Serre jean-pierre

Collège de France · Mathématiques

Let k be a field of characteristic 2 and let L/k be a finite Galois extension with Galois group G. We show the equivalence of the following two properties: (*) The group G is generated by elements of order 2 and by elements of odd order. (**) There exists an element x of L such that Tr(x) = 1 and T(x.g(x)) = 0 for every non trivial element g of G.

We give a criterion for the independence of a family of ℓ-adic Galois representations of a number field.

We show that the Hecke algebra for modular forms mod 2 of level 1 is isomorphic to the power series ring F2[[x,y]]F2[[x,y]], where x=T3x=T3 and y=T5y=T5.

The nilpotence order of the mod 2 Hecke operators. Let $\Delta=\sum_{m=0}^\infty q^{(2m+1)^2} \in F_2[[q]]$ be the reduction mod 2 of the $\Delta$ series. A modular form f modulo 2 of level 1 is a polynomial in $\Delta$. If p is an odd prime, then the Hecke operator Tp transforms f in a modular form Tp(f) which is a polynomial in $\Delta$ whose deg...

Modular forms mod 2 : structure of the Hecke ring We show that the Hecke algebra for modular forms mod 2 of level 1 is isomorphic to the power series ring F2[[x, y]], where x = T3 and y = T5.

If k is a commutative field and G a reductive (connected) algebraic group over k, we give bounds for the orders of the finite subgroups of G(k); these bounds depends on the type of G and on the Galois groups of the cyclotomic extensions of k. Comment: Notes of lectures published in "Group Representation Theory", Lausanne, EPFL Press 2007

We give a criterion for the " almost independence " of a family of l-adic representations. Comment: 12 pages (French)

Let Cr(k) be the Cremona group of rank 2 over a field k, i.e. the group of all k-automorphisms of k(X,Y). We determine the l.c.m. of the orders of the finite subgroups of Cr(k) of order prime to the characteristic of k.

The first part is expository: it explains how finite fields may be used to prove theorems on infinite fields by a reduction mod p process. The second part gives a variant of P.Smith's fixed point theorem which applies in any characteristic.

Let Cr(k) = Autk(X; Y ) be the Cremona group of rank 2 over a eld k. We give a sharp multiplicative bound M(k) for the orders of the

Representations of the quaternion group by 2×2 matrices with coefficients in the ring of integers of an imaginary quadratic field.

Disons qu’un groupe G est un amalgame s’il est isomorphe à une somme amalgamée G 1 *A G 2, avec G l ≠ A ≠ G 2 (pour la définition des sommes amalgamées, voir par exemple [1], §7 ou [3], §4.2). Dans ce qui suit, je montre que certains groupes, notamment SL3(Z), ne sont pas des amalgames; comme on le verra ci-dessous, cela revient à prouver que, lors...

The trace form of a central simple algebra of degree 4 Let k be a field of characteristic dierent from 2 containing a primitive 4-th root of unity. We show that the trace quadratic form of any central simple k-algebra A of degree 4 decomposes in the Witt group of k as the sum of a 2-fold Pfister form q2 and a 4-fold Pfister form q4 which are unique...

Memorial article with Knapp as editor and with the other nine people as authors

The theorem of Jordan which I want to discuss here dates from 1872. It is an elementary result on finite groups of permutations. I shall first present its translations in Number Theory and Topology.

Soit Kun corps local de caractéristique 0 et de caractéristique résiduelle p, et soit C la complétion d’une clôture algébrique de K. Soit T le module de Tate ([9], n° 2.4) associé à un groupe p-divisible F, défini sur l’anneau des entiers de K. Tate a montré ([9], § 4, cor. 2 au th. 3) que T⊗C possède une décomposition analogue à celle de Hodge pou...

Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field F_q come either from Serre's refinement of the Weil bound if the genus is small compared to q, or from Oesterle's optimization of the explicit formulae method if the genus is large...

We show that for all finite fields F_q, there exists a curve C over F_q of genus 3 such that the number of rational points on C is within 3 of the Serre-Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over F_q.

In this chapter, \(\mathfrak{g}\) denotes a complex semisimple Lie algebra, \(\mathfrak{h}\) a Cartan subalgebra of \(\mathfrak{g}\) and R the corresponding root system. We choose a base S = α1,…, αn of R, and we denote by R + the set of positive roots (with respect to S).

The Lie algebras considered in this chapter are finite-dimensional algebras over a field k. In Sees. 7 and 8 we assume that k has characteristic 0. The Lie bracket of x and y is denoted by [x, y], and the map y → [x, y] by ad x.

