January 3 - 5, 2011
Keywords of the presentation: p-adic periods, derived de Rham cohomology
I will show that Fontaine's ring of p-adic periods can be realized as the ring of universal p-adic constants in the sense of derived algebraic geometry, and discuss a possible new construction of the p-adic periods map.
Vector bundles and p-adic Galois representationsJanuary 5, 2011- Video(flv)
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- Slides(pdf)
Keywords of the presentation: $p$-adic Galois representations, $p$-adic Hodge theory, vector bundles
Let $F$ be a perfect field of characteristic $p>0$ equipped with a non trivial absolute value, $E$ a non archimedean locally compact field whose residue field is contained in $F$ and $pi$ a uniformizing parameter of $E$. We associate functorially to these datas a separated integral noetherian regular scheme $X=X_{F,E,pi}$ of dimension $1$ defined over $E$. There is an equivalence of categories between semi-stable vector bundles of slope $0$ over $X$ and continuous $E$-linear representations of the absolute Galois group $H_F$ of $F$.
When $F$ is algebraically closed, the closed points of $F$ can be described in terms of the Lubin-Tate formal group of $E$ corresponding to $pi$.
If $C$ is the $p$-adic completion of $overline Q_p$, one can associate to $C$ an algebraically closed field $F=F(C)$ as above and ${rm Gal)(overlineQ_p/Q_p)$ acts on the curve $X=X_{F(C),Q_p,p}$. The two main results of $p$-adic Hodge theory can be recovered from the classification of vector bundles over $X$.
(joint work with Laurent Fargues)
Read more at
http://www.math.u-psud.fr/~fargues/Prepublications.html.
Keywords of the presentation: Shimura varieties
I will explain some results towards the Langlands-Rapoport
conjecture which
predicts the structure of the mod p points of a Shimura
variety. A consequence
of the conjecture is that the isogeny class of every mod p
point contains a point which
admits a lifting to a special (ie CM) point of the Shimura
variety. One of the roots
of the subject is the work of John Tate on CM liftings and
endomorphisms of
abelian varieties mod p.
Elliptic curves: problems and applicationsJanuary 3, 2011- Video(flv)
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- Slides(pdf)
Keywords of the presentation: elliptic curve, integer factoring, smooth number
In the past three decades there have been some exciting
applications of elliptic curves over finite fields
to integer factoring, primality testing, and
cryptography. These applications in turn have raised some interesting
problems often of an unconventional flavor. For example, how
often is the order of an elliptic curve group prime, or how
often does it have all small prime factors? In this talk we will
visit problems such as these, as well as other analytic-type
problems relating to ranks of elliptic curves over function
fields and to elliptic divisibility sequences.
Random maximal isotropic subspaces and Selmer groupsJanuary 5, 2011- Video(flv)
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- Slides(pdf)
Keywords of the presentation: Selmer group, Shafarevich-Tate group, maximal isotropic, quadratic space
We show that the p-Selmer group of an elliptic curve is naturally the intersection of two maximal isotropic subspaces in an infinite-dimensional locally compact quadratic space over F_p. By modeling this intersection as the intersection of a random maximal isotropic subspace with a fixed compact open maximal isotropic subspace, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2-Selmer groups in certain families of quadratic twists, and the average size of 2- and 3-Selmer groups as computed by Bhargava and Shankar. The only distribution on Mordell-Weil ranks compatible with both our random model and Delaunay's heuristics for Sha[p] is the distribution in which 50% of elliptic curves have rank 0, and 50% have rank 1. We generalize many of our results to abelian varieties over global fields. This is joint work with Eric Rains.
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I will discuss the statement and proof of an Equivariant Main Conjecture (EMC) in the Iwasawa theory of arbitrary global fields.
This will be followed by applications of the EMC (via Iwasawa co-descent) towards proving various well known conjectures on special values of global L-functions. In the process, an important role will be played by
an explicit construction of ell-adic Tate sequences.
This is based on joint work with Cornelius Greither (Munich).
Keywords of the presentation: Rigid geometry, valuations, rigid-tale topology, semi- stable models
If we specialize algebraic equations having good
properties, we usually face degeneracies. Starting with a bad
specialization, we can try to improve it , performing modifications
under control. If we succeed to get a new specialization with the
initial good properties preserved,we get a permanence statement.
We shall present examples of permanence with particular interest
concerning semi-stable models.
We know, thanks to the Weil conjectures, that counting points of
varieties over finite fields yields purely topological information
about them. In this talk I will first describe how we may count the
number of points over finite fields on the character varieties
parameterizing certain representations of the fundamental group of a
Riemann surface into GL_n. The calculation involves an array of
techniques from combinatorics to the representation theory of finite
groups of Lie type. I will then discuss the geometric implications of
this computation and the conjectures it has led to.
This is joint work with T. Hausel and E. Letellier
Keywords of the presentation: elliptic curves, Selmer groups, ranks
In joint work with Barry Mazur, we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions for an elliptic curve to have twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) with a given 2-Selmer rank. As a consequence, under appropriate hypotheses there are many twists with Mordell-Weil rank zero, and (assuming the Shafarevich-Tate conjecture) many others with Mordell-Weil rank one. Another application of our methods, using ideas of Poonen and Shlapentokh, is that if the Shafarevich-Tate conjecture holds then Hilbert's 10th problem has a negative answer over the ring of integers of any number field.
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