abstract: I will report on the work of my student A. Marmora.
Fontaine defined a hierarchy on $p$-adic Galois representations: crystallin, semi-stable and de Rham
representations. He also introduced
numerical invariants that measure, for a potentially semi-stable
representation, the defect for being semi-stable, namely the Swan and the Artin conductors of its Weil-Deligne
representation. Recently, Berger has associated with any de Rham
representation a $p$-adic differential equation, that
provided one way for proving the $p$-adic monodromy
theorem: de Rham representations are potentially
semi-stable (and vice-versa).
Marmora's result compares the Swan conductor of a de Rham
representation with the irregularity of its p-adic differential equation.
Abstract: Coleman's iterated integral theory can be
formulated as saying that the space of paths between two points on a scheme X
over a field of characteristic p, has a canonical
element which is fixed under Frobenius. Here, a path means an isomorphism
between the fiber functors associated with these two points. In particular, all
these fiber functors are naturally isomorphic, so in a sense there is only once
such functor.
In trying to generalize this theory to higher order
forms one is faced with the problem of not having appropriate fiber functors.
Thus, it seems interesting, even in the case where its existence is already
proved, to try to define "the" fiber functor without using points. We
will report on this project.
There is a very beautiful analogy between the theory
of l-adic sheaves on curves over a finite field, and the theory of connections
(holonomic D-modules) on curves over a field k in characteristic 0. Pursuing
this analogy, we define local and global epsilon factors for such D-modules.
The local factors depend, just as in the l-adic case, on a choice of non-zero
meromorphic 1-form on the curve. There is a global product formula, and the
local factors satisfy exactly the same formulae as in the l-adic case, with q
(number of elements in the finite field) replaced by 2\pi i. This is joint work
with A. Beilinson, P. Deligne, and H. Esnault. It builds on earlier work by T.
Saito and T. Terasoma.
Jean-Louis
Colliot-Thélène, Rational
points on surfaces fibred into curves of genus one
Abstract : A couple of years ago, a
technique was devised by Swinnerton-Dyer to study the
existence of rational points on the total space of a surface fibred over the
projective line, the fibres being curves of genus one
(over a number field). I will describe the developments of the method and the
concrete results hitherto obtained.
Hélène Esnault, Vanishing cycles and local Fourier
transforms for holonomic D-modules in dimension 1
abstract: N.
Katz defined the notion of rigid l-adic reprensations
on the projective line over a finite field, and of rigid local systems over the
projective line over the complex numbers. There are those which are uniquely
recognized by their local data at the punctures. He shows the ramarkable theorem that those are all obtained from rank
one ones after convolution and Fourier transforms. He leaves aside the case of holonomic D-modules which are not regular singular. The
reason is that he does not have in this case Laumon's
equivalence of categories at disposal, between local representations. Those quivalence of categories come from his local Fourier tranforms. We try to construct those for holonomic (non-tame) D modules and to complete Katz'
theorem in this case.
Abstract: Over $p$-adic fields, nearby cycle construction
for $\ell$-adic sheaves are regarded as an analogue of microlocalization. For a
finer analysis of $\ell$-adic sheaves, it is quite useful to introduce the
viewpoint of rigid geometry. In this talk, we formulate a general conjecture on
$\ell$-adic sheaves generalizing Abbes-Saito, and also discuss the
function-sheaf dictionary in this case.
Abstract:
I will discuss how to combine ideas of Lichtenbaum
and Voevodsky to construct a cohomology theory for varieties over finite fields
and give a conjecture relationship with special values of zeta-functions.
Masaki Hanamura, Motivic sheaves over
varieties
In my previous work I gave constructions of
(1) the triangulated category of mixed
motives over a field, and
(2) the category of pure motivic sheaves over a base variety.
