arXiv subjects

Gabriel Andreas Dill

Publications and source records attributed to Gabriel Andreas Dill.

16 recordsLinked to original sources

arXiv2026

Arithmetic unlikely intersections in powers of the multiplicative group

Inspired by work of Bugeaud-Corvaja-Zannier, we formulate a conjecture about unlikely intersections in powers of the multiplicative group over the ring of integers in a number field. Broadly speaking, if an intersection with a subgroup scheme is unlikely for dimension reasons, its ``size" should not be too big compared to the ``complexity" of the subgroup scheme. We first obtain some results on likely intersections that serve as a benchmark for the unlikely case and generalize work of Barroero-Capuano-Mérai-Ostafe-Sha. We then show that our conjecture in dimension $1$ follows from work of Corvaja-Zannier, we obtain some partial result in dimension $2$, and we present some open problems that are special cases of the conjecture.

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arXiv2025

New evidence for Rémond's generalisation of Lehmer's conjecture

In this article, we generalise a result of Pottmeyer from the multiplicative group of the algebraic numbers to almost split semiabelian varieties defined over number fields. This concerns a consequence of Rémond's generalisation of Lehmer's conjecture. Namely, for a finite rank subgroup $Γ$ of an almost split semiabelian variety $G$, we consider the group of rational points of $G$ over a finite extension of the field generated by the saturated closure of $Γ$, i.e. the division closure of the subgroup generated by $Γ$ and all its images under geometric endomorphisms of $G$. We show that this becomes a free group after one quotients out the saturated closure of $Γ$. The proof uses, amongst other ingredients, a criterion of Pottmeyer, which relies on a result of Pontryagin, together with a result from Kummer theory, of which we reproduce a proof by Rémond.

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arXiv2025

Likely intersections in powers of the multiplicative group

We derive two finiteness properties as consequences of the geometrical non-degeneracy of an algebraic subvariety $W$ of a power of the multiplicative group, concerning the intersections of $W$ with translates of a subtorus $H$ of dimension greater than or equal to the codimension of $W$. The first one is that every translate of $H$ intersects $W$, unless $H$ is contained in one of finitely many proper subtori depending only on $W$. The second one is that every translate of $H$ by a torsion point intersects $W$, unless the translate is contained in one of finitely many proper algebraic subgroups, again depending only on $W$. We use methods from tropical geometry and equidistribution, as well as some very mild model theory.

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arXiv2025

Hecke orbits and the Mordell-Lang conjecture in distinguished categories

Inspired by recent work of Aslanyan and Daw, we introduce the notion of $Σ$-orbits in the general framework of distinguished categories. In the setting of connected Shimura varieties, this concept contains many instances of (generalized) Hecke orbits from the literature. In the setting of semiabelian varieties, a $Σ$-orbit is a subgroup of finite rank. We show that our $Σ$-orbits have useful functorial properties and we use them to formulate two general statements of Mordell-Lang type (one of them implying the other one). We prove an analogue of a recent theorem of Aslanyan and Daw in this general setting, which we apply to deduce an unconditional result about unlikely intersections in a fibered power of the Legendre family. In an appendix, we prove an unconditional Zilber-Pink result for subvarieties of $\mathcal{A}_g$ that cannot be defined over the algebraic numbers.

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arXiv2024

Distinguished categories and the Zilber-Pink conjecture

We propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura varieties. It allows us to define all basic concepts of the field and prove some fundamental facts about them, e.g. the defect condition. In some categories that we call very distinguished, we are able to show some implications between Zilber-Pink statements with respect to base change. This yields unconditional results, i.e. the Zilber-Pink conjecture for a complex curve in $\mathcal{A}_2$ that cannot be defined over $\bar{\mathbb{Q}}$, a complex curve in the $g$-th fibered power of the Legendre family, and a complex curve in the base change of a semiabelian variety over $\bar{\mathbb{Q}}$.

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arXiv2024

On a Galois property of fields generated by the torsion of an abelian variety

In this article, we study a certain Galois property of subextensions of $k(A_{\mathrm{tors}})$, the minimal field of definition of all torsion points of an abelian variety $A$ defined over a number field $k$. Concretely, we show that each subfield of $k(A_{\mathrm{tors}})$ which is Galois over $k$ (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of $k$. As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, i.e. does not contain any infinite set of algebraic numbers of bounded height.

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arXiv2024

Algebraic relations over finite fields that preserve the endomorphism rings of CM $j$-invariants

We characterise the integral affine plane curves over a finite field $k$ with the property that all but finitely many of their $\overline{k}$-points have coordinates that are $j$-invariants of elliptic curves with isomorphic endomorphism rings. This settles a finite field variant of the André-Oort conjecture for $Y(1)^2_\mathbb{C}$, which is a theorem of André. We use our result to solve the modular support problem for function fields of positive characteristic.

