Torsion hypersurfaces on abelian schemes and Betti coordinates

Abstract

In this paper we extend to arbitrary complex coefficients certain finiteness results on Unlikely intersections linked to torsion in abelian surface schemes over a curve, which have been recently proved for the case of algebraic coefficients; in this way we complete the solution of Zilber-Pink conjecture for abelian surface schemes over a curve. As experience has shown also in previous cases, the extension from algebraic to complex coefficients often requires entirely new arguments, whereas simple specialization arguments fail. The outcome gives as a byproduct new finiteness results when the base of the scheme has arbitrary dimension; another consequence is a proof of an expectation of Mazur concerning the structure of the locus in the base when a given section is torsion. Finally, we show the link with an old work of Griffiths and Harris on a higher dimensional extension of a theorem of Poncelet.

Unlikely intersections in families of abelian varieties and the polynomial Pell equation

“…There are at most finitely many specializations such that the point is torsion. As already mentioned in the Introduction, this was proved for g = 2 by Masser and Zannier in [37] and jointly with Corvaja in [14] when everything is defined over C, while the case of arbitrary g has been considered in [38], again with the abelian scheme and the curve defined over the algebraic numbers.…”

“…These will be used to define the set we will apply Proposition to. For more details, we refer to [, the Appendix] and [, Section 2].…”

“…To the authors' knowledge, [, Conjecture 6.1] (for the non‐isotrivial case) has been settled only in the case of a curve in a fibred power of an elliptic scheme when everything is defined over (as a combination of and ), and for a curve in an abelian surface scheme, where the codimension 2 algebraic subgroups of the fibres are torsion subgroups and the matter then reduces to the relative Manin–Mumford problem, settled by Masser and Zannier in a series of articles . The first two papers deal also with the case of a complex curve in a product of complex elliptic schemes, while the same authors with Corvaja handled complex curves in complex simple abelian surface schemes in .…”

Pfaffian definitions of Weierstrass elliptic functions

“…In these papers the maps are a tool in the study of unlikely intersection problems. Very recently, the maps themselves have been studied, for instance in a paper by Corvaja, Masser and Zannier [CMZ2], and a paper by Voisin [V], and also ongoing work by André, Corvaja and Zannier. These maps were also implicitly used in older work, in particular in Manin's famous proof of the Mordell conjecture over function fields [Man].…”

Torsion Points of Sections of Lagrangian Torus Fibrations and the Chow Ring of Hyper-Kähler Manifolds

“…It uses the easy Proposition 2.1 which says that the torsion points of ν are dense in B if the natural locally defined map f ν : B → H 1 (A t0 , R) obtained from ν by a real analytic trivialization of the family of complex tori A → B is generically submersive. This map is called the Betti map in [2], [13]. A key tool for our work is the following very useful result of André-Corvaja-Zannier [2].…”