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B. Edixhoven
Modular parametrisations
2 lectures
The first two lectures will be devoted to modular curves and
parametrizations of elliptic curves over the rationals by modular
curves. Apart from the basics, some things will be said about
reduction mod p and the image of the supersingular points, Manin
constants, and Stevens's conjecture on the relation between strong
parametrizations and modular height.
Non-triviality of Heegner points (after Cornut)
2 lectures
The remaining two lectures will be used to explain Cornut's proof of
non-triviality of Heegner points in anti-cyclotomic towers, using
Moonen's work on the André-Oort conjecture.
References :
Chapters II (Silverman) and III (Rohrlich) of Modular forms and
Fermat's Last Theorem, Gary Cornell and Glenn Stevens (editors),
Springer-Verlag, 1997.
Edixhoven's home page
Cornut's home page
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J. Oesterlé
La conjecture de Birch et Swinnerton-Dyer
4 exposés
Un premier exposé sera dévolu à un rappel des propriétés
arithmétiques des
variétés abéliennes définies sur les corps finis, locaux et globaux.
Nous considérerons ensuite une variété abélienne
définie sur un corps global, définirons sa fonction L
et son groupe de Tate-Shafarevich et formulerons les
conjectures de Birch et Swinnerton-Dyer : prolongement holomorphe de la
fonction
L, ordre d'annulation et coefficient dominant en s=1, finitude du
groupe de Tate-Shafarevich. Nous examinerons leur compatibilité
aux isogénies, à la restriction de Weil et leur lien avec les mesures
de Tamagawa.
Dans un dernier exposé, nous ferons le point sur
l'état actuel des connaissances en ce qui concerne ces conjectures.
Bibliographie :
1. Pour une introduction à la théorie des variétés
abéliennes :
Arithmetic Geometry, édité par G. Cornell et J. H. Silverman,
Springer-Verlag, 1986. (surtout les chapitres IV à VIII).
2. Pour la formulation des conjectures de Birch and
Swinnerton-Dyer :
J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a
geometric analog (1966), Séminaire Bourbaki, Vol. 9, Exp. 306, 415-440, S.M.F,
1995.
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P. Colmez
Fonctions L p-adiques de courbes elliptiques
Le but de l'exposé est d'énoncer la conjecture de
Mazur-Tate-Teitelbaum, analogue p-adique de la conjecture
de Birch et Swinnerton-Dyer. On passera donc rapidement en revue
les ingredients utilisés : distributions sur Z_p, symboles
modulaires, hauteurs p-adiques ...
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H. Darmon
Periods of modular forms and the arithmetic of elliptic curves
4 lectures
The Birch and Swinnerton-Dyer conjecture relates the arithmetic of
an elliptic curve to the behaviour of its associated L-series. When the
elliptic curve is modular, this L-series is related to periods of a
corresponding modular form. The Birch and Swinnerton-Dyer conjecture can
then be recast in terms of such periods. Viewed in this way, it
becomes
part of a more general and flexible framework in which a variety of
new
phenomena can be discovered and studied. Each of my lectures will be
devoted to exploring a different aspect of this simple unifying theme.
Much of what I will present is the fruit of a collaboration with Massimo
Bertolini.
1. Overview
2. p-adic integration and p-adic L-functions
3. Stark-Heegner points attached to real quadratic fields
4. The Mazur-Tate circle pairing
References :
Lectures 1 and 2 :
The notes of my NSF-CBMS lecture series which are
gradually being posted at
http://www.math.mcgill.ca/darmon/courses/cbms/cbms.html
Lecture 3 :
Henri Darmon :
Integration on Hp x H and arithmetic applications
Annals of Mathematics 154 (2001) 589-639.
Henri Darmon and Peter Green :
Elliptic curves and class fields of real quadratic fields : algorithms and
evidence,
Experimental Mathematics, 11:1, 2002, 37-55.
(Both these articles can be dowloaded from
http://www.math.mcgill.ca/darmon/pub/pub.html)
Lecture 4 :
Barry Mazur and John Tate : Canonical height pairings via
biextensions, Arithmetic and geometry, Vol. I, 195--237, Progr. Math.,
35,
Birkhauser Boston, Boston, MA, 1983.
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B. Mazur
The Mechanics of Kolyvagin systems
2 lectures
The technique of Kolyvagin used to control the size of arithmetic
Selmer groups has, as input, a certain system of cohomological classes
(or
rational points, or algebraic cycles, or elements in algebraic K-theory,
depending upon the setting) over abelian extensions of the base field.
Such
systems have been called "Euler Systems" by Kolyvagin. As an
intermediary
step in Kolyvagin's method, a given Euler System is used to construct a
specific collection of cohomology classes over the base field. It is this
collection of cohomology classes over the base field that directly
provide
sufficiently many "local linear relations" satisfied by elements in the
Selmer group so as to obtain good upper bounds for the size of the Selmer
group. Such "collections of cohomology classes over the base field" enjoy
(somewhat surprisingly) very rigid inter-relations. Karl Rubin and I
have a
manuscript in which we study this type of structure ; we call collections
of cohomology classes satisfying these relations "Kolyvagin Systems".
These Kolyvagin Systems turn out to have especially nice, and
interesting,
properties. For one thing, they seem more congenial to Galois
deformations
than are Euler systems, and they can be constructed in instances where we
have no construction, yet, of a corresponding Euler system. It is
helpful
to view the generator of the module of Kolyvagin Systems, when there is
only one generator, as something of a p-adic L-function (with values in
cohomology groups) governing the arithmetic of the situtation. The aim of
these two lectures is to elucidate this structure and its basic
arithmetic applications.
