arXiv subjects

Wojciech Gajda

Publications and source records attributed to Wojciech Gajda.

12 recordsLinked to original sources

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 $\mathrm{Gal}(Ω/K)$ and of certain closed normal subgroups thereof, where $Ω$ 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 étale 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 $Ω/K$ 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.

math.NT↗

arXiv2023

Remarks on a theorem of Pink in presence of bad reduction

In this note we prove new cases of the Mumford-Tate conjecture by extending a theorem of Richard Pink for abelian varieties without nontrivial endomorphisms and with bad semistable reduction. We use quadratic pairs introduced by J.G.Thompson in the seventies, an important tool in the program of classifying all simple finite groups. Proof of our main result applies the classification of the quadratic pairs as described by Premet and Suprunenko. Along the way we reprove and generalize a theorem of Chris Hall on the image of Tate module representation of abelian variety as above, to all possible values of its toric dimension.

math.NT↗

arXiv2023

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.

math.AG↗

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.

math.AG↗

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.

math.AG↗

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.

math.AG↗

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.

math.AG↗

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.

math.AG↗

arXiv2009

Linear dependence in Mordell-Weil groups

We consider a local to global principle for detecting linear dependence of nontorsion points, by reduction maps, in the Mordell-Weil group of an abelian variety over a number field.

math.NT↗

arXiv2004

On the image of l-adic Galois representations for abelian varieties of type I and II

In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate, for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre.

math.NT↗

arXiv2004

Detecting linear dependence by reduction maps

We consider the local to global principle for detecting linear dependence of points in groups of the Mordell-Weil type. As applications of our general setting we obtain corresponding statements for Mordell-Weil groups of non{-}CM elliptic curves and some higher dimensional abelian varieties defined over number fields, and also for odd dimensional K-groups of number fields.

math.NT↗

arXiv2002

A support problem for the intermediate Jacobians of l-adic representations

This is a revised version of ANT-0332: "A support problem for the intermediate Jacobians of l-adic representations", by G. Banaszak, W. Gajda & P. Krason, which was placed on these archives on the 29th of January 2002. Following a suggestion of the referee we have subdivided the paper into two separate parts: "Support problem for the intermediate Jacobians of l-adic representations", and "On Galois representations for abelian varieties with complex and real multiplications". Our results on the image of Galois and the Mumford-Tate conjecture for some RM abelian varieties are contained in the second paper. Both papers were accepted for publication.

math.NT↗