Jerson Caro

Jerson Caro
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Jerson verified their affiliation via an institutional email.
Boston University | BU · Department of Mathematics and Statistics

Doctor of Philosophy
Postdoctoral researcher at Boston University

About

14
Publications
670
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18
Citations
Introduction
My research area is Number Theory, with an emphasis on arithmetic geometry and algebraic number theory. In general terms, my main interest is in three topics: finiteness of rational points, the rank of elliptic curves, and the arithmetic of number fields.
Skills and Expertise

Publications

Publications (14)
Preprint
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Let E E be an elliptic curve over the rational numbers. Watkins [Experiment. Math. 11 (2002), pp. 487–502 (2003)] conjectured that the rank of E E is bounded by the 2 2 -adic valuation of the modular degree of E E . We prove this conjecture for semistable elliptic curves having exactly one rational point of order 2 2 , provided that they have an od...
Article
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Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit bound for the number of rational points of a curve $C$ of genus $g\ge 2$ defined over a number field $F$, with Jacobian of rank at most $g-1$. Namely, in the case $F=\mathbb{Q}$, if $p>2g$ is a prime of good reduction, then the number of rational points of $C$ is at most the...
Preprint
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In 2002 Watkins conjectured that given an elliptic curve defined over $\mathbb{Q}$, its Mordell-Weil rank is at most the $2$-adic valuation of its modular degree. We consider the analogous problem over function fields of positive characteristic, and we prove it in several cases. More precisely, every modular semi-stable elliptic curve over $\mathbb...
Preprint
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For an elliptic curve E defined over the field C of complex numbers, we classify all translates of elliptic curves in E 3 such that the x-coordinates satisfy a linear equation. This classification enables us to establish a relation between the rank of finite rank subgroups of E and triples in E whose x-coordinates are linearly related. The method o...
Preprint
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Watkins' conjecture asserts that the rank of an elliptic curve is upper bounded by the $2$-adic valuation of its modular degree. We show that this conjecture is satisfied when $E$ is any quadratic twist of an elliptic curve with rational $2$-torsion and prime power conductor. Furthermore, we give a lower bound of the congruence number for elliptic...
Preprint
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Caro and Pasten gave an explicit upper bound on the number of rational points on a hyperbolic surface that is embedded in an abelian variety of rank at most one. We show how to use their method to produce a refined bound on the number of rational points on the surface $W_2 := C+C$ in the case of a hyperelliptic curve $C$ of genus $3$ over $\mathbb{...
Article
For an elliptic curve E defined over the field of complex numbers, we classify all translates of elliptic curves in E^3 such that the x-coordinates satisfy a linear equation. This classification enables us to establish a relation between the rank of finite rank subgroups of E and triples in E whose x-coordinates are linearly related. The method of...
Article
Full-text available
In 2002 Watkins conjectured that given an elliptic curve defined over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {Q}}}$$\end{document}, its Mordell–Weil...
Article
Watkins's conjecture asserts that the rank of an elliptic curve is upper bounded by the $2$-adic valuation of its modular degree. We show that this conjecture is satisfied when $E$ is any quadratic twist of an elliptic curve with a rational point of order $2$ and prime power conductor, in particular, for the congruent number elliptic curves. Furthe...
Preprint
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The Hessian map is the rational map that sends a homogeneous polynomial to the determinant of its Hessian matrix. We prove that the Hessian map is birational on its image for ternary forms of degree $d\ge 4$, $d\neq 5$, by considering the action of the orthogonal group. In a previous paper we proved the analogous result for binary forms, with more...
Article
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For a nonconstant elliptic surface over $\mathbb {P}^1$ defined over $\mathbb {Q}$ , it is a result of Silverman [‘Heights and the specialization map for families of abelian varieties’, J. reine angew. Math. 342 (1983), 197–211] that the Mordell–Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If th...
Preprint
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For a non-constant elliptic surface over $\mathbb{P}^1$ defined over $\mathbb{Q}$, it is a result of Silverman that the Mordell--Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is non-isotrivial one expects that this bound is an equality for infinitely many fibres, although...
Preprint
Full-text available
Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit bound for the number of rational points of a curve $C$ of genus $g\ge 2$ defined over a number field $F$, with Jacobian of rank at most $g-1$. Namely, in the case $F=\mathbb{Q}$, if $p>2g$ is a prime of good reduction, then the number of rational points of $C$ is at most the...

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