Browsing by Author "Caro, Jerson"

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    A Chabauty-Coleman bound for surfaces
    (2023) Caro, Jerson; Pasten, Hector
    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 = 2 defined over a number field F, with Jacobian of rank at most g - 1. Namely, in the case F = Q, if p > 2g is a prime of good reduction, then the number of rational points of C is at most the number of F-p-points plus a contribution coming from the canonical class of C. We prove a result analogous to Coleman's bound in the case of a hyperbolic surface X over a number field, embedded in an abelian variety A of rank at most one, under suitable conditions on the reduction type at the auxiliary prime. This provides the first extension of Coleman's explicit bound beyond the case of curves. The main innovation in our approach is a new method to study the intersection of a p-adic analytic subgroup with a subvariety of A by means of overdetermined systems of differential equations in positive characteristic.
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    Watkins's conjecture for elliptic curves over function fields
    (2024) Caro, Jerson
    In 2002 Watkins conjectured that given an elliptic curve defined over Q\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 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 Fq(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_q(T)$$\end{document} after extending constant scalars, and every quadratic twist of a modular elliptic curve over Fq(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_q(T)$$\end{document} by a polynomial with sufficiently many prime factors satisfy the analogue of Watkins's conjecture. Furthermore, for a well-known family of elliptic curves with unbounded rank due to Ulmer, we prove the analogue of Watkins's conjecture.
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    WATKINS'S CONJECTURE FOR ELLIPTIC CURVES WITH NON-SPLIT MULTIPLICATIVE REDUCTION
    (2022) Caro, Jerson; Pasten, Hector
    Let E be an elliptic curve over the rational numbers. Watkins [Experiment. Math. 11 (2002), pp. 487-502 (2003)] conjectured that the rank of E is bounded by the 2-adic valuation of the modular degree of E. We prove this conjecture for semistable elliptic curves having exactly one rational point of order 2, provided that they have an odd number of primes of non-split multiplicative reduction or no primes of split multiplicative reduction.