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arXiv2022

On morphisms between connected commutative algebraic groups over a field of characteristic $0$

Let $K$ be a field of characteristic $0$ and let $G$ and $H$ be connected commutative algebraic groups over $K$. Let $\text{Mor}_0(G,H)$ denote the set of morphisms of algebraic varieties $G \to H$ that map the neutral element to the neutral element. We construct a natural retraction from $\text{Mor}_0(G,H)$ to $\text{Hom}(G,H)$ (for arbitrary $G$ and $H$) which commutes with the composition and addition of morphisms. In particular, if $G$ and $H$ are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If $G$ has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between $G$ and $H$. We also characterize all connected commutative algebraic groups over $K$ whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.

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arXiv2022

Around the support problem for Hilbert class polynomials

Let $H_D(T)$ denote the Hilbert class polynomial of the imaginary quadratic order of discriminant $D$. We study the rate of growth of the greatest common divisor of $H_D(a)$ and $H_D(b)$ as $|D| \to \infty$ for $a$ and $b$ belonging to various Dedekind domains. We also study the modular support problem: if for all but finitely many $D$ every prime ideal dividing $H_D(a)$ also divides $H_D(b)$, what can we say about $a$ and $b$? If we replace $H_D(T)$ by $T^n-1$ and the Dedekind domain is a ring of $S$-integers in some number field, then these are classical questions that have been investigated by Bugeaud-Corvaja-Zannier, Corvaja-Zannier, and Corrales-Rodrigáñez-Schoof.

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arXiv2021

Torsion points on isogenous abelian varieties

Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If everything is defined over the algebraic numbers and the abelian scheme has maximal variation, we prove that the geometric generic fiber of such a subvariety is a union of torsion cosets. We go on to prove fully or partially explicit versions of this result in fibered powers of the Legendre family of elliptic curves. Finally, we apply a recent result of Galateau-Martínez to obtain uniform bounds on the number of maximal torsion cosets in the Manin-Mumford problem across a given isogeny class. For the proofs, we adapt the strategy, due to Lang, Serre, Tate, and Hindry, of using Galois automorphisms that act on the torsion as homotheties to the family setting.

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arXiv2021

On the frequency of height values

We count algebraic numbers of fixed degree $d$ and fixed (absolute multiplicative Weil) height $\mathcal{H}$ with precisely $k$ conjugates that lie inside the open unit disk. We also count the number of values up to $\mathcal{H}$ that the height assumes on algebraic numbers of degree $d$ with precisely $k$ conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if $k \in \{0,d\}$ or $\gcd(k,d) = 1$. We therefore study the behaviour in the case where $0 < k < d$ and $\gcd(k,d) > 1$ in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function.

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arXiv2019

Unlikely intersections with isogeny orbits in a product of elliptic schemes

Fix an elliptic curve $E_0$ without CM and a non-isotrivial elliptic scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of a fixed finite-rank subgroup (of arbitrary rank) of $E_0^g$, also defined over the algebraic numbers, under all isogenies between $E_0^g$ and some fiber of the $g$-th fibered power $\mathcal{A}$ of the elliptic scheme, where $g$ is a fixed natural number. As a special case of a slightly more general result, we characterize the subvarieties (of arbitrary dimension) inside $\mathcal{A}$ that have potentially Zariski dense intersection with this set. In the proof, we combine a generalized Vojta-Rémond inequality with the Pila-Zannier strategy.

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arXiv2019

On the Zilber-Pink conjecture for complex abelian varieties

In this article, we prove that the Zilber-Pink conjecture for abelian varieties over an arbitrary field of characteristic $0$ is implied by the same statement for abelian varieties over the algebraic numbers. More precisely, the conjecture holds for subvarieties of dimension at most $m$ in the abelian variety $A$ if it holds for subvarieties of dimension at most $m$ in the largest abelian subvariety of $A$ that is isomorphic to an abelian variety defined over $\bar{ \mathbb{Q}}$.

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arXiv2019

Generalized Vojta-Rémond inequality

Following and generalizing unpublished work of Ange, we prove a generalized version of Rémond's generalized Vojta inequality. This generalization can be applied to arbitrary products of irreducible positive-dimensional projective varieties, defined over the field of algebraic numbers, instead of powers of one fixed such variety. The proof runs closely along the lines of Rémond's proof.

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arXiv2019

Unlikely intersections between isogeny orbits and curves

Fix an abelian variety $A_0$ and a non-isotrivial abelian scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of translates of a fixed finite-rank subgroup of $A_0$, also defined over the algebraic numbers, by abelian subvarieties of $A_0$ of codimension at least $k$ under all isogenies between $A_0$ and some fiber of the abelian scheme. We characterize the curves inside the abelian scheme which are defined over the algebraic numbers, dominate the base curve and potentially intersect this set in infinitely many points. Our proof follows the Pila-Zannier strategy.

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arXiv2016

Effective Approximation and Diophantine Applications

Using the Thue-Siegel method, we obtain effective improvements on Liouville's irrationality measure for certain one-parameter families of algebraic numbers, defined by equations of the type $(t-a)Q(t)+P(t)=0$. We apply these to some corresponding Diophantine equations. We obtain bounds for the size of solutions, which depend polynomially on $a$, and bounds for the number of these solutions, which are independent of $a$ and in some cases even independent of the degree of the equation.

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