Lower bounds for CM points and torsion in class groups

Lower bounds for CM points and torsion in class groups

Thu, 03/11/2011
16:00 		Jacob Tsimerman (Harvard) Logic Seminar Add to calendar
Number Theory Seminar Add to calendar 		L3
Lower bounds for CM points and torsion in class groups
Let $ x $ be a CM point in the moduli space $ \mathcal{A}_g(\mathbb{C}) $ of principally polarized complex abelian varieties of genus $ g $, corresponding to an Abelian variety $ A $ with complex multiplication by a ring $ R $. Edixhoven conjectured that the size of the Galois orbit of x should grow at least like a power of the discriminant $ {\rm Disc}(R) $ of $ R $. For $ g=1 $, this reduces to the classical Brauer-Siegel theorem. A positive answer to this conjecture would be very useful in proving the André-Oort conjecture unconditionally. We will present a proof of the conjectured lower bounds in some special cases, including $ g\le 6 $. Along the way we derive transfer principles for torsion in class groups of different fields which may be interesting in their own right.