Bibliography :
1. Chapter 1 of Karl Rubin's Euler Systems Annals of Math Studies,
Princeton Univ. Press, 2000.
2. Chapters 2 and 3 of the above.
3. The still-unfinished manuscript Kolyvagin Systems that Karl Rubin
and I are working on, and versions of this are on his web-page and on mine
(http://abel.math.harvard.edu/~mazur/projects.html).
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J. Nekovár
Skew-symmetric pairings on Selmer groups and their applications
4 lectures
The course will cover the following topics :
(1) The proof of the Parity conjecture for Selmer groups of elliptic
curves over Q (in the case of ordinary reduction).
(2) General duality theory for Selmer groups in Iwasawa theory
(in the case of (quasi)-ordinary reduction).
(3) A generalization of (1) to modular forms of higher weight
(if time permits).
Prerequisites :
(a) Basic homological algebra.
(b) Descent theory for elliptic curves (definitions and basic properties
of Selmer groups).
(c) Duality theorems for Galois cohomology over local and global fields
(Tate, Poitou).
References :
The course of Oesterlé and the standard texts on the
subject (Silverman : The arithmetic of elliptic curves ;
Serre : Cohomologie galoisienne;
Neukirch, Schmidt, Wingberg : Cohomology of number fields).
Nekovár's
home page
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W. Stein Constructing Shafarevich-Tate Groups Using Visibility
I will discuss three situations in which congruences between
modular forms have been used to construct subgroups of
Shafarevich-Tate groups in situations where a direct computation would
be very difficult. The first, due to Cremona, Mazur, Agashe, and
myself, gives new systematic evidence for the Birch and
Swinnerton-Dyer conjecture for certain rank 0 quotients of the modular
Jacobian J_0(N). The second gives the first examples, for each odd p
< 20000, of abelian varieties A such that the p-part of Sha(A) is not
a perfect square. The third, due to Dummigan, Watkins, and myself,
provides examples and evidence for the Bloch and Kato conjecture for
motives attached to modular forms. I will assume the audience members
are familiar with abelian varieties, know the definition of
Shafarevich-Tate groups and have some experience with modular forms,
but I will not assume prior exposure to visibility theory or modular
motives.
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E. Urban
Sur les déformations p-adiques des formes de Saito-Kurokawa
3 exposés
Etant donne une forme modulaire f de poids 2k de niveau N
et caractère trivial. Sa fonction zeta satisfait une équation
fonctionnelle du type L(f,s) = epsilon(f,s) L(f,2k-s). Soit T_f la
représentation galoisienne associée à f par Deligne.
Le résultat suivant est le fruit d'une collaboration avec C. Skinner :
Théorème (S-U) : Supposons que epsilon(f,k) = -1 (i.e.
L(f,s) s'annule en s=k avec un ordre impair), alors le groupe de Selmer
associé à T_f(k) est infini.
Un exposé sera consacré la construction de déformations
p-adiques de formes modulaires de Siegel p-adiques semi-ordinaires et
des représentations galoisiennes correspondantes. Dans l'autre exposé,
j'expliquerai comment l'étude des déformations p-adiques des
formes de Saito-Kurokawa permet de construire certaines extensions du
type :
0 ----> T_f ----> E ----> Q_p(-k) ----> 0
et de montrer le théorème ci-dessus.
Bibliographie :
M. Kisin : Overconvergent modular forms and the
Fontaine-Mazur conjecture, preprint.
G. Laumon : Fonctions zeta des variétés de
Siegel de dimension 3, preprint.
J. Tilouine, E. Urban : Several variable p-adic
families of Siegel-Hilbert cusp eigensystems and their Galois representations,
Ann. Sci. E.N.S., 4 srie, t. 32, p. 499-574, 1999.
E. Urban : Sur les représentations p-adiques
associées aux représentations cuspidales de (GSp_4)_Q,
preprint.
R. Weissauer : Four dimensional Galois
representations, preprint.
J.-L. Waldspurger : Correspondances de Shimura et
quaternions, Forum Math. 3, p. 219-307, 1991.
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S. Zhang Gross-Zagier formula for Shimura curves
3 lectures
1. Statement of the formulas
2. Rankin-Selberg convolution of L-series
3. Height pairings on a Shimura curve
Bibliography :
B. Gross and D. Zagier : Heegner points and derivatives of L-series, Invent. Math 84, (1986), 225-320
B. Gross : Heegner points on X_0(N), in Modular forms (Durham, 1983) 87-105, Ellis-Hoewood, Chichester (1984)
B. Mazur : Modular curves and Arithmetic, ICM, Warsaw (1983) 185-211
S. Zhang : Heights of Heegner points on Shimura curves, Annals of Mathematics, 153 (2001) 27-147
S. Zhang : Gross-Zagier formula for GL_2, Asian J. Math., 5 (2001) 183-290
H. Jacquet and R. Langlands : Automorphic forms on GL_2, Lect. Notes in
Math., 114, Springer-Verlag (1972)
H. Jacquet : Automorphic forms on GL_2 II, Lect. Notes in Math. 289, Springer-Verlag (1972)
P. Deligne : Travaux de Shimura, Seminaire Bourbaki, in Lect. Notes Math. 244 Springer-Verlag (1972), 123-165
H. Carayol : Sur la mauvaise reduction des courbes de Shimura, Comp. Math 59 (1986) 151-618
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