arXiv subjects

Sebastian Petersen

Publications and source records attributed to Sebastian Petersen.

13 recordsLinked to original sources

arXiv2025

Fibration theorems for varieties with the weak Hilbert property

The weak Hilbert property (WHP) for varieties over fields of characteristic zero was introduced by Corvaja and Zannier in 2017. There exist integral variants of WHP for arithmetic schemes. We present new fibration theorems for both the WHP and its integral analogue. Our primary fibration result, in a sense dual to the mixed fibration theorems of Javanpeykar and Luger, establishes for a smooth proper morphism $f: Y \to Z$ of smooth connected varieties, that if $Z$ has the strong Hilbert property (HP) and the generic fiber has WHP, then the total space $Y$ also has WHP. As an application, we use this result in combination with previous work by Corvaja, Demeio, Javanpeykar, Lombardo, and Zannier and in combination with recent work of Javanpeykar to show that certain non-constant abelian schemes over HP varieties possess WHP. For integral WHP, we prove a new fibration theorem for proper smooth morphisms with a section, which generalizes earlier product theorems of Javanpeykar and Wittenberg, and of Luger. A key lemma gives information about the structure of covers of $Y$ whose branch locus is not dominant over $Z$.

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arXiv2024

Finiteness properties of torsion fields of abelian varieties

Let $A$ be an abelian variety defined over a field $K.$ We study finite generation properties of the profinite group $\mathrm{Gal}(\Omega/K)$ and of certain closed normal subgroups thereof, where $\Omega$ is the torsion field of $A$ over $K$. In fact, we establish more general finite generation properties for monodromy groups attached to smooth projective varieties via \'etale cohomology. We apply this in order to give an independent proof and generalizations of a recent result of Checcoli and Dill about small exponent subfields of $\Omega/K$ in the number field case. We also give an application of our finite generation results in the realm of permanence principles for varieties with the weak Hilbert property.

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arXiv2023

Hilbert properties under base change in small extensions

We study the preservation of the Hilbert property and of the weak Hilbert property under base change in field extensions. In particular we show that these properties are preserved if the extension is finitely generated or Galois with finitely generated Galois group, and we also obtain some negative results.

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arXiv2022

Local to global principles for homomorphisms of abelian schemes

Let $A$ and $B$ be abelian varieties defined over the function field $k(S)$ of a smooth algebraic variety $S/k.$ We establish criteria, in terms of restriction maps to subvarieties of $S,$ for existence of various important classes of $k(S)$-homomorphisms from $A$ to $B,$ e.g., for existence of $k(S)$-isogenies. Our main tools consist of Hilbertianity methods, Tate conjecture as proven by Tate, Zarhin and Faltings, and of the minuscule weights conjecture of Zarhin in the case, when the base field is finite.

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arXiv2022

Ramified covers of abelian varieties over torsion fields

We study rational points on ramified covers of abelian varieties over certain infinite Galois extensions of $\mathbb{Q}$. In particular, we prove that every elliptic curve $E$ over $\mathbb{Q}$ has the weak Hilbert property of Corvaja-Zannier both over the maximal abelian extension $\mathbb{Q}^{\rm ab}$ of $\mathbb{Q}$, and over the field $\mathbb{Q}(A_{\rm tor})$ obtained by adjoining to $\mathbb{Q}$ all torsion points of some abelian variety $A$ over $\mathbb{Q}$.

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arXiv2019

Ranks of abelian varieties and the full Mordell-Lang conjecture in dimension one

Let $A$ be a non-zero abelian variety over a field $F$ that is not algebraic over a finite field. We prove that the rational rank of the abelian group $A(F)$ is infinite when $F$ is large in the sense of Pop (also called ample). The main ingredient is a deduction of the 1-dimensional case of the relative Mordell-Lang conjecture from a result of Rössler.

