Activities 2008


Seminar on ``Differential Equations on p-adic Analytic Spaces"

The topics discussed:
1. Foudations of the theory of non-archimedean analytic spaces, in the sense of Berkovich.
2. Smooth analytic spaces. Topologies.
3. General definitions on connections on p-adic vectors bundles and p-adic differential equations
4. The one-dimensional case. Some classical results: radii of convergence, growth of solutions, Frobenius structures.
5. Local systems on the pointed p-adic disk.
6. Christol-Mebkhout slopes and factorization.
7. Equations of Robba type. Exponents.
8. Semistable models and log-isocrystals.

The course will be held in Room 2AB40 of Math department

Mo-Tu-We 11.30-13.00

in 8 weeks from January 7th to March 12th 2008. First meeting: Monday, January 7, 2008 at 11.30.

The lectures on ``Smooth analytic spaces and topologies" will be delivered (as an independent course in 6 sessions of 90 minutes each) by Vladimir Berkovich (January 14 -23).

The lectures on ``Local systems on the pointed p-adic disk" will be delivered (as an independent course in 4 sessions of 90 minutes each) by Lorenzo Ramero during February.



-Vladimir Berkovich (Weizmann Institut, Rehovot) January 14 -23, 2008
``Non-archimedean analytic spaces" (5 sessions  of 90 minutes each)
Dates: January 14, 16, 21, 22, 23 11.30-13.00, Room 2AB/40

-Paolo Stellari (Milano) January 29, 2008,
“Equivalences of K3 surfaces and Orientation”
room 1BC/45, 13:45
Abstract: We will consider the problem of describing equivalences between the derived categories of smooth projective K3 surfaces. After recalling the `classical' Derived Torelli Theorem, we will prove a conjecture by Szendroi which improves the result and involves the preservation of the orientation of some 4-dimensional space in the real cohomology of the K3 surfaces. Our approach relies on the proof of the same conjecture for generic (non-projective) K3 surface. This is a joint work with D.Huybrechts and E. Macri.

-Michael McQuillan (IHES)
February 5, 2008
"Old and New methods in function field arithmetic"
room 2BC/60, 17:30
Abstract: Vojta conjectured that algebraic points, $f$, on quasi projective curves, $C$, over function fields, $K$, in characteristic zero, satisfy, for every $epsilon > 0$,
$$h_{K_C}(f) leq (1+epsilon) discr K(f) + Constant(epsilon)$$
The case of $P1 \backslash {4 points}$ was already a conjecture of Osterle, which he described as "the real `a,b,c' conjecture" over function fields.
The talk will describe a new method in function field arithmetic, from which the conjecture follows trivially.
The method will be explained from the point of view of applying the underlying functorial principle of functional analysis to spaces of algebraic cycles.
Comparisons will also be made with old methods, such as that of Vojta, himself, who had already obtained a similar result, but with $2+epsilon$ instead of $1+epsilon$.

-V. Di Proietto (Padova) February 18, 2007
“p-adic differential equations and log convergent isocrystals on semistable curves”
16:30, room 2BC/60
Abstract: We will consider a smooth and projective curve X_K over a p-adic field K and a semistable model X over a complete discrete valuation ring V, such that K is the fraction field of V. We will define a fully faithful functor between the category of log convergent isocrystals on  X  and the category of p-adic differential equations without singularities on X_K and we'll describe the essential image of this functor.

Lorenzo Ramero (Lille) February 1-April 30, 2008
will give a short course on
        ``Local systems on the p-adic punctured disk"
 The plan of the mini-course is as follows:
 a) basic ramification theory for rank two valuations (Huber's theory)
 b) applications to the study of finite etale coverings of a p-adic annulus: variation of the discriminant and proof of the p-adic Riemann existence theorem
 c) local systems on a p-adic punctured disc: Swan conductor and break decomposition.
 d) variation of the break decomposition and asymptotic slopes.

The lectures will be part of the ALGANT and PhD course ``Number Theory 2" and will take place in Hall 2AB/40 at 11.30-13.00 on
Monday, Tuesday and Wednesday February 18,19,20.

-Pieter Moree (MPI, Bonn)
"Large cyclotomic coefficients".
March 27, 2008, room 1A/150, 14:30

-Dajano Tossici (Roma 3), April, 7, 2008
“Schemi in gruppo di ordine $p^2$ ed estensione di torsori”
15:00, room 2BC/30

Hironori Shiga (Chiba University, Japan) Spring 2008
"Introduction to Elliptic Curves and Modular Forms"
Course (4 weeks)

Xavier Caruso (CNRS, Rennes) June 5-10
“Comparaison entre les poids de Hodge-Tate d'une représentation semi-stable et les poids de l'inertie modérée de sa semi-simplifiée mod p”,
June 9, 5:00 pm, in 2AB/40

Chia-Fu Yu (Academia Sinica, Taiwan) May 21- July 20
July 3, 14:30, room 1AD30.
“Superspecial abelian varieties with parahoric level structure”

Conference: ?p-adic differential equations: a conference in honor of Gilles Christol? September 6-9, 2008 - Bressanone

Kentaro Nakamura (Univ. Tokyo)
"Classification of 2-dimensional trianguline representations of p-adic fields"
October 17, 15:00 room 1AD50

Vasily Golyshev (Mosca) "Famiglie di varieta' di Calabi-Yau" October 24, 14:30, room 2BC60.