Related papers: Reduction of abelian varieties

We obtain necessary and sufficient conditions for abelian varieties to acquire semistable reduction over fields of low degree. Our criteria are expressed in terms of torsion points of small order defined over unramified extensions.

We study N\'eron models of pseudo-Abelian varieties over excellent discrete valuation rings of equal characteristic $p>0$ and generalize the notions of good reduction and semiabelian reduction to such algebraic groups. We prove that the…

We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame…

It is well known that the rational torsion of an abelian variety defined over a number field injects into the reduction modulo any sufficiently large prime, so the order of the torsion group divides the greatest common divisor of the sizes…

Let A be an abelian variety defined over a number field K, the number of torsion points rational over a finite extension L is bounded polynomially in terms of the degree [L : K]. When A is isogenous to a product of simple abelian varieties…

We study a certain class of simple abelian varieties of type $\mathrm{IV}$ (in Albert's classification) over number fields with Mumford-Tate groups of type $A$. In particular, we show that such abelian varieties have ordinary reduction away…

Let $K$ be a field which is complete for a discrete valuation. We prove a logarithmic version of the N\'eron-Ogg-Shafarevich criterion: if $A$ is an abelian variety over $K$ which is cohomologically tame, then $A$ has good reduction in the…

We study the special fibers of a certain class of absolutely simple abelian varieties over number fields with endomorphism rings $\bz$ and possessing $l$-adic monodromy groups of the least possible rank. We also study the Dirichlet density…

Let $A$ be an abelian variety defined over a number field $K$. The number of torsion points that are rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$ of $L$ over $K$. Under the following three…

Let $A$ be an abelian variety defined over a number field $K$, the number of torsion points rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$. When $A$ is isogenous to a product of simple abelian…

In this paper we investigate linear dependence of points in Mordell-Weil groups of abelian varieties via reduction maps. In particular we try to determine the conditions for detecting linear dependence in Mordell-Weil groups via finite…

The main result of the paper is that if $A$ is an abelian variety over a subfield $F$ of ${\bold C}$, and $A$ has purely multiplicative reduction at a discrete valuation of $F$, then the Hodge group of $A$ is semisimple. Further, we give…

We give a uniform bound on the degree of the maximal torsion cosets for subvarieties of an abelian variety. The proof combines algebraic interpolation and a theorem of Serre on homotheties in the Galois representation associated to the…

Let G be the product of an abelian variety and a torus defined over a number field K. Let R_1,..., R_n be points in G(K). Let l be a rational prime and let a_1,..., a_n be non-negative integers. Consider the set of primes p of K satisfying…

In this article, we investigate the possible torsion subgroups of twists of abelian varieties with good reduction. As an application, we prove a theorem concerning ramified primes over any quadratic extension where odd-order torsion growth…

Using the irreducibility of a natural irreducible representation of the theta group of an ample line bundle on an abelian variety, we derive a bound for the number of $n$-torsion points that lie on a given theta divisor. We present also two…

For an abelian variety $A$ over a number field we study bounds depending only on the dimension of $A$ for the minimal degree $d(A)$ of a field extension over which $A$ acquires semi-stable reduction. We first compute $d(A)$ in terms of the…

Let $A/K$ be an abelian variety over a number field $K$. We prove in this article that a good lower bound (in terms of the degree $[K(P):K]$) for the N\'eron-Tate height of the points $P$ of infinite order modulo every strict abelian…

We extend to the general codimension a lower bound for the essential minimum on abelian varieties found in a previous work, under a conjecture about ordinary primes in abelian varieties. This lower bound is the best expected, ``up to an…

We prove that abelian varieties of small dimension over discrete valuated, stricty henselian ground fields with perfect residue class field obtain semistable reduction after a tamely ramified extension of the ground field. Using this result…