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Jean-Pierre Serre

Publications and source records attributed to Jean-Pierre Serre.

18 recordsLinked to original sources
arXiv2025

Zéros de caractères

Let G be a compact real Lie group, and let f be an irreducible complex character of G, of degree > 1. We show that there exists an element g of G, of finite order, such that f(g)=0. We also give an unpublished result of Deligne, about replacing " of finite order " by " of order a power of a prime number ".

math.GR↗
arXiv2022

Groupes de Coxeter finis: involutions et cubes

We review the properties of the finite Coxeter groups which are most useful for applications to cohomological invariants, namely their classes of involutions and their "cubes" (abelian subgroups generated by reflections).

math.GR↗
arXiv2020

Index-stable compact $p$-adic analytic groups

A profinite group is index-stable if any two isomorphic open subgroups have the same index. Let $p$ be a prime, and let $G$ be a compact $p$-adic analytic group with associated $\mathbb{Q}_p$-Lie algebra $\mathcal{L}(G)$. We prove that $G$ is index-stable whenever $\mathcal{L}(G)$ is semisimple. In particular, a just-infinite compact $p$-adic analytic group is index-stable if and only if it is not virtually abelian. Within the category of compact $p$-adic analytic groups, this gives a positive answer to a question of C. Reid. In the Appendix, J-P. Serre proves that $G$ is index-stable if and only if the determinant of any automorphism of $\mathcal{L}(G)$ has $p$-adic norm 1.

math.GR↗
arXiv2019

Distribution asymptotique des valeurs propres des endomorphismes de Frobenius [d'après Abel, Chebyshev, Robinson, ...]

We consider unitary polynomials $P \in Z[X]$ whose roots $(x_i)$ belong to a given compact $K$ of $C$. To such a polynomial we associate the measure $μ_P$ on $K$ which is the mean value of the Dirac measures $δ_{x_i}$. What are the limits of the measures $μ_P$ when $P$ varies ? In particular, what are their supports? We give partial answers to such questions, especially when $K$ is contained in $R$.

math.NT↗
arXiv2018

On the mod p reduction of orthogonal representations

We show that the reduction mod p of an orthogonal linear representation is orthogonal, and we generalize this fact to representations of algebras with involution.The proofs make an essential use of the notion of " middle lattices ".

math.GR↗
arXiv2014

Bases normales autoduales et groupes unitaires en caractéristique 2

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.

math.AG↗
arXiv2012

Formes modulaires modulo 2 : l'ordre de nilpotence des opérateurs de Hecke

The nilpotence order of the mod 2 Hecke operators. Let $Δ=\sum_{m=0}^\infty q^{(2m+1)^2} \in F_2[[q]]$ be the reduction mod 2 of the $Δ$ series. A modular form f modulo 2 of level 1 is a polynomial in $Δ$. If p is an odd prime, then the Hecke operator Tp transforms f in a modular form Tp(f) which is a polynomial in $Δ$ whose degree is smaller than the degree of f, so that Tp is nilpotent. The order of nilpotence of f is defined as the smallest integer g = g(f) such that, for every family of g odd primes p1, p2, ..., pg, the relation Tp1Tp2... Tpg (f) = 0 holds. We show how one can compute explicitly g(f); if f is a polynomial of degree d in $Δ$, one finds that g(f) << d^(1/2).

math.NT↗
arXiv2010

Bounds for the orders of the finite subgroups of G(k)

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.

math.AG↗
arXiv2001

Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields

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. This paper presents three methods for improving these bounds. The arguments used are the indecomposability of the theta divisor of a curve, Galois descent, and Honda-Tate theory. Examples of improvements on the bounds include lowering them for a wide range of small genus when q=2^3, 2^5, 2^{13}, 3^3, 3^5, 5^3, 5^7, and when q=2^{2s}, s>1. For large genera, isolated improvements are obtained for q=3,8,9.

math.AG↗