Abstract for the seminar talks on 23.05.2008

Prof. Sergei Vostokov (St. Petersburg State Univ.)
The classical reciprocity law as an analog of an Abelian integral theorem
Time: 14.15 / Place: HWZ (HG G43)

Apparently Leopold Kronecker came up with the idea of an analogy between numbers and functions. David Hilbert was the first who began to study this analogy in algebraic number fields. He, in particular , observed that his reciprocity law for norm residue symbols resembles the Cauchy integral theorem. To make this analogy more transparent, we consider the classical reciprocity law for power residues. Class field theory connects the product of power residues with the product of local norm residue symbols and this relation must be an analog of integral theorem stating that the Abelian integral of a differential form on a Riemann surface equals the sum of residues of the form at the singular points. I obtain the explicit global reciprocity law for power residue and show that this explicit formula is an analog of the Abelian integral theorem.

Cécile Armana (Jussieu, Paris)
Rational points on Drinfeld modular curves
Time: 15.30 / Place: HWZ (HG G43)

Since the work of Mazur, Kamienny and Merel, we know the existence of a uniform bound for the torsion of elliptic curves over number fields. Their approach is based on the study of rational points on classical modular curves. I will discuss to what extent this method applies to Drinfeld modular curves over function fields and the uniform boundedness conjecture for torsion of rank-2 Drinfeld modules. When N is a prime polynomial in F_q[T] of degree 3, we show that the Drinfeld modular curve Y_1(N) has no rational points of small degree. For general N, we have a similar statement, under a hypothesis on the Hecke structure of Drinfeld modular forms.