring of endomorphisms defined over ${\mathbb{Q}}$. Interpret the extracted notation $X(A/{\mathbb{Q}})$ as the Tate--Shafarevich group \[ \mathcal{X}(A/{\mathbb{Q}})=\ker\!\left(H^1({\mathbb{Q}},A)\longrightarrow\prod_v H^1({\mathbb{Q}}_v,A)\right), \] where $v$ ranges over all places of ${\mathbb{Q}}$ and $H^1$ denotes Galois cohomology. For a prime $p$, its $p$-primary subgroup is \[ \mathcal{X}(A/{\mathbb{Q}})[p^\infty]=\{x\in\mathcal{X}(A/{\mathbb{Q}}):p^n x=0\text{ for some integer }n\geq 1\}. \] Write $L(A,s)$ for the Hasse--Weil $L$-function, with $L(A,1)$ denoting the value used in the excerpt; its analytic prerequisites are not specified there.

The excerpt labels the following as a proved result, "Theorem 5 (L.--Zerbes)": if $A/{\mathbb{Q}}$ is an abelian surface satisfying $\operatorname{End}_{{\mathbb{Q}}}(A)={\mathbb{Z}}$, is "deformable at $p$", satisfies "various other conditions", and has $L(A,1)\ne 0$, then \[ \#A({\mathbb{Q}})<\infty \qquad\text{and}\qquad \#\mathcal{X}(A/{\mathbb{Q}})[p^\infty]<\infty. \] This is only a transcription of an incompletely specified source claim. Neither deformability at $p$ nor the additional conditions are defined in the supplied record, so this implication cannot be promoted to a fully specified theorem or conjecture.

The accompanying discussion concerns weight $(2,2)$ eigenforms associated with modular abelian surfaces and an eigenvariety parametrising $p$-adic families for $\operatorname{GSp}_4$. The excerpt says these nonregular weights are not immediately covered by an unspecified preceding theorem and that the eigenvariety need not be $\acute{e}$tale over weight space. It reports reasons to expect failure of $\acute{e}$taleness at certain points associated with surfaces induced from elliptic curves over imaginary quadratic fields. Deformability at $p$ is described as slightly weaker than $\acute{e}$taleness and expected to hold for imaginary-quadratic lifts; these descriptions are not definitions or established universal assertions. The source also reports then-ongoing work by James Rawson, described as the speaker's PhD student, on a computationally practical test for deformability.

The preceding, disconnected fragment attributes to Rubin a CM converse theorem in the $p$-ordinary case, explained there as $p$ splitting in an imaginary quadratic field $K$, under finiteness of the $p$-part of a Tate--Shafarevich group. The object and converse theorem are not identified. The subsequent words "Recently Burungale-Tian removed" are incomplete and specify no removed hypothesis. Consequently, no precise open assertion can be recovered from that fragment either.

Short English statement

The record contains an incomplete, source-attributed finiteness theorem for rational points and a Tate--Shafarevich group of an abelian surface, together with a disconnected CM converse-theorem fragment. Missing hypotheses and an undefined deformability condition prevent a precise standalone formulation. The supplied open label does not identify a recoverable open conjecture.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005568$, exact source code catalogue entry, title "Unspecified Abelian Surface BSD Conjecture". The registry supplies status "open", statement status "unrecoverable", and a verification note explaining that the extraction joins a proved Gross--Zagier converse discussion to an abelian-surface BSD application theorem without stating a separate open conjecture. These are supplied registry classifications, not independently verified mathematical conclusions.
• Supplied report citation: Algebraische Zahlentheorie (2023), Oberwolfach Reports, DOI 10.4171/owr/2023/28, pp. 1543--1606. Supplied source URL and literature-source link, labelled "Oberwolfach Report 28/2023": https://doi.org/10.4171/owr/2023/28. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No precise passage location or complete theorem citation is supplied.
• Attribution retained from the extraction: "Theorem 5 (L.--Zerbes)", Rubin, Burungale--Tian, and James Rawson. The initial "L." is not expanded in the supplied record; no separate references for these attributions are supplied.
• Supplied literature assessment checked on 2026-08-22: status "open"; the extracted fragment does not identify the conjecture, and surrounding conditional and case-specific results leave the intended incomplete claim unresolved for triage. This is supplied dated curation, not independently verified source consultation or a new literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.