This chapter contains no proofs. All the Lie groups considered (except in Sec. 7) are complex groups.

Throughout this chapter, \(\mathfrak{g}\)denotes a complex semisimple Lie algebra, and \(\mathfrak{h}\) a Cartan subalgebra of \(\mathfrak{g}\) (cf. Chap. III).

In this chapter, the base field k is a field of characteristic zero.The Lie algebras and vector spaces considered have finite dimension over k.

In this chapter (apart from Sec. 6) the ground field is the field C of complex numbers. The Lie algebras considered are finite dimensional.

In this chapter (apart from Sec. 6) the ground field is the field C of complex numbers.

In this chapter (apart from Sec. 17) the ground field is the field R of real numbers. The vector spaces considered are all finite dimensional.

This chapter summarizes standard results in commutative algebra. For more details, see [Bour], Chap. II, III, IV.

In this section, A is a commutative noetherian ring; all A-modules are assumed to be finitely generated.

Let A be a ring (commutative, with a unit element).

Let A be a commutative ring (which is not assumed to be noetherian for the time being) and let x be an element of A. We denote by K(x), or sometimes K A (x), the following complex:

p) (cf. e.g. Hecke [6], p. 644--671). This implies the following properties, which have been conjectured by Ramanujan [16] and proved by Mordell [14]: (mn) = (m) (n); if (m; n) = 1 (4) (p n+1 ) = (p n ) (p) Gamma p 11 (p nGamma1 ); if p is prime: (5) These formulas allow us to compute (n) from the values of (p) for primes p. 1 2 CONGRUENCES INVOLVI...

The aim of this course is the study of rational and integral points on algebraic varieties, especially on curves or abelian varieties. Before the end of the last century only special cases had been considered. The first general results are found, around 1890, in the work of Hurwitz and Hilbert [HH] where they introduced the, nowadays natural, viewp...

Als Dokument aus den Anfängen der Garben- und Cohomologietheorie veröffentlichen wir hier Briefe von J-P. Serre an H. Cartan aus den Jahren 1952/53. In diesen Noten werden erstmals die gerade von Cartan entwickelten algebraischen Methoden auf klassische Fragen der Funktionentheorie mehrerer Veränderlicher angewendet, u.a. auf die Probleme von P. Co...

This chapter contains the construction and elementary study of the generalized Jacobians of an algebraic curve. We will follow closely the paper of Rosenlicht [64] on this subject, itself inspired by Weil’s Variétés abéliennes [89], where the case of the usual Jacobian is treated. We will make use, as they did, of the method of “generic points”. Th...

Let k be a finite field with q = p n elements and let V be an algebraic variety defined over k (or, as one also says, a k-variety). Suppose that V is defined by charts U i (isomorphic to affine k-varieties) and changes of coordinates u ij (with coefficients in k). If x = (x 1, …, xr ) is a point of an affine space, we write Fx, or x q , for the poi...

La fonction η de Dedekind est définie par où , Im( z )>0. C'est une forme modulaire parabolique de poids 1/2. Si r est un entier, la puissance r –ième de η s'écrit; où les coefficients p r ( n ) sone définis par l'identité .

The goal of this book is to present local class field theory from the cohomological point of view, following the method inaugurated by Hochschild and developed by Artin-Tate. This theory is about extensions—primarily abelian—of “local” (i.e., complete for a discrete valuation) fields with finite residue field. For example, such fields are obtained...

Let H denote the upper half plane of C, i.e. the set of complex numbers z whose imaginary part Im(z) is > 0.

First recall the general notion of a quadratic form (see Bourbaki, Alg., chap. IX, 3, n° 4).

1. Enonc6 du r6sultat Soit A un anneau de Dedekind, de corps des fractions K, et soit L une extension galoisienne finie de K, de groupe de Galois G. Soit A L la fermeture int6grale de A dans L; c'est un anneau de Dedekind. Dans tout ce qui suit, nous supposerons que les extensions r~siduelles de A L sont s~parables; cela signifie que, pour tout id6...

Sans rsum

Le mathematicien suisse Armand Borel est mort le 11 aout 2003, ` a Princeton, des suites d'un canceraevolution rapide. Iletait membre de notre compagnie depuis 1981. Peu de mathematiciensetrangers ont eu autant de relations avec la France. Il aeteeleve de Leray, il a pris part au seminaire Cartan et il a publied e nombreux articles (plus de vingt)...