Bounding tangencies of sections on elliptic surfaces

Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces

We consider elliptic surfaces $\mathcal{E}$ over a field $k$ equipped with zero section $O$ and another section $P$ of infinite order. If $k$ has characteristic zero, we show there are only finitely many points where $O$ is tangent to a multiple of $P$. Equivalently, there is a finite list of integers such that if $n$ is not divisible by any of them, then $nP$ is not tangent to $O$. Such tangencies can be interpreted as unlikely intersections. If $k$ has characteristic zero or $p>3$ and $\mathcal{E}$ is very general, then we show there are no tangencies between $O$ and $nP$. We apply these res…

椭圆曲面上的新的不可期交集

New unlikely intersections on elliptic surfaces

本文将复数域上关于 Jacobi 椭圆曲面不可期交集的相切结果扩展至正特征 $p$，并提出了一种研究代数曲线相切性质的新方法，发现了与 Manin 同态相关的新交集族。

On the geometry of K3 surfaces with finite automorphism group and no elliptic fibrations

Nikulin and Vinberg proved that there are only a finite number of lattices of rank $\geq 3$ that are the Néron-Severi group of projective K3 surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric description of such K3 surfaces $X$, when these surfaces have moreover no elliptic fibrations. In that case we show that such K3 surface is either a quartic with special hyperplane sections or a double cover of the plane branched over a smooth sextic curve which has special tangencies properties with some lines, conics or cuspidal cubic curves. We then study th…

New methods for bounding the number of points on curves over finite fields

We provide new upper bounds on N_q(g), the maximum number of rational points on a smooth absolutely irreducible genus-g curve over F_q, for many values of q and g. Among other results, we find that N_4(7) = 21 and N_8(5) = 29, and we show that a genus-12 curve over F_2 having 15 rational points must have characteristic polynomial of Frobenius equal to one of three explicitly given possibilities. We also provide sharp upper bounds for the lengths of the shortest vectors in Hermitian lattices of small rank and determinant over the maximal orders of small imaginary quadratic fields of class numbe…

Multiple points in P^2 and degenerations to elliptic curves

We consider the problem of bounding the dimension of the linear system of curves in ${\bf P}^2$ of degree $d$ with prescribed multiplicities $m_1,...,m_n$ at $n$ general points (\cite{Hir1},\cite{Hir2}). We propose a new method, based on the work of Ciliberto and Miranda (\cite {CM1}, \cite {CM2}), by specializing the general points to an elliptic curve in ${\bf P}^2$.

Non-archimedean equidistribution on elliptic curves with global applications

Let $E$ be an elliptic curve over an algebraically closed, complete, non-archimedean field $K$, and let ${\mathsf E}$ denote the Berkovich analytic space associated to $E/K$. We study the $μ$-equidistribution of finite subsets of $E(K)$, where $μ$ is a certain canonical unit Borel measure on ${\mathsf E}$. Our main result is an inequality bounding the error term when testing against a certain class of continuous functions on ${\mathsf E}$. We then give two applications to elliptic curves over global function fields: we prove a function field analogue of the Szpiro-Ullmo-Zhang equidistribution …