Regulator Finiteness for Elliptic Curves from ABC

Definitions and mathematical statement

Let $K$ be a fixed number field. For an elliptic curve $E/K$, write \[ r_K(E)=\operatorname{rank}_{{\mathbb{Z}}}\left(E(K)/E(K)_{\mathrm{tors}}\right). \] Choose a basis $P_1,\ldots,P_r$ of this free abelian group, where $r=r_K(E)>0$. Using the Néron--Tate height $\widehat h_E$ associated with the origin divisor, fix the pairing convention \[ \langle P,Q\rangle_E=\frac{\widehat h_E(P+Q)-\widehat h_E(P)-\widehat h_E(Q)}{2}, \qquad \operatorname{Reg}_K(E)=\det\left(\langle P_i,P_j\rangle_E\right)_{1\leq i,j\leq r}. \] The determinant is independent of the chosen integral basis. The supplied record does not specify its height normalization; the displayed convention makes the regulator used here explicit.

For every integer $R\geq1$ and real number $B>0$, put \[ \mathcal E_K(R,B)=\{E/K:\ E\text{ is an elliptic curve},\ 1\leq r_K(E)\leq R,\ \operatorname{Reg}_K(E)\leq B\}. \] The intended conditional finiteness assertion has the following form, once the hypothesis and equivalence relation below are clarified: \[ \text{Assuming ABC},\qquad \forall R\in{\mathbb{Z}}_{\geq1}\ \forall B\in{\mathbb{R}}_{>0},\qquad \#\left(\mathcal E_K(R,B)/{\sim}\right)<\infty. \] Here the source explicitly calls $\sim$ "${\mathbb{Q}}$-isomorphism." For arbitrary elliptic curves defined over $K$, this does not specify whether one means isomorphism over $K$, isomorphism after forgetting to ${\mathbb{Q}}$-schemes, or isomorphism of models defined over ${\mathbb{Q}}$. The last interpretation would require a descent hypothesis absent from the record. Consequently the displayed assertion is a formulation template, not a fully specified conjecture; no equivalence relation has been silently substituted. The source also does not specify whether ABC means its rational or number-field formulation, or precisely which fields its hypothesis must cover.

The older original_statement additionally records a Lang--Silverman conditional finiteness assertion for simple abelian varieties over a fixed number field, with fixed dimension, bounded positive Mordell--Weil rank, and bounded regulator, again using the phrase ${\mathbb{Q}}$-isomorphism classes. The supplied verification says that this duplicated assertion belongs to OWR-14603-007 and was removed from the present target. The present record therefore targets only the ABC consequence for elliptic curves. Its supplied literature assessment still discusses the Lang--Silverman hypothesis; that assessment does not resolve the formulation issues above.

Short English statement

Assuming the intended version of ABC, the current target asks whether elliptic curves over a fixed number field, with positive rank bounded above and regulator bounded above, have only finitely many classes. The source's meaning of ${\mathbb{Q}}$-isomorphism and its version of ABC require clarification. The supplied open label and conditional-result descriptions are source assertions, not independent mathematical verification.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30003110$, exact source code catalogue entry, title "Regulator Finiteness for Elliptic Curves from ABC." No individual proposer is supplied; the proposed year is recorded as 2016.
• Supplied report citation: Diophantische Approximationen (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/21, pp. 1099--1187. https://doi.org/10.4171/owr/2016/21. The report title/year and DOI are reproduced as supplied.
• Supplied literature source, "OWR report": https://ems.press/content/serial-article-files/46625?nt=1.
• Supplied literature source, "Silverman notes": https://swc-math.github.io/aws/2024/2024SilvermanNotes.pdf.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem open and describes the recorded theorem as conditional on Lang--Silverman, with special consequences established but the hypothesis unresolved. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its verification note reports that the official report was checked and that the duplicated Lang--Silverman consequence was removed in favor of the distinct ABC consequence for elliptic curves. This reports the registry's verification claim; no source visit or independent verification is asserted here.