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 C̃ of C which has a surjection onto C'. Thanks to Bely theorem we can show that any curve C' defined over Q is dominated by one of the curves Cn, yn-1 = x2. Over Fp any curve in fact is dominated by C6 which is in a way also a minimal possible curve with such a property. Conjecturally the same holds over 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.
Fedor Alekseyevich Bogomolov was born on 26 September 1946 is Moscow, graduated from Moscow State University, Faculty of Mechanics and Mathematics, and earned his doctorate in 1973, in Steklov Institute, advised by Sergei Novikov. Bogomolov worked at Steklov Institute in Moscow, and in 1994 he became a full professor at the Courant Institute.
Bogomolov is known for his pioneering work on algebraic geometry. He worked extensively on Kaehler manifolds, especially Calabi-Yau manifolds, contributing to the foundation of Mirror Symmetry and String Theory. He is the author of about 100 papers, many of them containing milestone results and deep conjectures in algebraic geometry.
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