Torsion Specializations of Sections of Abelian Schemes

Definitions and mathematical statement

Let $C$ be a complex algebraic curve, let $\pi:A\to C$ be a family of abelian varieties of relative dimension at least two, and let $s:C\to A$ be an algebraic section, so that $\pi\circ s=\operatorname{id}_C$. For $p\in C({\mathbb{C}})$, write $A_p$ for the fiber and $0_p$ for its identity. A point $x\in A_p({\mathbb{C}})$ is torsion if $[n]_{A_p}x=0_p$ for some integer $n\geq1$.

For a scheme-theoretic formulation, interpret the family as an abelian scheme over $C$. Interpret an algebraic subgroup scheme as a closed subgroup scheme $H\hookrightarrow A$ over $C$, with the group operations inherited from $A$; it need not be connected. Here "proper subgroup scheme" means $H\neq A$, not a condition on the properness of its structure morphism. The supplied record does not explicitly specify these conventions or whether $C$ is smooth, irreducible, or complete, so these aspects require confirmation against the source.

Question under this interpretation. Suppose that \[ s(C)\not\subseteq H \qquad\text{for every closed subgroup scheme }H\subsetneq A\text{ over }C. \] Must the set \[ T_s=\left\{p\in C({\mathbb{C}}):\exists n\in{\mathbb{Z}}_{\geq1},\ [n]_{A_p}s(p)=0_p\right\} \] be finite? The integer $n$ may depend on $p$. The question concerns arbitrary complex coefficients; the supplied formulation imposes no additional hypothesis of non-isotriviality or descent to $\overline{{\mathbb{Q}}}$. This is a statement-only question, not an asserted finiteness theorem.

Short English statement

For a section of a family of abelian varieties of dimension at least two over a complex algebraic curve, ask whether avoiding every proper algebraic subgroup scheme forces torsion values to occur at only finitely many complex points of the base. The precise family and subgroup-scheme conventions require source review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30003117$, title "Torsion Specializations of Sections of Abelian Schemes," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2016.
• Supplied source citation: Diophantische Approximationen (2017), Oberwolfach Reports, pp. 1099--1187, DOI 10.4171/owr/2016/21. https://doi.org/10.4171/owr/2016/21. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, "OWR report": https://ems.press/content/serial-article-files/46625?nt=1.
• Supplied literature source, "Corvaja--Masser--Zannier": https://doi.org/10.1007/s00208-017-1537-z.
• Supplied literature source, "Masser--Zannier": https://doi.org/10.4310/ACTA.2020.v225.n2.a2.
• Literature assessment dated exactly 2026-08-21: this is supplied dated curation, not independently verified. The registry labels the problem partially solved and reports that the curve and abelian-surface cases over complex numbers and several algebraic-number cases are proved, that later work treats arbitrary relative dimension over $\overline{{\mathbb{Q}}}$, and that the full arbitrary-complex-coefficient statement remains unsettled. The supplied background calls this a dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. This supplied label does not constitute independent mathematical verification of this draft.