Mon Sep 20: 10h-12h (IHP, salle Darboux) -- Mon Sep 27: 10h-12h (IHP, salle Darboux)
Mon Oct 18: 10h-12h (IHP, salle Darboux) -- Mon Oct 25: 10h-12h (IHP, salle Darboux)

Nous donnerons un sens et une démonstration à l'assertion suivante : l'hypothèse de Riemann équivaut au fait que toute suite équilibrée non nulle est d'énergie infinie.

Wed Sep 08: 10h-12h -- 14h-16h (IHP, salle Darboux)
Thu Sep 09: 10h-12h -- 14h-16h (IHP, salle Darboux)
Fri Sep 10: 14h-16h (IHP, salle Darboux)

The course is an introduction to the computer algebra system PARI/GP.

We shall present

• basic concepts, general hints and tricks
• the GP programming language (and the gp interpreter)
• the number fields module

Fri Sep 10: 10h-12h (IHP, salle Darboux) -- Fri Sep 17: 10h-12h (IHP, salle Darboux)
Fri Sep 24: 10h-12h (IHP, salle Darboux) -- Fri Oct 01: 10h-12h (IHP, salle Darboux)

In these lectures, we discuss the various approaches to solve this type of diophantine equations. In particular, we give some more detailed attention to the cases when there are infinitely many solutions, for example the full solution of x² + y³ = z⁵ using invariant theory.

Download the course notes.

Thu Oct 07: 14h-15h30 (Collège de France) -- Thu Oct 14: 14h-15h30 (Collège de France)
Thu Oct 21: 14h-16h (Collège de France) -- Thu Oct 28: 14h-16h (Collège de France)

Mon Nov 08: 10h-12h (IHP, salle Darboux) -- Mon Nov 15: 10h-12h (IHP, salle Darboux)
Mon Nov 22: 10h-12h (IHP, salle Darboux) -- Mon Nov 29: 10h-12h (IHP, salle Darboux)

Wed Oct 20: 14h-16h (IHP, salle Darboux) -- Wed Oct 27: 14h-16h (IHP, salle Darboux)
Thu Oct 28: 10h-12h (IHP, salle Darboux) -- Thu Nov 04: 14h-16h (IHP, salle Darboux)
Fri Nov 12: 10h-12h (IHP, salle Darboux)

The aim of this series of lectures is to give an account of the theory of curves with complex multiplication, with a special emphasis on the efficient effective construction of such curves. Starting from the most classical case of elliptic curves, for which a rich variety of results are known since the mid nineteenth century, we also provide a general description in higher genus and give numerical examples for all these. The last talk is an illustration of the power of CM curves in algorithmic number theory, and more specifically for proving large numbers to be prime.

We have split the lectures into four parts:

1. (A. Enge -- 2 hours) CM in genus 1: the classical approach.

The lecture deals with the construction of elliptic curves with complex multiplication using the absolute modular invariant j. We briefly present the theory of elliptic functions and elliptic curves over the complex numbers (referring, where possible, to the course by Cremona, but having followed this course is not a prerequisite). The relationship to the explicit class field theory of imaginary quadratic fields is exhibited. The resulting algorithm is presented together with numerical examples.

Prerequisites: basic complex analysis, basic algebraic number theory, finite fields.

Bibliography:
David Cox, "Primes of the form x² + n y²", Wiley, New York 1989
Neal Koblitz, "Introduction to elliptic curves and modular forms", Graduate Texts in Mathematics 97, 2nd ed., Springer, New York 1993

2. (A. Weng -- 2 x 2 hours) CM in higher genus: theory and practice.

This talk deals with the construction of curves of low genus whose Jacobian has complex multiplication by an order in a CM field. This generalizes the generation of elliptic curves with complex multiplication to higher dimensions.

We discuss the theory of complex multiplication due to G. Shimura, the algorithmic features and present examples for genus 2 and 3.

We require some knowledge in algebraic number theory and basic complex algebraic geometry (I recommend the book by Lang, Introduction to Algebraic and Abelian functions.).

Bibliography:
G. Shimura, Complex multiplication of abelian varieties and its applications to number theory. 1961 (or the revised edition from 1998)
S. Lang, Complex multiplication, Springer 1983.

3. (A. Enge -- 2 hours) CM in genus 1: the modern approach.

The aim of this lecture is to show how elliptic curves with complex multiplication can be constructed more efficiently by replacing the j-function by modular functions of higher level. Besides presenting the necessary theory of modular functions for Γ₀(N) and of Shimura's reciprocity law, we also discuss a few tricks to speed up the implementation.

