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We reinterpret a conjecture of Breuil on the locally analytic $mathrm{Ext}^1$ in a functorial way using $(varphi,Gamma)$-modules (possibly with $t$-torsion) over the Robba ring, making it more accurate. Then we prove several special or partial cases of this improved conjecture, notably for ${rm GL}_3(mathbb{Q}_p)$.
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On the rank of Jacobians over function fields.} Let $f:mathcal{X}to C$ be a projective surface fibered over a curve and defined over a number field $k$. We give an interpretation of the rank of the Mordell-Weil group over $k(C)$ of the jacobian of the generic fibre (modulo the constant part) in terms of average of the traces of Frobenius on the fibers of $f$. The results also give a reinterpretation of the Tate conjecture for the surface $mathcal{X}$ and generalizes results of Nagao, Rosen-Silverman and Wazir.
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).
Consider the ring of holomorphic function germs in $C^n$ and denote by $M$ the maximal ideal of this ring. For any a holomorphic function germ $f$ with an isolated critical point, the finite determinacy theorem (Mather-Tougeron) asserts that there exists some $k$, such that $f+g$ can be brought back to $f$, via a holomorphic change of variables, for any $g in M^k$. In this paper, a generalisation of this theorem for functions defined in a neighbourhood of a Stein compact subset and for an arbitrary ideal is given.
In this short note, I explain how the non-degeneracy condition of the KAM can be bypassed. The first version of the note has been submitted for publication back in 2010 and this version in 2012.
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 modular jacobian $J_{0}(N)$, and we derive non-trivial remarks about the conjecture.
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