This is a project in Arithmetic Algebraic Geometry comprising two parts. The first part deals with the study of values of L-functions of varieties (motives) over number fields. In previous work the investigator and his collaborator have extended the conjectures of Bloch Kato, Fontaine, Perrin-Riou to motives with (possibly noncommutative) coefficients. At the same time they have established a precise link between these conjectures and classical Galois module theory, thereby generalizing various conjectures and theorems in this latter area to arbitrary motives. With the conjectural picture firmly in place, the main task now is to prove more cases. What seems within reach of current techniques are Tate motives over number fields, CM elliptic curves (with CM by non-maximal orders) and the adjoint of a modular form (with action of the integral Hecke algebra). The last two are currently looked into by students of the investigator. Equally within reach seems to be a proof of the compatibility of the equivariant special value conjectures with the functional equation of the L-function. The second part of the project is a rather concrete question in deformation theory (of schemes, vector bundles or representations of profinite groups). Using the theory of the cotangent complex one can define higher Kodaira Spencer maps and the investigator proposes to study the injectivity of these maps. In degree 1 this is known and leads to a criterion for smoothness of the deformation ring. The case of most interest is degree 2 where a similar injectivity would lead to a simple criterion for the deformation ring to be a local complete intersection.
This is a project in number theory which has been part of the mathematical heritage ever since the Babylonians discovered that there can be triangles with all sides of integer length and one angle of ninety degrees. By the theorem of Pythagoras this gives integer solutions of an algebraic equation. Modern number theory still looks for integer solutions of algebraic equations but this search is informed and enriched by much deeper connections with ideas from geometry and topology than the ones alluded to in this introductory example. The theory of special values of L-functions is a case in point. While defined by counting solutions of equations modulo prime number it turns out that values of these functions can sometimes be expressed in terms of integrals of differential forms, or volumes of certain lattices. Nobody knows exactly why such a relationship should hold. The examples one can prove all seem to rely on some happy coincidences, although they follow a pattern that leads one to guess ("conjecture") what happens in general. This is much like the situation in an experimental science. The investigator and his collaborator have generalized this picture to situations where the system of equations has some additional symmetries and they now try to collect further evidence for (or indeed falsify!) their conjectures. As far as applications are concerned, number theorists would probably agree that L-functions are a key concept in their field. On the other hand, number theory as a whole no longer needs to be defensive about its applicability, with much of cryptography and coding, on the internet and otherwise, being based on its results.
