Rational points of abelian varieties with values in towers of number fields

1972 ◽  
Vol 18 (3-4) ◽  
pp. 183-266 ◽  
Author(s):  
Barry Mazur
Author(s):  
Chien-Hua Chen

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.


2004 ◽  
Vol 2004 (52) ◽  
pp. 2795-2808 ◽  
Author(s):  
Victor Rotger

We consider principally polarized abelian varieties with quaternionic multiplication over number fields and we study the field of moduli of their endomorphisms in relation to the set of rational points on suitable Shimura varieties.


2018 ◽  
Vol 154 (4) ◽  
pp. 850-882
Author(s):  
Yunqing Tang

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.


2018 ◽  
Vol 14 (10) ◽  
pp. 2673-2685
Author(s):  
Kaoru Sano

We provide an explicit formula on the growth rate of ample heights of rational points under iteration of endomorphisms of smooth projective varieties over number fields. As an application, we give a positive answer to a variant of the Dynamical Mordell–Lang conjecture for pairs of étale endomorphisms, which is also a variant of the original one stated by Bell, Ghioca, and Tucker in their monograph.


2017 ◽  
Vol 13 (07) ◽  
pp. 1881-1894 ◽  
Author(s):  
Jesse Patsolic ◽  
Jeremy Rouse

Given a quintic number field K/ℚ, we study the set of irreducible trinomials, polynomials of the form x5 + ax + b, that have a root in K. We show that there is a genus 4 curve CK whose rational points are in bijection with such trinomials. This curve CK maps to an elliptic curve defined over a number field, and using this map, we are able (in some cases) to determine all the rational points on CK using elliptic curve Chabauty.


2012 ◽  
Vol 08 (01) ◽  
pp. 255-264
Author(s):  
ANTONELLA PERUCCA

Let G be the product of an abelian variety and a torus defined over a number field K. The aim of this paper is detecting the dependence among some given rational points of G by studying their reductions modulo all primes of K. We show that if some simple conditions on the order of the reductions of the points are satisfied then there must be a dependency relation over the ring of K-endomorphisms of G. We generalize Larsen's result on the support problem to several points on products of abelian varieties and tori.


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