Intercity Number TheoryNovember 1, Leiden. Room 312 of the Snellius building
|11:00-12:00||Martin Bright (Leiden), A walk on the wild side: p-torsion in the Brauer group|
Abstract. The Brauer group of a variety over a number field is a powerful tool for studying failures of the Hasse principle. To apply this, one needs to understand the function obtained by evaluating a given element of the Brauer group at the p-adic points of the variety. For elements of order coprime to p, this evaluation map factors through reduction modulo p. For elements of order p, the situation is much more intricate: one may need to look at the points to higher p-adic precision. We relate the resulting filtration on the Brauer group to one defined by Bloch–Kato, and show that Kato's refined Swan conductor controls the local behaviour of the evaluation map. This is joint work with Rachel Newton.
|13:30-14:30||Leila Schneps (Jussieu, Paris), Grothendieck-Teichmüller theory, a crossroads between geometry and number theory.|
Abstract. Grothendieck-Teichmüller theory was originated by Alexander Grothendieck as a way to study the
absolute Galois group of the rationals by considering its action on fundamental groups of varieties,
in particular of moduli spaces of curves with marked points: the special properties of the Galois
action with respect to inertia generators and the fact of respecting the relations in the fundamental
group gave rise to the definition of the group GT which contains GQ.
The group GT is profinite, but its defining relations can also be used to give a pro-unipotent avatar,
and an associated graded Lie algebra grt. The study of the Lie algebra grt reveals many unexpected
relations with number theory that are completely invisible in the profinite situation. We will show how
Bernoulli numbers, cusp forms on SL2(Z) and multiple zeta values arise in the Lie algebra context.
|14:50-15:50||Pierre Lochak (Jussieu, Paris), A topological version of Grothendieck-Teichmüller theory.|
|16:10-17:10||Igor Shparlinski (Sydney), Integers of prescribed arithmetic structure in residue classes|
Abstract. We give an overview of recent results about the distribution some special integers
in residues classes modulo a large integer q. Questions of this type were introduced by
Erdos, Odlyzko and Sarkozy (1987), who considered products of two primes as a relaxation
of the classical question about the distribution of primes in residue classes. Since that time,
numerous variations have appeared for different sequences of integers. The types of
numbers we discuss include smooth, square-free, square-full and almost primes integers.
We also expose the wealth of different techniques behind these results: sieve methods,
bounds of short Kloosterman sums, bounds of short character sums and many others.
Belgian Dutch Algebraic Geometry DayNovember 8, Antwerpen.
DIAMANT SymposiumNovember 29, De Bilt. The is part of a two-day event, November 28-29.
Intercity Number Theory SeminarDecember 13, UvA and VU Amsterdam. At the VU.
Intercity Number Theory Seminar / Getaltheorie in het vlakke landDecember 11, Utrecht. (2020)