## Research themes

**Galois representations and modular forms:**

A modern way to look at algebraic number theory is to study the group G_Q of symmetries of all finite extension of the rational number Q, i.e. of all number fields. One way of doing this is via p-adic (or complex) Galois representations. These are homomorphisms from G_Q to GL_n(K) for (the complex or) a p-adic field K. If one would understand all these, one could deduce many results in number theory. The reason why this approach is promising is that the Langlands program predicts (conjectures) that many interesting Galois representation can be found in (arithmetic) geometry, for instance by studying modular forms or elliptic curves. The most visible success of this method has been the proof of Fermat's last theorem by Wiles and Taylor-Wiles building on work of many others.

One concrete example of a problem considered by our group is the study of local universal deformation rings (A.-K. Juschka, partially joint with G. Böckle). This is foundational in the understanding of Galois representations of local fields, i.e., the local constituents of number fields. We would like to derive certain ring theoretic properties which in turn have direct consequences for representations. The main emphasis is on theoretical results. But for small residue characteristics computational methods from commutative algebra are necessary, such as factoring in multivariate power series rings.

A project that applies modular forms to the study of quadratic lattices is pursued by Dr. J.M. Cervino. A main aim in his research is to find invariants that allow one to distinguish non-isomorphic integral lattices that carry a quadratic form. To such forms one can attach theta-series which are certain classical or Siegel modular forms. This approach in general has a long tradition. Jointly with G. Hein at Essen, J.M. Cervino has developed a systematic method to give a new infinite series of such invariants. Their invariants are classical and not Siegel modular forms, and so computations with these are considerably simpler. This has for instance allowed them to resolve a question by Conway and Sloane in lattice theory. A far hope would be to find a complete list of invariants. While the final results of Cervino and Hein are theoretical, for certain aspects experiments on the computer were essential.

Questions directly in the area of modular forms is the focus of research of Dr. T. Centeleghe. Using computational and theoretical methods he investigates the set of mod p Galois representations attached to modular forms and derives (conjectural) asymptotic formulas for their number as p tends to infinity. Recently he started to consider the question of determining the image of Frobenius automorphisms of G_Q under (intergral) p-adic Galois representations (with image for instance in GL_2(Z_p)) coming from modular forms. For weight 2 forms related to elliptic curves the question can be answered. The general weight 2 case is still open. Also in the area of Galois representations is the research of Dr. Kumar Cheraku. He works on Hida families (for the prime 2) and on the irreducibility of local Galois representations. Recently he has started exploring questions on p-adic L-functions and L-invariants.**Drinfeld modular varieties and Drinfeld modular forms:**

It has been observed that questions in number theory often have analogs over global function fields; then the ring of integers Z is replaced by rings like the polynomial ring F_p[t], and the field of rational numbers Q by the field of rational functions over the finite field F_p of p elements, where p is a prime number. Because methods from algebraic geometry can be applied to function fields, many questions over the latter are more tractable than the corresponding questions over number fields. One particular instance of this is Drinfeld's proof of the global Langlands correspondence for GL_2 over function fields. In his proof Drinfeld introduced what are now called Drinfeld modular varieties; these are function field analogs of certain Shimura varieties. Their cohomology gives rise to Galois representations and these varieties have an interesting geometry coming from their moduli interpretation.

The research of Dr. P. Hubschmid is centered around the analog of the André-Oort conjecture for function fields, with a particular emphasis on the case of Drinfeld modular varieties. Drinfeld modular as well as Shimura varieties carry special points and special subvarieties on them that are derived from their moduli interpretation. The André-Oort conjecture asserts that every irreducible component of the Zariski closure of a set of special points in a Drinfeld modular variety is a special subvariety. In his thesis P. Hubschmid proves the conjecture for special points with separable reflex field over the base field using Galois-theoretic and geometric methods. His ongoing work concentrates on equidistribution statements of Hecke and Galois orbits of special points in Drinfeld modular varieties. It is hoped that they can be used to prove the general André-Oort conjecture for Drinfeld modular varieties.

The research of R. Butenuth (partially joint with G. Böckle) is on Drinfeld modular forms which were originally introduced by D. Goss. These forms are global (rational) functions on bundles over Drinfeld modular varieties. They are useful for various purposes: they provide embeddings of Drinfeld modular varieties into projective spaces; they have attached Galois representations; Hecke theory provides arithmetically interesting invariants of such forms. The work of R. Butenuth is concerned with the algorithmic computation of such forms. This leads to theoretical work following J. Teitelbaum that gives a combinatorial description, as well as to implementation aspects of how to compute quotients of Bruhat-Tits trees by certain arithmetic groups. The groups considered are related to quaternion algebras. The natural next step in this research is to pass from trees to buildings. A closely related project is the research of the PhD student Y. Bermudez. He considers automorphic forms over function fields which are combinatorially very similar to Drinfeld modular forms. His aim is to find (and implement) an efficient algorithm to determine, given such an automorphic form with rational Hecke eigenvalues, the corresponding elliptic curve (i.e., the strong Weil curve). This approach might allow one also to effectively compute Heegner points on the elliptic curves.**L-functions and characteristic p geometry:**

L-functions are a clever way to package much information about (certain) quantities from algebraic or arithmetic geometry. A simple example is the L-function of a smooth projective curve over a finite field, say F_p, for instance the L-function of an elliptic curve. This L-function contains all information necessary to compute the number of points of the curve over any finite extension of F_p. Another example is the Riemann zeta-function. In the context of function field arithmetic, such a theory of L-functions was developed by G. Böckle and R. Pink. As it turns out, the formalism developed here encompasses étale sheaves with F_p coefficients of schemes in characteristic p. An alternative way to look at the latter question was developed by M. Emerton and M. Kisin. In ongoing work with M. Blickle at Mainz, the relation between these two approaches is clarified in detail.

In a direction closely related to étale sheaves but with different tools, Dr. A. Maurischat considers (iterative) differential structures on varieties in positive characteristic, or in more geometric terms, stratified bundles. A current aim of his research is the development of a de Rham cohomology. This is non-trivial, since the usual de Rham complex in positive characteristic has rather degenerate properties as observed by Cartier in the 1950's. Another emphasis in his work is on differential Galois theory. Given a bundle with a differential structure, one can seek a covering on which the differential structure is "trivial" in a way. This leads to Galois covers that are compatible with the differential structure, i.e., to differential Galois theory where the Galois groups are now algebraic groups. He also has a project in invariant theory of algebraic groups with E. Dufresne at Basel in which they aim to determine invariants of actions of the additive group by computational methods.

Directly in the area of L-function is the research of the PhD student Y. Qiu. She considers Goss' L-functions over global function fields with one marked place. This leads to interesting questions about entire functions in one variable in positive characteristic and their distribution of zeros. One approach to such questions is via the above cohomological theory of L-functions of Böckle-Pink. It is hoped that one can rather completely describe the pattern of these zero distributions. There are many formal similarities (but also differences) to questions about distributions of zeros of the Riemann zeta-function. To formulate hypotheses, the project of Y. Qiu strongly relies on computer experiments.