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.

scholarly journals The Sato-Tate conjecture for Hilbert modular forms

2011 ◽  
Vol 24 (2) ◽  
pp. 411-411 ◽  
Author(s):  
Thomas Barnet-Lamb ◽  
Toby Gee ◽  
David Geraghty
Keyword(s):  
Modular Forms ◽  
Hilbert Modular Forms ◽  
Tate Conjecture ◽  
Hilbert Modular

scholarly journals Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

2016 ◽  
Vol 68 (2) ◽  
pp. 361-394
Author(s):  
Francesc Fité ◽  
Josep González ◽  
Joan-Carles Lario
Keyword(s):  
Abelian Variety ◽  
Complex Multiplication ◽  
Cyclotomic Field ◽  
Explicit Method ◽  
Fermat Curve ◽  
Tate Conjecture ◽  
Tate Group ◽  
Fermat Curves ◽  
Prime Exponent ◽  
Simple Abelian Variety

AbstractLet denote the Fermat curve over ℚ of prime exponent ℓ. The Jacobian Jac() of splits over ℚ as the product of Jacobians Jac(k), 1 ≤ k ≤ ℓ −2, where k are curves obtained as quotients of by certain subgroups of automorphisms of . It is well known that Jac(k) is the power of an absolutely simple abelian variety Bk with complex multiplication. We call degenerate those pairs (ℓ, k) for which Bk has degenerate CM type. For a non-degenerate pair (ℓ, k), we compute the Sato–Tate group of Jac(Ck), prove the generalized Sato–Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of whether (ℓ, k) is degenerate, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the ℓ-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

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