4. (F. Morain -- 2 hours) Use of CM curves in primality proving.

The elliptic curve primality proving algorithm (ECPP) -- due originally to Atkin -- is one of the most practical algorithms for giving an exact checkable proof of primality for large numbers. Heuristically, it runs in time O((log N)^4+c) and the known implementations can handle numbers with up to several thousands decimal digits. The core of the algorithm is the construction of CM elliptic curves associated to quadratic fields of huge class numbers (10⁴, say). We will describe the theory and practice of this algorithm.

Fri Oct 22: 10h-12h (IHP, salle Darboux) -- Fri Oct 29: 10h-12h (IHP, salle Darboux)
Fri Nov 05: 10h-12h (IHP, salle Darboux) -- Fri Nov 05: 14h-16h (IHP, salle Darboux)

Audience: First half will assume nothing but elementary complex variable. In particular, it will assume no number theory. The second half will assume some algebraic number theory, at the level of knowing what Dirichlet L-series and ray clas groups are.

Bibliography: Any of the basic textbooks on algebraic number theory, such as Lang, Froehlich-Taylor, Neukirch or Koch.

Subject (Summary). Around 1900 Barnes defined the multiple gamma function, a higher-dimensional generalization of Euler's gamma function. This special function has since become important in various areas of mathematics, but its study has been uninviting due to the extensive and unilluminanting calculations involved. Following recent joint work with mathematical physicist Simon Ruijsenaars, I will give a clean treatment of the multiple gamma function and its basic properties. This should cover about half of the course.

The second half of the course will be devoted to Shintani's attack of Stark's conjecture (in the first order zero case) using Barnes' multiple gamma function.

Fri Nov 26: 14h-16h (IHP, salle Darboux) -- Fri Dec 03: 14h-16h (IHP, salle Darboux)

Le but de ce cours est de présenter les différents algorithmes qui permettent de déterminer la fonction Zeta d'une courbe sur un corps fini. Selon le type de courbe (genre petit ou grand, hyperelliptique ou non, ...) et le type de corps fini (petite ou grande caractéristique), les méthodes utilisent des outils différents.

Le cours inclura une description plus ou moins détaillée des algorithmes suivants:

1. Les algorithmes basés sur un relèvement canonique p-adique (Satoh, AGM).
2. L'algorithme de Kedlaya qui utilise un calcul explicite dans un espace de cohomologie de Monsky-Washnitzer.
3. Les algorithmes basés sur le calcul de l'action de l'endomorphisme de Frobenius sur la l-torsion (algorithme de Schoof et extensions).

Mon Sep 27: 16h30-18h30 (IHP, salle Darboux) -- Wed Sep 29: 16h30-18h30 (IHP, salle Darboux)
Fri Oct 01: 16h30-18h30 (IHP, salle Darboux)

The aim of the course is to become familiar with the computer algebra system Magma with a view to computations in algebraic number theory and algebraic geometry. It is intended to look at

• the basic concepts and the programming language of Magma,
• general hints and tricks,
• the number field and function field modules of Magma.

Thu Sep 23: 10h-12h (IHP, salle Darboux) -- Fri Sep 24: 14h-16h (IHP, salle Darboux)
Wed Sep 29: 14h-16h (IHP, salle Darboux) -- Thu Sep 30: 10h-12h (IHP, salle Darboux)
Fri Oct 01: 14h-16h (IHP, salle Darboux)

We present conjectures and results on the asymptotic of the counting function for number fields with given Galois group.

We introduce the counting function Z(k,G;x) and the associated zeta-function and look at some examples with cyclic and symmetric groups. We then introduce two group theoretic constants and study their behaviour under wreath products and direct products. With these we formulate a conjecture and give supporting evidence. We also present results of explicit computations on number fields of small degree and discriminant.

In a second part we prove that some of the conjectures are true for nilpotent groups. Furthermore we show that the asymptotic conjectures for solvable groups are related to class group questions. E.g. the asymptotic conjecture for dihedral groups is related to the Cohen-Lenstra heuristic for class groups of quadratic number fields.

In the last part we determine the analytic behaviour of the associated zeta-function for generalized quaternion groups and some wreath products. This implies that for all even degrees there exists a group which has linear asymptotic.

Required prerequisites are some basic algebraic number theory and elementary group theory.

References:

1. J. Klüners, G. Malle: Counting nilpotent Galois extensions, J. reine angew. Math. 572 (2004), 1-26.
2. G. Malle: On the distribution of Galois groups, J. Number Theory 92 (2002), 315-329.
3. G. Malle: On the distribution of Galois groups II, Experiment. Math. 13 (2004), 129-135.

