Long-term special program on Arithmetics and Geometry
数论和代数几何预备课程
Time: 6-27 (9:00-12:00), 7-4 (9:00-12:00, 14:00-17:00), 7-11 (9:00-12:00, 14:00-17:00), 7-18 (9:00-12:00, 14:00-17:00), 7-25 (14:00-17:00)
Place: 北京国际数学研究中心 82J12 教室
Speaker: 梁志斌(首都师范大学), 唐舜(Max Planck Institute for Mathematics)
Abstract: 本预备课程讲述代数数论，代数几何和 Arakelov 几何的基础知识，帮助听众学习正式课程。
Contents: p-adic absolute value, number field, extension of absolute values, product formula, algebraic and projective varieties, divisor and line bundle, Arakelov degree and height.
Introduction to Lehmer's Problem
Time: from 2012-7-12 to 2012-8-9, Tuesday and Thursday 9:00-11:00
Place: 北京国际数学研究中心 82J12 教室
Speaker: Sinnou David (Institut de Mathématiques de Jussieu)
Abstract: In this course, we shall introduce Lehmer's problem, claiming a universal lower bound for the Mahler measure of an algebraic number.
After an introductory lecture explaining the original problem in its historical context, and explaining the earlier results obtained towards this still open conjecture, we shall cover the methods and results obtained during the last decade.
During the course, we shall cover classical tools from diophantine geometry, such as Siegel's lemma (both the classical versions and the absolute ones over algebraic numbers), Schwarz lemmas (with an emphasis on the ultrametric case). We shall pay special attention to zero estimates.
We shall then give an overview of the situation over abelian extensions and explain the natural generalizations of the classical problem to the higher dimension. We shall then give complete proofs of at least one partial result towards these more general conjectures.
The series of lectures will be concluded by an overview of the situation over abelian varieties and an introduction to Bogomolov's problem.
Prequisites: we shall make utmost efforts to make the lectures as self contained as possible. However, standard algebraic number theory, some notions of height theory will be assumed to be known. For part of the lectures, we shall also need the audience to know some commutative algebra and basic algebraic geometry. The last lectures will assume that the audience has had some exposition to abelian varieties.
Reading: some complementary reading will be suggested as the course develops.
Abelien varieties with special endomorphismes, modular forms and Galois representations
Time: 2012-8-10, 9:00-10:00
Place: 北京国际数学研究中心 82J12 教室
Speaker: Jerome William Hoffman (Louisiana State University)
Abstract: Abstract of the talk
Poncelet's theorem and genus two curves with real multiplication
Time: 2012-8-10, 10:00-11:00
Place: 北京国际数学研究中心 82J12 教室
Speaker: Yukiko Sakai (Waseda Universityy)
Abstract: In 1936, G. Humbert investigated some relations between Poncelet's
Closure Theorem and algebraic curves of genus two with real
multiplication (=RM).
In the 1980s, F. Mestre studied them again in connection with
$n$-division points of
elliptic curves.
In this talk, we disscuss the same subject from different point of
view. For discriminant
$\Delta=5$ (resp. $\Delta=8$), we give a concrete description of a
family of such
curves with RM as a double cover $X$ of a conic $C$ in $\mathbb{P}^2$ associated
with Poncelet's $n$-polygon where $n$=5 (resp. $n$=4).
Our study is based on the explicit computation of the lifting of Poncelet's
algebraic correspondence on $C$ to $X$.
This yields a nontrivial endomorphism of $\mathrm{Pic}^0(X)$ which is identified
with the Jacobian variety of $X$. This enables us to construct a family of
curves of genus two whose Jacobian varieties are of $\mathrm{GL}_2$-type.
Lecture on modulo p representation of GL2
Time: 9-4, 9-11, 9-18, 9-25, 10-9, 10-16 (14:00--16:00)
9-6, 9-13, 9-20, 9-27, 10-11, 10-18 (10:00--12:00)
Place: 北京国际数学研究中心 82J12 教室
Speaker: 胡永泉 (Universié de Rennes I)
Abstract: Abstract of the course
Varieties over global fields: rational points, integral points, zero-cycles
Time: 9-11, 9-13, 9-19, 9-21, (9:00--11:00)
Place: 北京国际数学研究中心 82J12 教室, 9-19 在全斋全 9 教室
Speaker: Jean-Louis Colliot-Thélène (Université Paris Sud)
Abstract: I shall discuss some of the following topics.
-- Hasse principle, weak approximation, elementary fibration method, arithmetic surjectivity.
-- The Picard and the Brauer group of a variety over a field, how to compute it.
-- Over a global field : definition of the Brauer-Manin obstruction for rational points. How to compute it.
-- Torsors, descent. Connexion with the Brauer-Manin obstruction. Unconditional results for rational points : conic bundles with few degenerate fibres (Châtelet surfaces). Harari's formal lemma with applications. Some recent results on values of a polynomial represented by a norm form.
-- Results conditional on Schinzel's hypothesis. Some surfaces with a fibration into curves of genus one, some K3-surfaces.
-- Curves : connexion with the section conjecture.
-- Beyond the Brauer-Manin obstruction : the étale descent obstruction (Skorobogatov, Harari-Skorobogatov, Demarche). Poonen's new type of example.
-- Zero-cycles, the index of varieties, the Brauer-Manin obstruction for zero-cycles. Conjecture of CT, Sansuc, Kato et Saito; results of Salberger, the speaker, Swinnerton-Dyer, Skorobogatov; recent results by Wittenberg and by Yongqi Liang.
-- Strong approximation and local-global theorems for integral points. The case of homogeneous spaces of linear algebraic groups (Xu and the speaker, Harari, Demarche, Borovoi). Strong approximation for the total space of some families of homogenenous spaces.
Introduction to p-adic Hodge theory
Time: 9-17 (14:00--16:00), 9-19 (9:00--11:00), 9-24 (14:00--16:00), 9-26 (9:00--11:00)
Place: 北京国际数学研究中心 82J12 教室
Speaker: Farid Mokrane (Institut Galilée)
Abstract: In this course we will present the foundations of p-adic Hodge theory starting with the notion of periods rings and then the construction of p-adic Galois representations of the Galois group of a local field of unequal characteristic $(0, p)$. We will first discuss the non ramified case with Hodge-Tate weights between 0 and p-1 (Fontaine-Laffaille case) then we explain the general case using methods of Breuil and Kisin.
p-adic functionnal analysis and applications
to galois representations
Time: 9-19, 9-20, 9-26, 9-27, (14:00--16:00)
Place: 北京国际数学研究中心 82J12 教室
Speaker: Pierre Colmez (CNRS Institut de Mathématiques de Jussieu)