Семинар лаборатории алгебраической геометрии: Ф.Богомолов

На семинаре с докладом Geometry of sets of torsion points on elliptic curves выступит Ф.А.Богомолов (Courant Institute and HSE)

In this talk I introduce and discuss geometry of curves parametrizing subset of points in $P^1$ obtained as projections of torsion points of elliptic curves. For every subset of different $k$ points in $P^1$ we can define it's image in the moduli  $M_{0,k}$ of $k$-tuples of points which is essentially  a quotient of projective space $S^kP^1= P^k$ by the action of $PGL(2)$. Thus $M_{0,k}$ is a rational variety of dimension $k-3$. If we consider the images of points of finite order in different elliptic curves under natural projections then we obtain an( infinite) system of modular typoe curves with maps into $M_{0,k}$ I will formulate three conjectures (semi theorems) about properties of such maps which provide a possiblity of realistic universal estimate for intersections between subset  of torsion points for different elliptic curves.