arXiv subjects

Nuno Hultberg

Publications and source records attributed to Nuno Hultberg.

5 recordsLinked to original sources

arXiv2026

New gap principle for semiabelian varieties using globally valued fields

Hrushovski observed that the new gap principle of Gao-Ge-Kühne is essentially equivalent to the Bogomolov conjecture over arbitrary globally valued fields of characteristic $0$. Building on this observation, we prove a new gap principle for semiabelian varieties by reducing the Bogomolov conjecture for semiabelian varieties to the Bogomolov conjecture for abelian varieties over arbitrary GVFs. This reduction remains valid in positive characteristic; however, the corresponding Bogomolov conjecture for abelian varieties is not yet known in that setting. We prove an unconditional new gap principle in positive characteristic for semiabelian varieties whose abelian quotient is an elliptic curve.

math.NT↗

arXiv2024

Continuity of heights in families and complete intersections in toric varieties

We study the variation of heights of cycles in flat families over number fields or, more generally, globally valued fields. To a finite type scheme over a GVF we associate a locally compact Hausdorff space which we refer to as its GVF analytification. For a flat projective family, we prove that the height of fibres is a continuous function on the GVF analytification of the base. As an application, we prove Roberto Gualdi's conjecture on limit heights of complete intersections in toric varieties.

math.NT↗

arXiv2024

Arakelov geometry of toric bundles: Okounkov bodies and BKK

This article introduces the study of toric bundles and the morphisms between them from the perspective of adelic fibre bundles, as introduced by Chambert-Loir and Tschinkel. We study the Okounkov bodies and Boucksom-Chen transforms of suitable adelic line bundles on toric bundles. Finally, we prove an arithmetic analogue of a formula for intersection numbers due to Hofscheier, Khovanskii and Monin. We apply this to the study of compactifications of semiabelian varieties, whose height and successive minima we compute. This extends computations of Chambert-Loir to arbitrary toric compactifications.

math.NT↗

arXiv2024

A linear AFL for quaternion algebras

We prove new fundamental lemma and arithmetic fundamental lemma identities for general linear groups over quaternion division algebras. In particular, we verify the transfer conjeture and the arithmetic transfer conjecture from arXiv:2307.11716 in cases of Hasse invariant 1/2.

math.NT↗

arXiv2024

Fields with few small points

Let $X$ be a projective variety over a number field $K$ endowed with a height function associated to an ample line bundle on $X$. Given an algebraic extension $F$ of $K$ with a sufficiently big Northcott number, we can show that there are finitely many cycles in $X_{\bar{\mathbb{Q}}}$ of bounded degree defined over $F$. Fields $F$ with the required properties were explicitly constructed in arXiv:2107.09027 and arXiv:2204.04446, motivating our investigation. We point out explicit specializations to canonical heights associated to abelian varieties and selfmaps of $\mathbb{P}^n$. We apply similar methods to the study of CM-points. As a crucial tool, we introduce a refinement of Northcott's theorem.

math.NT↗