Tue Nov 02: 14h-16h (IHP, salle Darboux) -- Tue Nov 09: 14h-16h (IHP, salle Darboux)
Tue Nov 23: 14h-16h (IHP, salle Darboux) -- Tue Nov 30: 14h-16h (IHP, salle Darboux)

This series of lectures concentrates on deterministic algorithms for finite fields. The emphasis is not on developing algorithms for practical use, but on viewing the quest for polynomial-time algorithms as a challenge of our structural understanding of finite fields. The topics to be treated include: representing finite fields; recognizing finite fields; constructing finite fields; constructing maps between finite fields. In addition, a selection of the following will be addressed: multiplicative algorithms; solving diagonal equations; factoring polynomials; applications to primality testing. Prerequisites: mathematical maturity appropriate for an advanced graduate course, including basic algebra and algebraic number theory.

References:

R. Lidl, H. Niederreiter, Introduction to finite fields and their applications, Cambridge University Press, Cambridge, 1986.
H.W. Lenstra, Jr., Finding isomorphisms between finite fields, Math. Comp. 56 (1991), 329-347.

Thu Nov 18: 14h-16h (IHP, salle Darboux) -- Wed Nov 24: 14h-16h (IHP, salle Darboux)
Thu Dec 02: 14h-16h (IHP, salle Darboux) -- Mon Dec 13: 10h-12h (IHP, salle Darboux)

Wed Nov 3: 14h-16h (IHP, salle Darboux) -- Thu Nov 4: 10h-12h (IHP, salle Darboux)

Prerequistes: Attendance of John Cremona's course on elliptic curves. Previous knowledge of modular forms is NOT assumed but the reader has to take some deep theorems on trust.

Synopsis:

1. Newforms and level lowering.
2. Frey curves.
3. Proof of Fermat's Last Theorem.
4. Eliminating Newforms.
5. Kraus' Method of eliminating exponents.
6. Sieving for exponents.
7. Diagonal version of eliminating newforms.

Wed Nov 10: 14h-16h (IHP, salle Darboux) -- Tue Nov 16: 14h-16h (IHP, salle Darboux)
Thu Nov 18: 10h-12h (IHP, salle Darboux) -- Thu Nov 25: 10h-12h (IHP, salle Darboux)
Wed Dec 01: 14h-16h (IHP, salle Darboux) -- Thu Dec 02: 10h-12h (IHP, salle Darboux)

Download the outline notes (uploaded Dec. 12) and the basic facts and notations (please check for updated version).

PREREQUISITES

1. First course in Algebraic Number Theory (for number fields): integers, ideals, absolute values, class groups, units, Dirichlet's Theorem, behaviour of primes in Galois extensions, basic theory of Cyclotomic fields.
2. Acquaintance with main theorems of (abelian) class-field theory in terms of ideals: Mainly ray-class groups/fields and Artin Isomorphism. (See e.g. appendix to Washington's book below).
3. Basic understanding of representation theory of finite groups over a field of characteristic 0, almost exclusively abelian groups over C. For these: characters, orthogonality relations, idempotents, connection with the ring/module theory of the group-ring etc.
4. Basic familiarity with Riemann zeta-function and Dirichlet L-functions (definitions, Euler product and acquaintance with the functional equation will probably suffice.)
5. General Algebra: Basic theory of rings and modules. Tensor product and exterior powers over commutative rings. Group-rings.
6. Basic notions of complex analysis (analytic continuation, Dirichlet series, Gamma function)
7. Knowledge of p-adic numbers and very basic p-adic analysis.

1. PRELIMINARIES
   • ζ_K(s)
   • L_K/k,S(s,χ) for K/k an abelian extension of number fields. Formulation in terms of Galois characters and ray-class characters.
   • Partial zeta-functions.
   • Units and S-units.
   • Class number formulae.
2. BASIC COMPLEX STARK CONJECTURES (complex, at s=0 without integrality conditions)
   • The function Θ_K/k,S(s).
   • The 0th order abelian stark conjecture (Thm. of Siegel)
   • The 1st order abelian stark conjecture
   • Cyclotomic examples in thes two cases.
   • The rth order abelian Stark conjecture (à la Rubin)
   • Discussion of what's known.
3. REFINED 0th AND 1ST ORDER STARK CONJECTURES (complex, at s=0 with integrality conditions)
   • Formulation, discussion of what's known.
4. THE BRUMER-STARK CONJECTURE
   • Statement
   • Cyclotomic example (Stickelberger)
   • Discussion of what's known.
5. THE FUNCTION Φ_K/k(s)
   • Twisted zeta-functions.
   • Definition of Φ_K/k(s)
   • Relation between Φ_K/k(1-s) and Θ_K/k(s).
   • Reformulation of complex abelian stark
   • conjectures at s=1.
6. p-ADIC STARK CONJECTURES
   • p-adic interpolation of L-functions and Φ_K/k(s).
   • p-adic conjecture at s=1.
   • Gross' p-adic conjecture at s=0.
   • Cyclotomic examples.
   • Discussion of what's known.
7. FURTHER TOPICS
   As time allows, chosen from:
   • rth order REFINED abelian Stark conjectures (à la Rubin/Popescu)
   • Conjecture for Φ_K/k(s) at s=0 (or Θ_K/k(s) at s=1)
   • Computational methods and examples
   • Connection with Hilbert's 12th problem
   • General (ie non-abelian) Stark's conjecture for Artin L-functions (Tate's formulation)

