Seminar on Arithmetic Algebraic Geometry: 3 talks the reduction of genus 2 curves with complex multiplication

2013-2-5 10:00 - 2013-2-6 16:00
Faculty of Science Building #4 Room 501
Eyal Goren (McGill University)
Abstract :
The theory of the mod p reduction of an elliptic curve with complex multiplication is well understood and some of the key results were already obtained by Deuring. The theory of the mod p reduction of two different elliptic curves with complex multiplication is encased in the Gross-Zagier theorem, although our understanding there is not yet complete. This theory has applications to explicit class field, and in particular to the problem of constructing units in class fields of quadratic imaginary fields, and to cryptography through the problem of generating elliptic curves mod p with a given number of points.
In contrast the theory of genus two curves with complex multiplication - namely, whose Jacobians have complex multiplicaiton - is poorly understood. One of the main problems is the possibility of bad reduction (even stable bad reduction). As in the case of elliptic curves, the theory has applications to the construction of S-units in class fields, now of quartic CM fields, and to cryptography. At the same time, it is also related to a very general problem of understanding intersection of Shimura subvarieties contained in a given ambient Shimura variety.
The first talk will be an introduction to this circle of ideas and problems, discussion of the difficulties that arise and some of the results we have at present.
In the second talk I will state certain theorems proved in collaboration with Kristin Lauter (Microsoft Research) concerning genus 2 curves with complex multiplication and indicate the
methods used to prove them, to the extent time allows.
In the third talk I will explain the intersection theory point of view and, in that context, some theorems and conjectures obtained by Bruinier and Yang, and discuss to the extent time allows recent work with Fabrizio Andreatta (Milano), Ben Howard (Boston College) and Kirti Madapusi-Pera (Harvard) concerning this point of view.