Regulator Finiteness from Lang–Silverman Heights

Definitions and mathematical statement

Let $K$ be a number field and let $g\geq 1$ be an integer. For an abelian variety $A/K$, its Mordell--Weil rank is \[ r_K(A)=\operatorname{rank}_{{\mathbb{Z}}}\left(A(K)/A(K)_{\mathrm{tors}}\right). \] An abelian variety is simple over $K$ if it has no nonzero proper abelian subvariety defined over $K$; geometric simplicity means simplicity after extension to an algebraic closure. The supplied record does not specify which meaning of simplicity is intended.

To explain the regulator qualification, if a symmetric ample line bundle $L$ on $A$, defined over $K$, is chosen, its Néron--Tate height gives the pairing \[ \langle P,Q\rangle_L=\frac{1}{2}\left(\widehat h_L(P+Q)-\widehat h_L(P)-\widehat h_L(Q)\right). \] For representatives $P_1,\ldots,P_r$ of a ${\mathbb{Z}}$-basis of $A(K)/A(K)_{\mathrm{tors}}$, where $r=r_K(A)\geq1$, the corresponding regulator is \[ \operatorname{Reg}_L(A/K)=\det\left(\langle P_i,P_j\rangle_L\right)_{1\leq i,j\leq r}. \] This determinant is independent of the chosen free basis, but depends on the height data and normalization. The supplied record does not specify a polarization, a line bundle, a rule for choosing one, or a normalization. This definition therefore explains the missing data and does not identify the intended source regulator.

Conditional finiteness statement recorded by the source. Assume the Lang--Silverman height conjecture in the form required by the cited proposition. For every number field $K$, every integer $g\geq1$, every integer rank bound $M\geq1$, and every real regulator bound $B>0$, the intended assertion is that \[ \#\left( \left\{A/K:\ \dim A=g,\ A\text{ simple},\ 1\leq r_K(A)\leq M,\ \operatorname{Reg}(A/K)\leq B\right\}\big/\!\sim_{{\mathbb{Q}}} \right)<\infty. \] Here $\operatorname{Reg}$, the meaning of simplicity, and $\sim_{{\mathbb{Q}}}$ retain the unresolved source conventions. In particular, $\sim_{{\mathbb{Q}}}$ records the supplied phrase "${\mathbb{Q}}$-isomorphism classes"; it has not been replaced by $K$-isomorphism. For varieties specified only over $K$, that phrase needs an explanation of the objects and morphisms being compared. The exact Lang--Silverman hypothesis, including its uniformity and height conventions, is also absent. Thus the display records the intended quantified form, but these missing conventions prevent a fully determined mathematical proposition.

The cleaned statement uses bounded positive rank. The original extraction instead says both that the rank $m\geq1$ is fixed and that the rank is bounded by $m$. Finiteness for every fixed positive rank and finiteness under every finite positive rank bound are equivalent by a finite union, provided all other conventions and bounds agree.

The supplied dated assessment describes the displayed conditional implication as proved and its Lang--Silverman hypothesis as unresolved. It labels the corresponding unconditional extracted problem open. The explicit statements supplied here retain the conjectural hypothesis; removing it would be a separate unconditional target, not an established conclusion of this draft.

Short English statement

The record asserts conditional finiteness for simple abelian varieties of fixed dimension over a fixed number field, with positive Mordell--Weil rank bounded above and regulator bounded above, assuming Lang--Silverman. Its regulator conventions, simplicity convention, exact conjectural hypothesis, and meaning of ${\mathbb{Q}}$-isomorphism require clarification. The supplied curation calls the conditional implication proved and an unconditional extracted problem open; neither assessment is independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003109$, title "Regulator Finiteness from Lang--Silverman Heights," Oberwolfach Reports Open Problems collection, exact source problem code catalogue entry. No individual proposer is supplied; the recorded proposed year is 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 record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labelled "OWR report": https://ems.press/content/serial-article-files/46625?nt=1.
• Supplied literature source, labelled "Silverman notes": https://swc-math.github.io/aws/2024/2024SilvermanNotes.pdf.
• Literature assessment date, copied exactly: 2026-08-21. This is supplied dated curation, not independently verified. It labels the unconditional extracted problem open, describes the displayed conditional implication as proved, and describes its Lang--Silverman hypothesis as unresolved. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry assigns the statement status corrected_verified and reports that the official report was checked and the rank wording corrected. These are supplied verification claims, not verification performed for this draft.