We introduce an algebra given by quadratic relations in an algebra of polynomials in an infinite number of variables. Using this algebra, we prove some explicit formulas for the Sturm sequence of a polynomial.
Using a patching module constructed in recent work of Caraiani, Emerton, Gee, Geraghty, Pa{v{s}}k{=u}nas and Shin we construct some kind of analogue of an eigenvariety. We can show that this patched eigenvariety agrees with a union of irreducible com
ponents of a space of trianguline Galois representations. Building on this we discuss the relation with the modularity conjectures for the crystalline case, a conjecture of Breuil on the locally analytic socle of representations occurring in completed cohomology and with a conjecture of Bellaiche and Chenevier on the complete local ring at certain points of eigenvarieties.
Let F be a finite extension of Qp, O_F its ring of integers and E a finite extension of Fp. The natural action of the unit group O_F* on O_F extends in a continuous action on the Iwasawa algebra E[[O_F]]. In this work, we show that non zero ideals of
E[[O_F]] which are stable under O_F* are open. As a consequence, we deduce the fidelity of the action of E[[U]], with U the subgroup of upper unipotent matrices in GL2(O_F) on an irreducible admissible smooth E-representation of GL2(F). ----- Soit F une extension finie de Qp, danneau des entiers O_F et E une extension finie de Fp. Laction naturelle du groupes des unites O_F* sur O_F se prolonge alors en une action continue sur lalg`ebre dIwasawa E[[O_F]]. Dans ce travail, on demontre que les ideaux non nuls de E[[O_F]] stables par O_F* sont ouverts. En particulier, on en deduit la fidelite de laction de lalg`ebre dIwasawa des matrices unipotentes superieures de GL2(O_F) sur une representation lisse irreductible admissible de GL2(F).
The aim of this paper is to study a conjecture predicting a lower bound on the canonical height on abelian varieties, formulated by S. Lang and generalized by J. H. Silverman. We give here an asymptotic result on the height of Heegner points on the m
odular jacobian $J_{0}(N)$, and we derive non-trivial remarks about the conjecture.
The legacy of Jordans canonical form on Poincares algebraic practices. This paper proposes a transversal overview on Henri Poincares early works (1878-1885). Our investigations start with a case study of a short note published by Poincare on 1884 : S
ur les nombres complexes. In the perspective of todays mathematical disciplines - especially linear algebra -, this note seems completely isolated in Poincares works. This short paper actually exemplifies that the categories used today for describing some collective organizations of knowledge fail to grasp both the collective dimensions and individual specificity of Poincares work. It also highlights the crucial and transversal role played in Poincares works by a specific algebraic practice of classification of linear groups by reducing the analytical representation of linear substitution to their Jordans canonical forms. We then analyze in detail this algebraic practice as well as the roles it plays in Poincares works. We first provide a micro-historical analysis of Poincares appropriation of Jordans approach to linear groups through the prism of the legacy of Hermites works on algebraic forms between 1879 and 1881. This mixed legacy illuminates the interrelations between all the papers published by Poincare between 1878 and 1885 ; especially between some researches on algebraic forms and the development of the theory of Fuchsian functions. Moreover, our investigation sheds new light on how the notion of group came to play a key role in Poincares approach. The present paper also offers a historical account of the statement by Jordan of his canonical form theorem. Further, we analyze how Poincare transformed this theorem by appealing to Hermites
In the Nineties, Michel Herman conjectured the existence of a positive measure set of invariant tori at an elliptic diophatine critical point of a hamiltonian function. I construct a formalism for the UV-cutoff and prove a generalised KAM theorem which solves positively the Herman conjecture.