Geometria Unramified correspondences Fedorov Bogomolov 21-04-2017 - 14:30 AULA 311 (SEMINARI) Largo San L. Murialdo,1
I will report on the results of an ongoing project which we began some years ago with Yuri Tschinkel and continue with Hang Fu and Jin Qian. We say that a smooth projective curve $C$ dominates $C'$ if there is nonramified covering $ ilde C$ of $C$ which has a surjection onto $C'$. Thanks to Bely theorem we can show that any curve $C'$ defined over $ar Q$ is dominated by one of the curves $C_n, y^n-1= x^2$. Over $ar F_p$ any curve in fact is dominated by $C_6$ which is in way also a minimal possible curve with such a property. Conjecturally the same holds over $ar Q$ but at the moment we can prove only partial results in this direction. There are not many methods to establish dominance for a particular pair of curves and the one we use is based on the study of torsion points and finite unramified covers of elliptic curves. |