Arithmetic algebraic geometry

Researchers

PhD students

Visiting PhD students

Former postgraduate students

Research topics

Arithmetic algebraic geometry allows us to study arithmetic properties of algebraic varieties by applying geometric and number-theoretic ideas at the same time. As an example, consider a 2-dimensional torus as the quotient of the complex plane C={u+iv | u,v in R} by a discrete subgroup generated by 1 and w in C\R. This is a topological group and it can be studied by topological methods. At the same time it can be provided with a complex-analytic structure and can be studied in a more precise way using methods from complex analysis. We can also provide this torus with the structure of an algebraic curve defined by Y2=X3+aX+b (for some a,b in C) in the 2-dimensional complex plane C2={(X,Y) |X,Y in C} and apply methods of algebraic geometry to study its algebraic invariants. The advantage of the last approach is that we can choose the coefficients a, b, from any field K. If K=Q, the field of rational numbers, then we can study points of the torus with coordinates in Q. They appear as solutions of a diophantine equation, so that we are in the area of arithmetic algebraic geometry. We can also choose K=Fp, the finite field with p elements, and still apply topological and complex-analytic ideas when studying the curve.

Various parts of arithmetic algebraic geometry are represented in the department, and we describe them below in more detail.

Finite group schemes (V. Abrashkin) This topic can be made more precise in several directions. Let me give some general idea. The theory of finite group schemes appears as a part of arithmetic algebraic geometry. Roughly speaking, such schemes appear as finite sets of points given by equations with integral coefficients with a composition law, where the coordinates of the sum of given points appear as rational functions of the coordinates of these points. This combination of two basic algebraic concepts, namely the concept of a finite algebraic set (i.e., the set of solutions of a system of algebraic equations) and the concept of an abstract group, gives a very rich theory. For example, there is a non-trivial description of group schemes of order 2. Finally, it is worth pointing out that a lot of important arithmetic invariants are obtained in terms of finite group schemes and that this area still attracts attention of leading arithmetic geometers.

Basic reading:

Crystalline representations (V. Abrashkin) The concept of a finite group scheme as above described is often used in arithmetic algebraic geometry to give an interpretation of Galois modules coming from the first etale cohomology groups of algebraic varieties. The study of Galois modules coming from higher cohomology groups can be done via the methods of crystalline representations, which was developed by Jean-Marc Fontaine.

Basic reading:

Polylogarithms, and multiple zeta values (H. Gangl) Polylogarithms are generalisations of the logarithm, which can be obtained from logarithms by iterated integration. A truncated version of the (n-1)-st higher logarithm, called the finite (n-1)-logarithm, satisfies the same functional equations as the differential of the n-th higher logarithm, and they can be obtained from the equations for the n-th logarithm by some derivation procedure due to Cathelineau. The proof makes use of Amnon Besser's version of the p-adic polylogarithm where he encounters the finite polylogarithm as the "first non-zero layer". The finite polylogs have already occurred in the literature as the Mirimanoff polynomials, and it turns out that many of Mirimanoff's criteria for Fermat's last theorem have their reinterpretation as functional equations for finite polylogs.

There is also a multi-variable version of the polylogarithm, and according to a conjecture of Alexander Goncharov, those multiple polylogarithms should give all "mixed Tate motives over a field". Their differential structure has been related to polygons and trees, giving a connection to the Connes-Kreimer Hopf algebra in renormalization theory. Specializing all the arguments of multiple polylogs to 1 produces multiple zeta values which arise in a variety of research topics. As an example, there are several very direct connections between the double zeta values (multiple zeta values with two arguments) and modular forms.

Basic reading:

Selected publications

Finite group schemes

Polylogarithms, and multiple zeta functions

Crystalline representations