A brief account of my research
Posted by Martin Orr on Thursday, 20 September 2012 at 17:39
I recently had to write an account of my research for a non-specialist. I think I took it more seriously than necessary -- I am not sure that anyone will read it. But it might interest some readers of this blog, so I shall put it here. I have just put the paper containing this work on Arxiv.
My research concerns subvarieties of Shimura varieties. Shimura varieties are geometrical objects whose points parameterise certain nice classes of objects from complex geometry. The simplest examples of Shimura varieties are the Siegel modular varieties, which parameterise abelian varieties of a given dimension. (Abelian varieties are varieties with a rule for adding points inside the variety. Abelian varieties are very simple from the point of view of complex geometry, but very rich from the point of view of number theory.)
By a subvariety of a Shimura variety, I mean the set of points inside the Shimura variety which solve some chosen polynomial equations. Sometimes the subvariety is itself a Shimura variety, in which case it is called a "special subvariety" or "weakly special subvariety".
My aim is to prove that if a subvariety of a Shimura variety contains lots of points satisfying a certain condition (namely, that these points lie in a single "isogeny class"), then it is a weakly special subvariety. I have succeeded in proving this under the assumptions
- The Shimura variety is a Siegel modular variety.
- The subvariety has dimension 1.
I am now trying to remove these assumptions, especially assumption 2. Assumption 1 is difficult to avoid because my proof relies upon a deep result from the number theory of abelian varieties. Assumption 2 is closely linked to assumption 1. In order to remove assumption 2, one would like to use an inductive argument, showing that the conjecture for a subvariety of a given dimension can be reduced to the case of a subvariety of smaller dimension. In order to perform this reduction however, we may be forced to replace our original Shimura variety by another which is not a Siegel modular variety, breaking assumption 1.