Umberto Zannier Scuala Normale Superiore (Pisa)
Representing isogeny classes of elliptic curves over $\overline\Q$.

ABSTRACT: It may be asked how "small" a set of representatives for isogeny classes (up to isomorphism) of elliptic curves over $\overline\Q$ may be found. For instance, any open set in the $j$-plane $\C$ suffices to represent every isogeny class, but no finite set does. In joint work with Masser we have proved in particular that no real algebraic curve in the $j$-plane (i.e. defined by a polynomial equation $f(\Re j, \Im j)=0$) may suffice (though a transcendental curve may do). This remains true also if we do not require to represent isogeny classes of "special", i.e. CM, curves.
The method of proof extends to give new and different answers to questions of Katz-Oort, compared to work by Tsimerman; such context concerns proving the existence of principally polarized abelian varieties of given dimension, defined over $\overline\Q$ and not isogenous to any abelian variety in a given subvariety of the moduli space.