Related papers: Abelian Varieties and Finitely Generated Galois Gr…

This paper studies the structure of the Mordell--Weil groups of semiabelian varieties over algebraic extensions of number fields whose absolute Galois group is finitely generated, with particular emphasis on that generated by a single…

We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of $\mathbb{Q}$ all of whose decomposition groups are cyclic (resp., abelian). This…

We prove new cases of the Tate conjecture for abelian varieties over finite fields, extending previous results of Dupuy--Kedlaya--Zureick-Brown, Lenstra--Zarhin, Tankeev, and Zarhin. Notably, our methods allow us to prove the Tate…

We prove that the finiteness of a finitely generated category of irreducible algebraic varieties over a field of characteristic zero is decidable. We also obtain a Burnside finiteness criterion for such a category, with applications to…

We study Galois representations attached to nonsimple abelian varieties over finitely generated fields of arbitrary characteristic. We give sufficient conditions for such representations to decompose as a product, and apply them to prove…

We prove that if $f:X \rightarrow A$ is a morphism from a smooth projective variety $X$ to an abelian variety $A$ over a number field $K$, and $G$ is a subgroup of automorphisms of $X$ satisfying certain properties, and if a prime $p$…

We affirmatively answer a conjecture in the preprint ``Essential dimension and algebraic stacks,'' proving that the essential dimension of an abelian variety over a number field is infinite.

In this paper we produce unconditionally new instances of Galois number field extensions exhibiting strong discrepancies in the distribution of Frobenius elements among conjugacy classes of the Galois group. We first prove an inverse Galois…

We realize infinitely many covering groups $2.A_n$ (where $A_n$ is the alternating group) as the Galois group of everywhere unramified Galois extensions over infinitely many quadratic number fields. After several predecessor works…

Let A be an abelian variety of positive dimension defined over a number field K and let Kbar be a fixed algebraic closure of K. For each element sigma of the absolute Galois group Gal(Kbar/K), let Kbar(sigma) be the fixed field of sigma in…

We prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero.

We consider three isogeny invariants of abelian varieties over finite fields: the Galois group, Newton polygon, and the angle rank. Motivated by work of Dupuy, Kedlaya, and Zureick-Brown, we define a new invariant called the weighted…

We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising from Galois representations. In…

In this paper, we prove the geometric Bombieri-Lang conjecture for projective varieties which have finite morphisms to abelian varieties of trivial traces over function fields of characteristic 0. The proof is based on the idea of…

We provide an infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $SL_2(7)$. To my knowledge, this is the first instance of infinitely many such realizations for a perfect group which is…

In this article we study the Galois group of field generated by division points of special class of formal group laws and prove an equivalent condition for the group to be abelian. Further, we explore relations between the endomorphism ring…

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…

Let $K$ be a field, $L$ a finite Galois extension of $K$, and $X$ an abelian variety defined over $L$. If $X$ is isogenous over $L$ to an abelian variety defined over $K$, then the $\ell$-adic Galois representations associated to $X$ extend…

We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms.

We study group extensions of Finite Abelian Groups using matrices. We also prove a Theorem for equivalence of extensions using matrices.