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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 t
Externí odkaz:
http://arxiv.org/abs/2406.05382
Autor:
Caro, Jerson, Garcia-Fritz, Natalia
For an elliptic curve $E$ defined over the field $\mathbb{C}$ 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 b
Externí odkaz:
http://arxiv.org/abs/2310.17592
Autor:
Caro, Jerson, Pasten, Hector
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 surfac
Externí odkaz:
http://arxiv.org/abs/2210.14181
Autor:
Caro, Jerson
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
Externí odkaz:
http://arxiv.org/abs/2206.10008
Autor:
Caro, Jerson
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
Externí odkaz:
http://arxiv.org/abs/2203.10932
Autor:
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 $g\ge 2$ defined over a number field $F$, with Jacobian of rank at most $g-1$. Namely, in the case $F=\mathbb{Q
Externí odkaz:
http://arxiv.org/abs/2102.01055
Autor:
Caro, Jerson1 (AUTHOR), Pasten, Hector1 (AUTHOR) hpasten@gmail.com
Publikováno v:
Inventiones Mathematicae. Dec2023, Vol. 234 Issue 3, p1197-1250. 54p.
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