scholarly journals A Variational Tate Conjecture in crystalline cohomology

2019 ◽  
Vol 21 (11) ◽  
pp. 3467-3511 ◽  
Author(s):  
Matthew Morrow
Keyword(s):  
Crystalline Cohomology ◽  
Tate Conjecture

scholarly journals Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz theorem

2019 ◽  
Vol 155 (5) ◽  
pp. 1025-1045
Author(s):  
Christopher Lazda ◽  
Ambrus Pál
Keyword(s):  
Perfect Field ◽  
Global Function ◽  
Crystalline Cohomology ◽  
Tate Conjecture ◽  
Global Function Fields ◽  
Counter Example ◽  
Rigid Cohomology ◽  
De Rham ◽  
First Chern Class ◽  
Special Fibre

In this paper we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that for $k$ a perfect field of characteristic $p$ , a rational (logarithmic) line bundle on the special fibre of a semistable scheme over $k\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham–Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter-example to show that the variational Tate conjecture for divisors cannot hold with $\mathbb{Q}_{p}$ -coefficients.

Monodromy and the Tate Conjecture: Picard numbers and Mordell-Weil ranks in families

2000 ◽  
Vol 120 (1) ◽  
pp. 47-79 ◽  
Author(s):  
A. Johan de Jong ◽  
Nicholas M. Katz
Keyword(s):  
Tate Conjecture

scholarly journals The Sato-Tate conjecture on average for small angles

2008 ◽  
Vol 361 (04) ◽  
pp. 1811-1832 ◽  
Author(s):  
Stephan Baier ◽  
Liangyi Zhao
Keyword(s):  
Tate Conjecture

scholarly journals Cycles in the de Rham cohomology of abelian varieties over number fields

2018 ◽  
Vol 154 (4) ◽  
pp. 850-882
Author(s):  
Yunqing Tang
Keyword(s):  
Endomorphism Ring ◽  
Abelian Varieties ◽  
Number Fields ◽  
De Rham Cohomology ◽  
Tate Conjecture ◽  
De Rham ◽  
Prime Dimension

In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of$\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus’ prediction for some families of abelian varieties. These families include geometrically simple abelian varieties of prime dimension that have non-trivial endomorphism ring. The proof uses a crystalline analogue of Faltings’ isogeny theorem due to Bost and the known cases of the Mumford–Tate conjecture.

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

scholarly journals On the Sato-Tate conjecture for QM-curves of genus two

1999 ◽  
Vol 68 (228) ◽  
pp. 1649-1663 ◽  
Author(s):  
Ki-ichiro Hashimoto ◽  
Hiroshi Tsunogai
Keyword(s):  
Tate Conjecture ◽  
Genus Two

Period relations and the Tate conjecture for Hilbert modular surfaces

1987 ◽  
Vol 89 (2) ◽  
pp. 319-345 ◽  
Author(s):  
V. Kumar Murty ◽  
Dinakar Ramakrishnan
Keyword(s):  
Tate Conjecture ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular

scholarly journals Integral canonical models for Spin Shimura varieties

2015 ◽  
Vol 152 (4) ◽  
pp. 769-824 ◽  
Author(s):  
Keerthi Madapusi Pera
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
K3 Surfaces ◽  
Shimura Varieties ◽  
Good Reduction ◽  
Intersection Numbers ◽  
Orthogonal Groups ◽  
Canonical Models ◽  
Tate Conjecture ◽  
Precise Sense

We construct regular integral canonical models for Shimura varieties attached to Spin and orthogonal groups at (possibly ramified) primes$p>2$where the level is not divisible by$p$. We exhibit these models as schemes of ‘relative PEL type’ over integral canonical models of larger Spin Shimura varieties with good reduction at$p$. Work of Vasiu–Zink then shows that the classical Kuga–Satake construction extends over the integral models and that the integral models we construct are canonical in a very precise sense. Our results have applications to the Tate conjecture for K3 surfaces, as well as to Kudla’s program of relating intersection numbers of special cycles on orthogonal Shimura varieties to Fourier coefficients of modular forms.

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