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arXiv2019

On the semisimplicity of reductions and adelic openness for $E$-rational compatible systems over global function fields

Let $X$ be a normal geometrically connected variety over a finite field $κ$ of characteristic~$p$. Let $E$ be a number field. Using automorphic methods over global function fields, we derive properties of the geometric monodromy groups of arbitrary connected $E$-rational semisimple compatible systems $(ρ_λ)$ of $n$-dimensional representations of the arithmetic fundamental group $π_1(X)$, where $λ$ ranges over the finite places of $E$ not above $p$: Let $Λ_λ$ be any $π_1(X)$-stable lattice in $E_λ^n$ under $ρ_λ$. Then for almost all $λ$, the schematic closure of the geometric monodromy $ρ_λ(π_1(X_{\overlineκ}))$ in $\mathrm{Aut}_{\mathcal{O}_λ}(Λ_λ)$ is a semisimple $\mathcal{O}_λ$-group scheme, and its special fiber agrees with the Nori envelope of the geometric monodromy of the mod-$λ$ reduction of $ρ_λ$. A comparable result under different hypotheses was recently proved by Cadoret, Hui and Tamagawa by other methods. We also provide natural criteria for the image of $π_1(X_{\overlineκ})$ under $\prod_λρ_λ$ to have adelic open image in an appropriate sense.

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arXiv2017

A variational open image theorem in positive characteristic

In this note we prove a variational open adelic image theorem for the Galois action on the cohomology of smooth proper $S$-schemes where $S$ is a smooth variety over a finitely generated field of positive characteristic. A central tool is a recent result of Cadoret, Hui and Tamagawa.

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arXiv2017

Group theoretical independence of $\ell$-adic Galois representations

Let $K/\mathbb{Q}$ be a finitely generated field of characteristic zero and $X/K$ a smooth projective variety. Fix $q\in\mathbb{N}$. For every prime number $\ell$ let $ρ_\ell$ be the representation of $\mathrm{Gal}(K)$ on the étale cohomology group $H^q(X_{\overline{K}}, \mathbb{Q}_\ell)$. For a field $k$ we denote by $k_{\mathrm{ab}}$ its maximal abelian Galois extension. We prove that there exist finite Galois extensions $k/\mathbb{Q}$ and $F/K$ such that the restricted family of representations $(ρ_\ell|\mathrm{Gal}(k_{\mathrm{ab}} F))_\ell$ is group theoretically independent in the sense that $ρ_{\ell_1}(\mathrm{Gal}(k_{\mathrm{ab}} F))$ and $ρ_{\ell_2}(\mathrm{Gal}(k_{\mathrm{ab}} F))$ do not have a common finite simple quotient group for all prime numbers $\ell_1\neq \ell_2$.

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arXiv2013

On varieties of Hilbert type

A variety X over a field K is of Hilbert type if the set of rational points X(K) is not thin. We prove that if f: X\to S is a dominant morphism of K-varieties and both S and all fibers f^{-1}(s), s in S(K), are of Hilbert type, then so is X. We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Th'el`ene and Sansuc on algebraic groups.

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arXiv2012

Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion

In this paper we prove the Geyer-Jarden conjecture on the torsion part of the Mordell-Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian varieties with big monodromy, i.e., such that the image of Galois representation on l-torsion points, for almost all primes l, contains the full symplectic group.

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arXiv2012

Independence of $\ell$-adic Galois representations over function fields

Let $K$ be a finitely generated extension of $\mathbb{Q}$. We consider the family of $\ell$-adic representations ($\ell$ varies through the set of all prime numbers) of the absolute Galois group of $K$, attached to $\ell$-adic cohomology of a smooth separated scheme of finite type over $K$. We prove that the fields cut out from the algebraic closure of $K$ by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question asked by Serre in 1991.

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arXiv2012

Big monodromy theorem for abelian varieties over finitely generated fields

An abelian variety over a field K is said to have big monodromy, if the image of the Galois representation on l-torsion points, for almost all primes l contains the full symplectic group. We prove that all abelian varieties over a finitely generated field K with endomorphism ring Z and semistable reduction of toric dimension one at a place of the base field K have big monodromy. We make no assumption on the transcendence degree or on the characteristic of K. This generalizes a recent result of Chris Hall.

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