Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1209.3653

Families of abelian varieties with many isogenous fibres

Martin Orr
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let Z be a subvariety of the moduli space of principally polarised abelian varieties of dimension g over the complex numbers. Suppose that Z contains a Zariski dense set of points which correspond to abelian varieties from a single isogeny class. A generalisation of a conjecture of André and Pink predicts that Z is a weakly special subvariety. We prove this when dim Z = 1 using the Pila--Zannier method and the Masser--Wüstholz isogeny theorem. This generalises results of Edixhoven and Yafaev when the Hecke orbit consists of CM points and of Pink when it consists of Galois generic points.

Explore related subjects

math.AGmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Martin Orr. 2016-09-13. Families of abelian varieties with many isogenous fibres. https://doi.org/10.1515/crelle-2013-0058

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

G3-Criteria and Applications

The G3-property of a subvariety was introduced by Hironaka-Matsumura, and plays an important role for deducing connectedness and extension results. Unfortunately, it's a rather elusive notion, which is not always easy to establish. Most of the existing work is concentrated on subvarieties of homogeneous varieties. The first goal of this article is to show that mobility assumptions on the subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a certain partial positivity property, slightly stronger than G3, previously introduced by the author. Second, we apply the result to prove that, in numerous situations, the splitting of the normal bundle of a smooth two-codimensional subvariety implies that it is a complete intersection.

math.AG↗
arXiv2026

Nodal degeneration of chiral algebras I: Global structure and gluing formula

We define a natural extension of a universal factorization algebra $\mathcal{A}$ to families of stable punctured curves, by integrating over all semistable modifications. We prove that the resulting sheaf of factorization homology satisfies a natural gluing formula, by tensoring over a certain derived associative algebra $\mathfrak{Z}_{\mathcal{A}}^0$, generalizing the Verlinde formula for gluing of conformal blocks.

math.AG↗