Tue Sep 28: 10h-12h (IHP, salle Darboux) -- Wed Sep 29: 10h-12h (IHP, salle Darboux)
Wed Oct 20: 10h-12h (IHP, salle Darboux) -- Wed Oct 27: 10h-12h (IHP, salle Darboux)
Wed Nov 03: 10h-12h (IHP, salle Darboux) -- Wed Nov 10: 10h-12h (IHP, salle Darboux)
Wed Nov 24: 10h-12h (IHP, salle Darboux) -- Wed Dec 01: 10h-12h (IHP, salle Darboux)
Wed Dec 15: 10h-12h (IHP, salle Darboux)

We will develop the basic theory of elliptic curves over a general field, motivated by the desire to understand and make as explicit as possible the arithmetic of elliptic curves over number fields. This will involve additional study of curves over local fields, finite fields, and the real and complex fields. The emphasis throughout will be on Explicit Methods: wherever possible we will discuss algorithms and use software packages to provide examples. Some more specialised topics will only be covered briefly, as they will be treated in more detail by others during the semester; in particular, descent (see the short course by Stoll starting October 21) and complex multiplication (see the short course by Enge starting October 20). Conversely, we hope to include material on the computation of isogenies and Heegner points.

Familiarity with the basic notions and terminology from algebraic geometry, particularly for curves over fields which are not necessarily algebraically closed, will be a prerequisite; the first two chapters of Silverman's book "The Arithmetic of Elliptic Curves", which will be covered in the opening lectures by Bjorn Poonen, contain everything needed. We will also assume basic knowledge of algebraic number theory (including Galois-theoretic aspects and localization).

We do not plan to include modular aspects such as the explicit computation of modular elliptic curves; no knowledge of modular forms will be required.

Tue Sep 21: 10h-12h (IHP, salle Darboux) -- Wed Sep 22: 10h-12h (IHP, salle Darboux)
Tue Oct 19: 10h-12h (IHP, salle Darboux) -- Tue Oct 26: 10h-12h (IHP, salle Darboux)
Tue Nov 02: 10h-12h (IHP, salle Darboux) -- Tue Nov 09: 10h-12h (IHP, salle Darboux)
Tue Nov 16: 10h-12h (IHP, salle Darboux) -- Tue Nov 23: 10h-12h (IHP, salle Darboux)
Tue Nov 30: 10h-12h (IHP, salle Darboux) -- Tue Dec 14: 10h-12h (IHP, salle Darboux)

The course will focus on the explicit determination of rational points on curves defined over number fields. Among the topics to be covered will be

• basics on varieties over perfect fields (as in Chapters I and II of Silverman, The Arithmetic of Elliptic Curves)
• some Galois cohomology (H^0 and H^1)
• explicit models of low genus curves and hyperelliptic curves
• Jacobians of curves
• endomorphism rings of Jacobians
• Mordell-Weil groups of Jacobians
• étale covers and descent
• Chabauty's method

1. Galois theory, including the theory for infinite algebraic extensions (e.g., Chapter VI of Lang, Algebra)
2. Basic algebraic number theory (e.g., Chapters I-V of Lang, Algebraic Number Theory or Chapters I-III of Cassels & Fröhlich, Algebraic Number Theory)
   Class field theory will not be assumed.
3. Some familiarity with algebraic geometry, the more the better (as a minimum: Chapter I of Hartshorne, Algebraic Geometry, or parts of Shafarevich, Basic Algebraic Geometry, for instance)
   Knowledge of schemes will not be assumed, but may be helpful.

• Cassels & Fröhlich: Algebraic number theory
• Lang: Algebraic number theory
• Serre: Local fields
• Serre: Galois cohomology
• Shafarevich: Basic algebraic geometry (originally in one volume, now in two)
• Hartshorne: Algebraic geometry
• Silverman: The arithmetic of elliptic curves
• Cornell & Silverman (eds.): Arithmetic geometry
• Serre: Lectures on the Mordell-Weil Theorem
• Liu: Algebraic geometry and arithmetic curves
• Bosch, Lütkebohmert, Raynaud: Néron models
• Manin: Cubic forms
• Skorobogatov: Torsors and rational points

Tue Oct 26: 17h-18h30 (ENS, Salle W) -- Tue Nov 02: 17h-18h30 (ENS, Salle W)
Tue Nov 09: 17h-18h30 (ENS, Salle W) -- Tue Nov 16: 17h-18h30 (ENS, Salle W)
Tue Nov 23: 17h-18h30 (ENS, Salle W) -- Tue Nov 30: 17h-18h30 (ENS, Salle W)

À l'École Normale Supérieure (salle W, toits du DMA)

Liste indicative des thèmes traités:

• Caractères et séries L de Dirichlet.
• Formes quadratiques binaires.
• Théorie des genres.
• Liens entre formes quadratiques binaires et idéaux des corps quadratiques.
• Théorie de la réduction et fractions continues.
• Multiplication complexe.

Mon Oct 04: 16h15-18h15 (Collège de France) -- Mon Oct 25: 16h15-18h30 (Collège de France)
Mon Nov 08: 16h15-18h30 (Collège de France) -- Mon Nov 15: 16h15-18h30 (Collège de France)
Mon Nov 22: 16h15-18h30 (Collège de France) -- Mon Nov 29: 16h15-18h30 (Collège de France)

Au Collège de France