
Jerson CaroBoston University | BU · Department of Mathematics and Statistics
Jerson Caro
Doctor of Philosophy
Postdoctoral researcher at Boston University
About
14
Publications
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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)
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...
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...
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...
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...
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...
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{...
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...
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...
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...
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...
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...
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...
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...