Elliptic curves and unramified correspondences

Fedor Bogomolov (NYU)

Abstract

We define two different ( but related) notions of dominance. We will mostly consider them for curves defined over number fields of ${\overline{F}}_p$ though they can be defined for curves over any field.

Definition 1: For a curve $C$ of genus $g\ge 2$ we will say that $C$ is dominant over $C^{\prime }$ if there is an unramified covering $\tilde{C}$ of $C$ with a surjection onto $C^{\prime }$. In the case of elliptic curves we have a different notion ( assuming $p\ne 2$ ) There is a involution $x\to -x$ on elliptic curve $E$ if we fix $0$ and the quotient of this involution is $P^1$. Thus we have projection map $p:E\to P^1$ of degree $2$ with $4$ branch points $(a,b,c,d)$ corresponding to points of order $2$ on $E$. Such a map is unique modulo projective autmorphism of $P^1$. Vice versa we can associate to any quadruple of points in $P^1$ modulo projective autmorphism of $P^1$ unique elliptic curve $E$ modulo isomorphism. Moreover since the curve $E$ is an abelian group we can also define the subset $P_E(tors)\subset P^1$ which is the image of torsion points in $E$ in $P^1$.

Definition 2 We will say that $E$ dominates $E^{\prime }$ if $E^{\prime }$ corresponds to a quadruple of points contained in $P_E(tors)$. In my talk I will the relation between these two notions and nontrivial results relating them. The talk is base on my works with Yuri Tschinkel and our more recent results with Hang Fu and Jin.