Each theory is useful as it is, but there is limitation until they
are further developed into the theory of mixed motivic
sheaves. (Another source of limitation are several
conjectures on algebraic cycles: the standard conjecture by Grothendieck, the
filtration conjecture by Bloch-Beilinson, and the
vanishing conjecture by Soul\'e-Beilinson. As
consequences of these conjectures, the category of motives would have quite
desirable structures.)
In this talk, I will discuss mixed motivic
sheaves over a base variety -- the construction of the category, the
properties, and conjectures.
Annette Huber-Klawitter, From
Bloch-Kato to the Main Conjecture of Iwasawa Theory
Let $k$ be a perfect field, $DM_{gm}^{eff}(k,Q)$ Voevodsky's
triangulated category of effective geometric motives with rational coefficients
and $M_1(k,Q)$ Deligne's
category of $1$-isomotives (it is abelian). Following
an announcement of Voevodsky, Orgogozo has
constructed a fully faithful triangulated functor
$$D^b(M_1(k,Q))\to
DM_{gm}^{eff}(k,Q)$$
whose
essential image is the thick subcategory of $DM_{gm}^{eff}(k,Q)$ generated by motives of smooth curves. We prove that
this functor has a left adjoint
$LAlb$. This generalises
the classical construction of the Albanese variety and Albanese map associated
to a smooth variety.
Applying $LAlb$
to motives of varieties or their duals, we get natural complexes of
1-isomotives (up to quasi-isomorphism) whose homology 1-isomotives give back
the 1-(iso)motives constructed earlier by Barbieri-Viale--Srinivas and
(presumably) by Barbieri-Viale--Rosenschon--Saito.
This is joint work with Luca Barbieri-Viale.
How to formulate Riemann Roch type formulas for
$ell$-adic sheaves is a very interesting old problem. Using the K-theoretic localized intersection
theory, Takeshi Saito and I have obtained formulas of Riemann-Roch type for
$\ell$-adic sheaves. For a morphism of schemes $f : X \to Y$ and an $\ell$-adic
sheaf $F$ on $X$, our result compares the Swan conductor of $F$ and the Swan
conductors of the higher direct images of $F$ under $f$, under certain
assumptions on $f$ and $F$. I have to remark that we think our formulas are not
the final desired ones. I hope that better formulas are
found basing on the joint work of
Just as the fact that
``a
commutative ring may be represented by means of a locally ringed space''
underlies much of scheme theory, certain results
recently obtained by the speaker concerning ``representations of arithmetic log
schemes by means of categories'' allow one to set up a new kind of geometry in
which various operations not possible in conventional scheme theory become
possible. Such results concerning the category-theoretic representation of
arithmetic log schemes are motivated by the possibility of application to
Diophantine geometry and build on the ideas and techniques of anabelian
geometry --- a geometry originally proposed by Grothendieck as a possible
candidate for a new approach to Diophantine geometry. In the present lecture, we hope to
discuss some of these results and, in particular, some of the curious new
objects arising from these results --- such as an absolute Frobenius morphism
on a number field and the construction of a ``global multiplicative subspace''
of the module of torsion points of an elliptic curve over a number field.
I will discuss a positive characteristic version of
the non--abelian Hodge theory developed by Simpson, Katzarkov, Pantev, and Toen. In
particular I will explain how this theory yields restrictions on fundamental
groups as well as a positive characteristic version of the formality theorem
for homotopy types of smooth proper complex algebraic
varieties.
Michael
Spiess, Galois representations and $p$-adic periods of quaternionic automorphic forms.
The classical uniformization theorem of Koebe asserts
that all hyperbolic Riemann surfaces have a common (analytic) universal
covering, i.e., the complex upper half-plane. In
contrast to this, a theorem of Mochizuki (J. Pure Appl. Algebra 131 (1998))
implies that there are at most finitely many isomorphism classes of algebraic
curves over an algebraically closed field of characteristic 0 that have a
prescribed (algebraic) universal covering. In this talk, we discuss what
happens in positive characteristic.