scholarly journals Integral and rational points on algebraic curves of certain types and their Jacobian varieties over number fields

1997 ◽  
Vol 4 (4) ◽  
pp. 1-49
Author(s):  
Masami FUJIMORI
Keyword(s):  
Algebraic Curves ◽  
Number Fields ◽  
Rational Points ◽  
Jacobian Varieties

scholarly journals Galois criterion for torsion points of Drinfeld modules

2021 ◽  
pp. 1-12
Author(s):  
Chien-Hua Chen
Keyword(s):  
Maximal Ideal ◽  
Abelian Varieties ◽  
Number Fields ◽  
Rational Points ◽  
Drinfeld Module ◽  
Torsion Point ◽  
Drinfeld Modules ◽  
Torsion Points ◽  
Ideal Formula ◽  
Almost All

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.

scholarly journals Growth rate of ample heights and the Dynamical Mordell–Lang conjecture

2018 ◽  
Vol 14 (10) ◽  
pp. 2673-2685
Author(s):  
Kaoru Sano
Keyword(s):  
Growth Rate ◽  
Explicit Formula ◽  
Number Fields ◽  
Rational Points ◽  
Positive Answer ◽  
Projective Varieties ◽  
Smooth Projective Varieties

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.

scholarly journals Trinomials defining quintic number fields

2017 ◽  
Vol 13 (07) ◽  
pp. 1881-1894 ◽  
Author(s):  
Jesse Patsolic ◽  
Jeremy Rouse
Keyword(s):  
Elliptic Curve ◽  
Number Field ◽  
Number Fields ◽  
Rational Points

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.

scholarly journals On the Steinitz module and capitulation of ideals

2000 ◽  
Vol 160 ◽  
pp. 1-15
Author(s):  
Chandrashekhar Khare ◽  
Dipendra Prasad
Keyword(s):  
Number Field ◽  
Algebraic Curve ◽  
Class Group ◽  
Algebraic Curves ◽  
Finite Extension ◽  
Number Fields ◽  
Line Bundles ◽  
Exterior Power ◽  
Ring Of Integers ◽  
And Function

AbstractLet L be a finite extension of a number field K with ring of integers and respectively. One can consider as a projective module over . The highest exterior power of as an module gives an element of the class group of , called the Steinitz module. These considerations work also for algebraic curves where we prove that for a finite unramified cover Y of an algebraic curve X, the Steinitz module as an element of the Picard group of X is the sum of the line bundles on X which become trivial when pulled back to Y. We give some examples to show that this kind of result is not true for number fields. We also make some remarks on the capitulation problem for both number field and function fields. (An ideal in is said to capitulate in L if its extension to is a principal ideal.)

Sign in / Sign up

Export Citation Format

Share Document