Related papers: Arithmetic of abelian varieties with constrained t…

We count abelian number fields ordered by arbitrary height function whose generator of tame inertia is restricted to lie in a given subset of the Galois group, and find an explicit formula for the leading constant. We interpret our results…

Faltings proved that there are finitely many abelian varieties of genus $g$ over a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many…

For a prime $\ell$, let $h_\ell(K)$ denote the $\ell$-part of the class number of the number field $K$. We investigate upper bounds for $h_\ell(K)$ when $K$ is quadratic or cubic, particularly in the case in which the discriminant of $K$ is…

In this paper, we prove the non-existence of certain semistable Galois representations of a number field. Our consequence can be applied to some geometric problems. For example, we prove a special case of a Conjecture of Rasmussen and…

We study the $\ell$-torsion subgroup in Jacobians of curves of the form $y^{\ell} = f(x)$ for irreducible $f(x)$ over a finite field $\mathbf{F}_{q}$ of characteristic $p \neq \ell$. This is a function field analogue of the study of…

Let $[X,\lambda]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce…

We prove that the quotient of the Brauer group of a product of varieties over k by the sum of the images of the Brauer groups of factors has finite exponent. The bulk of the proof concerns p-primary torsion in characteristic p. Our approach…

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields. First, we show that if $L/K$ is a finite Galois extension of number fields such that $\Gal(L/K)$ does not have an…

Let T be a free Z_p-module of finite rank equipped with a continuous Z_p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a…

We provide a simple method of constructing isogeny classes of abelian varieties over certain fields $k$ such that no variety in the isogeny class has a principal polarization. In particular, given a field $k$, a Galois extension $\ell$ of…

We prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects `discrete Selmer groups' and…

Given a $p$-adic field $K$ and a prime number $\ell$, we count the total number of the isomorphism classes of $p^\ell$-extensions of $K$ having no intermediate fields. Moreover for each group that can appear as Galois group of the normal…

Heavenly abelian varieties are abelian varieties defined over number fields that exhibit constrained $\ell$-adic Galois representations for some rational prime $\ell$. At the ICMS Workshop held in November 2024, we presented evidence for…

We carry out a study of groups $G$ in which the index of any infinite subgroup is finite. We call them restricted-finite groups and characterize finitely generated not torsion restricted-finite groups. We show that every infinite…

There is a well known theorem by Deuring which gives a criterion for when the reduction of an elliptic curve with complex multiplication (CM) by the ring of integers of an imaginary quadratic field has ordinary or supersingular reduction.…

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…

Let $K/\mathbf{Q}$ be a finite Galois extension. The P\'olya group of $K$ is the subgroup of the class group $Cl(K)$, generated by the classes of ambiguous ideals of $K$. In this note, among other results, we prove that every finite abelian…

In this paper we consider the problem of Galois descent for suitably completed algebraic K-theory of fields. One of the main results is a suitable form of rigidity for Borel-style generalized equivariant cohomology with respect to certain…

Let $A$ be an abelian variety with commutative endomorphism algebra over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by a Weil polynomial $f_A$ without multiple roots. We give a classification of the groups of…

The aim of this paper is to extend our old results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.