scholarly journals A filtration on the higher Chow group of zero cycles on an abelian variety

2020 ◽  
Vol 72 (4) ◽  
pp. 595-619
Author(s):  
Buntaro Kakinoki
Keyword(s):  
Abelian Variety ◽  
Chow Group ◽  
Higher Chow Group

scholarly journals Beilinson's Hodge Conjecture for Smooth Varieties

2013 ◽  
Vol 11 (2) ◽  
pp. 243-282 ◽  
Author(s):  
Rob de Jeu ◽  
James D. Lewis
Keyword(s):  
Projective Variety ◽  
Chow Group ◽  
Hodge Conjecture ◽  
Generic Point ◽  
Higher Chow Group ◽  
Cycle Class

AbstractLet U/ℂ be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, andclr,m: CHr (U,m) ⊗ ℚ → homMHS (ℚ(0), H2r−m (U, ℚ(r)))the cycle class map. Beilinson once conjectured clr,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of clr,m in more detail (as well as at the “generic point” of U) in terms of kernels of Abel-Jacobi mappings. When r = m, we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of clm,m at the generic point is the same for integral or rational coefficients.

A CONSTRUCTION OF SURFACES WITH LARGE HIGHER CHOW GROUPS

2018 ◽  
Vol 236 ◽  
pp. 311-331
Author(s):  
TOMOHIDE TERASOMA
Keyword(s):  
Power Series ◽  
Linear Algebra ◽  
Chow Groups ◽  
Chow Group ◽  
Spectral Sequences ◽  
Configurations Of Lines ◽  
Higher Chow Group ◽  
Extension Class ◽  
Higher Chow Groups

In this paper, we construct surfaces in $\mathbf{P}^{3}$ with large higher Chow groups defined over a Laurent power series field. Explicit elements in higher Chow group are constructed using configurations of lines contained in the surfaces. To prove the independentness, we compute the extension class in the Galois cohomologies by comparing them with the classical monodromies. It is reduced to the computation of linear algebra using monodromy weight spectral sequences.

scholarly journals Higher Chow cycles on Abelian surfaces and a non-Archimedean analogue of the Hodge--conjecture

2014 ◽  
Vol 150 (4) ◽  
pp. 691-711
Author(s):  
Ramesh Sreekantan
Keyword(s):  
Local Field ◽  
Chow Group ◽  
Abelian Surface ◽  
Hodge Conjecture ◽  
Boundary Map ◽  
Localization Sequence ◽  
Hyperelliptic Jacobians ◽  
Higher Chow Group ◽  
Classical Construction ◽  
Ordinary Reduction

AbstractWe construct new indecomposable elements in the higher Chow group $CH^2(A,1)$ of a principally polarized Abelian surface over a $p$-adic local field, which generalize an element constructed by Collino [Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. 6 (1997), 393–415]. These elements are constructed using a generalization, due to Birkenhake and Wilhelm [Humbert surfaces and the Kummer plane, Trans. Amer. Math. Soc. 355 (2003), 1819–1841 (electronic)], of a classical construction of Humbert. They can be used to prove a non-Archimedean analogue of the Hodge-${\mathcal{D}}$-conjecture – namely, the surjectivity of the boundary map in the localization sequence – in the case where the Abelian surface has good and ordinary reduction.

scholarly journals On the dimension of voisin sets in the moduli space of abelian varieties

2021 ◽  
Author(s):  
E. Colombo ◽  
J. C. Naranjo ◽  
G. P. Pirola
Keyword(s):  
Abelian Variety ◽  
Moduli Space ◽  
Abelian Varieties ◽  
Chow Group ◽  
Hyperelliptic Locus

AbstractWe study the subsets $$V_k(A)$$ V k ( A ) of a complex abelian variety A consisting in the collection of points $$x\in A$$ x ∈ A such that the zero-cycle $$\{x\}-\{0_A\}$$ { x } - { 0 A } is k-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that $$\dim V_k(A) \le k-1$$ dim V k ( A ) ≤ k - 1 and $$\dim V_k(A)$$ dim V k ( A ) is countable for a very general abelian variety of dimension at least $$2k-1$$ 2 k - 1 . We study in particular the locus $${\mathcal {V}}_{g,2}$$ V g , 2 in the moduli space of abelian varieties of dimension g with a fixed polarization, where $$V_2(A)$$ V 2 ( A ) is positive dimensional. We prove that an irreducible subvariety $${\mathcal {Y}} \subset {\mathcal {V}}_{g,2}$$ Y ⊂ V g , 2 , $$g\ge 3$$ g ≥ 3 , such that for a very general $$y \in {\mathcal {Y}}$$ y ∈ Y there is a curve in $$V_2(A_y)$$ V 2 ( A y ) generating A satisfies $$\dim {\mathcal {Y}}\le 2g - 1.$$ dim Y ≤ 2 g - 1 . The hyperelliptic locus shows that this bound is sharp.

scholarly journals ON THE CHOW RING OF CERTAIN LEHN–LEHN–SORGER–VAN STRATEN EIGHTFOLDS

2021 ◽  
pp. 1-24
Author(s):  
CHIARA CAMERE ◽  
ALBERTO CATTANEO ◽  
ROBERT LATERVEER
Keyword(s):  
Chow Group ◽  
Chow Ring ◽  
Similar Statement ◽  
Intersection Product ◽  
Finite Dimensional

Abstract We consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

Computing Galois representations of modular abelian surfaces

2014 ◽  
Vol 17 (A) ◽  
pp. 36-48 ◽  
Author(s):  
Jinxiang Zeng
Keyword(s):  
Abelian Variety ◽  
Efficient Algorithm ◽  
Galois Representations ◽  
Abelian Surfaces ◽  
Genus Two

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f\in S_2(\Gamma _0(N))$ be a normalized newform such that the abelian variety $A_f$ attached by Shimura to $f$ is the Jacobian of a genus-two curve. We give an efficient algorithm for computing Galois representations associated to such newforms.

scholarly journals There are genus one curves of every index over every infinite, finitely generated field

2019 ◽  
Vol 2019 (749) ◽  
pp. 65-86
Author(s):  
Pete L. Clark ◽  
Allan Lacy
Keyword(s):  
Elliptic Curve ◽  
Abelian Variety ◽  
Finitely Generated ◽  
Weil Theorem ◽  
Genus One

Abstract We show that a nontrivial abelian variety over a Hilbertian field in which the weak Mordell–Weil theorem holds admits infinitely many torsors with period any given n>1 that is not divisible by the characteristic. The corresponding statement with “period” replaced by “index” is plausible but open, and it seems much more challenging. We show that for every infinite, finitely generated field K, there is an elliptic curve E_{/K} which admits infinitely many torsors with index any given n>1 .

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