scholarly journals On the $p$-adic distribution of torsion values for a section of an abelian scheme

2020 ◽  
Vol 31 (2) ◽  
pp. 285-295
Author(s):  
Brian Lawrence ◽  
Umberto Zannier
Keyword(s):  
Abelian Scheme

scholarly journals Duality for the de Rham cohomology of an abelian scheme

1998 ◽  
Vol 48 (5) ◽  
pp. 1379-1393 ◽  
Author(s):  
Robert F. Coleman
Keyword(s):  
De Rham Cohomology ◽  
Abelian Scheme ◽  
De Rham

scholarly journals The realization of the degree zero part of the motivic polylogarithm on abelian schemes in Deligne–Beilinson cohomology

2017 ◽  
Vol 13 (09) ◽  
pp. 2471-2485 ◽  
Author(s):  
Danny Scarponi
Keyword(s):  
Chow Groups ◽  
Degree Zero ◽  
Deligne Cohomology ◽  
Abelian Scheme ◽  
Logarithmic Singularities ◽  
Abelian Schemes

In 2014, Kings and Rössler showed that the realization of the degree zero part of the abelian polylogarithm in analytic Deligne cohomology can be described in terms of a class of currents which was previously defined by Maillot and Rössler and strongly related to the Bismut–Köhler higher torsion form of the Poincaré bundle. In this paper we show that, if the base of the abelian scheme is proper, Kings and Rössler’s result can be refined to hold already in Deligne–Beilinson cohomology. More precisely, by means of Burgos’ theory of arithmetic Chow groups, we prove that the class of currents defined by Maillot and Rössler has a representative with logarithmic singularities at the boundary and therefore defines an element in Deligne–Beilinson cohomology. This element coincides with the realization of the degree zero part of the motivic polylogarithm on abelian schemes in Deligne–Beilinson cohomology.

scholarly journals ON THE RESIDUE OF EISENSTEIN CLASSES OF SIEGEL VARIETIES

2017 ◽  
Vol 60 (3) ◽  
pp. 539-553
Author(s):  
FRANCESCO LEMMA
Keyword(s):  
Motivic Cohomology ◽  
Abelian Scheme ◽  
Arbitrary Genus ◽  
Direct Corollary

AbstractEisenstein classes of Siegel varieties are motivic cohomology classes defined as pull-backs by torsion sections of the polylogarithm prosheaf on the universal abelian scheme. By reduction to the Hilbert–Blumenthal case, we prove that the Betti realization of these classes on Siegel varieties of arbitrary genus have non-trivial residue on zero-dimensional strata of the Baily–Borel–Satake compactification. A direct corollary is the non-vanishing of a higher regulator map.

scholarly journals The Betti map associated to a section of an abelian scheme

2020 ◽  
Vol 222 (1) ◽  
pp. 161-202
Author(s):  
Y. André ◽  
P. Corvaja ◽  
U. Zannier
Keyword(s):  
Abelian Scheme

scholarly journals Relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano fibrations

2013 ◽  
Vol 155 (1) ◽  
pp. 129-153 ◽  
Author(s):  
ANA CRISTINA LÓPEZ MARTÍN ◽  
DARÍO SÁNCHEZ GÓMEZ ◽  
CARLOS TEJERO PRIETO
Keyword(s):  
Isometric Isomorphism ◽  
Abelian Scheme ◽  
Arbitrary Base ◽  
Fibre Products ◽  
Fano Fibrations ◽  
Abelian Schemes

AbstractWe study the group of relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano or anti-Fano fibrations. For Weierstraß and Fano or anti-Fano fibrations we describe this group completely. For abelian schemes over an arbitrary base we prove that if two of them are relative Fourier–Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. If the base is normal and the slope map is surjective we show that these two conditions are equivalent. Moreover in this situation we completely determine the group of relative Fourier–Mukai transforms and we prove that the number of relative Fourier–Mukai partners of a given abelian scheme over a normal base is finite.

scholarly journals Generic rank of Betti map and unlikely intersections

2020 ◽  
Vol 156 (12) ◽  
pp. 2469-2509
Author(s):  
Ziyang Gao
Keyword(s):  
Shimura Varieties ◽  
Connected Subset ◽  
Geometric Criterion ◽  
The Real ◽  
Formula One ◽  
Abelian Scheme ◽  
Generic Rank ◽  
Irreducible Variety ◽  
Simply Connected ◽  
Bogomolov Conjecture

Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

On the structure of flat families of complete homogeneous varieties

2017 ◽  
Vol 28 (11) ◽  
pp. 1750079
Author(s):  
Preena Samuel
Keyword(s):  
Automorphism Group ◽  
Homogeneous Space ◽  
Group Scheme ◽  
Base Change ◽  
Neutral Component ◽  
Natural Action ◽  
Abelian Scheme ◽  
Closed Fields ◽  
Algebraically Closed Fields ◽  
Homogeneous Varieties

In this paper, we investigate flat families of complete homogeneous varieties. Over a reduced, noetherian base of characteristic 0, such a family turns out to be a homogeneous space under the natural action of the neutral component of its automorphism group scheme; further, after an étale base change, such a family can be expressed as a product of an abelian scheme and a Borel scheme. The structure of the neutral component of the automorphism group scheme of such a family is also obtained. These results extend already known structure results for complete homogeneous varieties over algebraically closed fields.

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