Richard Pink
Research Interests
An Overview

Beginning with my Diplomarbeit and my Dissertation under the supervision of Prof. Dr. G. Harder at Bonn University I have studied several aspects of number theory and, especially, of arithmetic geometry. I find myself particularly fascinated by and drawn towards areas where very different approaches - from arithmetic, algebraic geometry, topology, etc. - join together to pose exciting questions and by interacting enable one to prove important and deep results. One such area is the arithmetic of Shimura varieties, where certain results and conjectures (would) have very important consequences for number theory as a whole. A wonderful example for this is Wiles' recently found proof of the Taniyama conjecture in the semistable case, which finally establishes the so-called Fermat's Last Theorem that was conjectured more than 350 years ago.

On the other hand it is not rare that research in number theory leads to specific questions in neighboring fields, which can be formulated purely within their own framework, and may turn out very interesting in themselves and of any conceivable level of difficulty. In my work I have repeatedly met such questions: for instance, of topological-geometric nature as in my contribution to the Lefschetz trace formula, or of algebraic nature as in my joint work with Michael Larsen, especially in our recent result on finite subgroups of linear groups.

On the whole, I seem to lean more towards problems of algebraic than of analytic flavor. For example, the algebro-geometric approach to Shimura varieties is based on their interpretation as moduli spaces. For different purposes one would like to have good compactifications which are as canonical as possible. The problems here are caused by the fact that the abelian varieties that are parametrized by the Shimura variety may degenerate. A central theme is that of semi-stable degeneration. The language of logarithmic schemes that was developed within the last ten years provides a conceptual framework to describe this phenomenon. One of my interests lies in this area: to describe toroidal compactifications of (Siegel) Shimura varieties as moduli spaces of certain polarized logarithmic abelian varieties.

My interest in abelian varieties on the one hand, and in Drinfeld modules and their generalizations on the other hand, has been aroused through the study of their moduli spaces, but also in other ways. One fundamental problem is to determine the Galois action on the associated Tate modules. Here I was able to verify the Mumford-Tate conjecture in many cases, although the general case still remains open. One interesting aspect of Galois representations lies in the fact that the representations on different Tate modules associated to the same arithmetic object are related to each other in a subtle way, although they are defined over completely different fields. This compatibility property can be exploited in many different ways. My ongoing cooperation with Prof. M. J. Larsen is devoted to this topic; our general goal is to determine the image of the Galois group in the whole adelic representation.

The role that abelian varieties play for the arithmetic of number fields is taken over by Drinfeld modules and Anderson's t-motives for the arithmetic of function fields. In the last few years I have developed a very strong interest in these objects. Some fundamental problems here are analogues of those that I have studied in characteristic zero, such as compactifications of moduli spaces and the arithmetic of Galois representations. But I have also discovered what I find are very exciting connections between arithmetic and analytic aspects of these `motives'. The analytic theory relies on a new kind of Hodge structures over function fields, whose basic theory I have already developed in a preprint. Based on this I can build a theory of analytic uniformization of t-motives which, for uniformizable t-motives, yields Hodge structures of this new kind, and which includes the proof of an analogue of the classical Hodge conjecture. These results have grown to such proportions that I have decided to present them in a book on the general algebraic and analytic theory of motives over function fields. If space permits, another part of this book will deal with the arithmetic of the associated Galois representations.

One aspect of motives over function fields that has been missing in the literature so far is that of cohomology. It is my dream that, one day, motives over function fields can be associated to non-linear objects of arithmetic geometry in positive characteristic in much the same way as motives in characteristic zero ought to be associated to algebraic varieties over number fields. As such, they should contain very important information about those non-linear objects; for instance, through their L-functions. First, and presumably vital, steps in this direction are taken in a joint project with Gebhard Böckle. Based on degenerating families of a kind of t-motives on an algebraic variety in characteristic p, we construct a derived category with three of the usual six functors, namely, with pullback, tensor product, and direct image with compact support. We can also prove a Lefschetz trace formula for Frobenius in this setting, which can be viewed as a generalization of the known Lefschetz trace formula in étale cohomology with p-torsion coefficients. As a consequence, we can show that the L-function of a family of t-motives coincides with that of its total cohomology, provided that the cohomology is taken along the ``geometric'' direction, for instance, for a family of motives in fixed ``special'' characteristic. Given the role of cohomology throughout arithmetic and, indeed, pure mathematics in general, and in view of these partial results, I find it not completely unreasonable to hope that this derived category or a similar one may hold an important key to solving many questions in the arithmetic of function fields in a conceptual way. Finally, my motivation to study the function field case includes the hope, however far-flung it may seem, that the knowledge gained here may re-fertilize the number field case from which originally all inspirations were drawn.