1. A. Frohlich and M. Taylor, Algebraic Number Theory. Cambridge University Press , 1991. (Useful for introductory material on L-functions etc.)
2. A .Frohlich (ed.), `Algebraic number fields (L-functions and Galois properties)', Proceedings of a Symposium, London, Academic Press , 1977. (See e.g. Martinet's article therein on characters and &L&-functions.)
3. Serge Lang, `Algebraic number theory' 2nd ed, New York, London Springer-Verlag , 1994, (Graduate texts in mathematics 110) (general reference)
4. J-P. Serre, `Représentations Linéaires de Groupes Finis' Paris, Hermann, 1971 or the English translation `Linear Representations of Finite Groups' GTM 42, Springer Verlag, 1977 (for representation theory)
5. L. Washington, `Introduction to Cyclotomic Fields', 2nd Ed., Graduate Texts in Math. 83, Springer-Verlag, New York, 1996. (for cyclotomic fields, (Dirichlet) L-functions, Summary of Global Class Field Theory.)

1. J. Tate, `Les Conjectures de Stark sur les Fonctions L d'Artin en s=0', Birkhaüser, Boston, 1984.
2. D. Burns, C. Popescu, J. Sands, D. Solomon, `Stark's Conjectures: Recent Work and New Directions', Amer. Math. Soc. (Proceedings of Baltimore conference, 2001). To appear shortly in the `Contemporary Mathematics' series of the AMS.
3. Lecture notes by D. Hayes available here.

Thu Oct 21: 10h-12h (IHP, salle Darboux) -- Fri Oct 22: 14h-16h (IHP, salle Darboux)
Tue Oct 26: 14h-16h (IHP, salle Darboux) -- Fri Oct 29: 14h-16h (IHP, salle Darboux)

Let E be an elliptic curve over Q (or, more generally, a number field). Then on the one hand, we have the finitely generated abelian group E(Q), on the other hand, there is the Shafarevich-Tate group (Q,E). Descent is a general method of getting information on both of these objects --- ideally complete information on the Mordell-Weil group E(Q), and usually partial information on (Q, E).

What descent does is to compute (for a given n > 1) the n-Selmer group Sel⁽ⁿ⁾(Q, E); it sits in an exact sequence

0 --> E(Q)/nE(Q) --> Sel⁽ⁿ⁾(Q, E) --> (Q, E)[n] --> 0

Tue Sep 21: 14h-16h (IHP, salle Darboux) -- Tue Sep 28: 14h-16h (IHP, salle Darboux)
Tue Oct 19: 14h-16h (IHP, salle Darboux) -- Tue Oct 20: 16h30-18h30 (IHP, salle Darboux)

Theme: how many/how few rational points can such a curve have? An emphasis on the cases with genus 1, 2 and 3, maybe something on point counting algorithms.

Fri Nov 26: 10h-12h (IHP, salle Darboux) -- Fri Dec 3: 10h-12h (IHP, salle Darboux)
Tue Dec 14: 14h-16h (IHP, salle Darboux) -- Fri Dec 17: 10h-12h (IHP, salle Darboux)

The goal of the lectures is to explain how one can use the theory of irreducible representations of G = GL_n(k) with k a finite field to count homomorphisms of the fundamental group of genus g Riemann surface to G. This calculation yields the zeta function of the variety parameterizing such representations from which we hope to extract geometric information about the analogous variety over C. The calculation involves a significant amount of combinatorics and the theory of symmetric functions, which we will cover.

(This is joint work with Tamas Hausel)

References: Macdonald, I. G., Symmetric functions and Hall polynomials, The Clarendon Press / Oxford University Press, New York, 1995 (MR96h:05207).

Notes for this short course are available here.