Représentations cristallines irréductibles de ${\rm GL}_2(\mathbf{Q}_p)$

Sur quelques représentations potentiellement cristallines de GL_2(Q_p)

To each 2-dimensional irreducible p-adic representation of Gal(Qpbar/Qp) which becomes crystalline over an abelian extension of Q_p, we associate a Banach space B(V) endowed with a linear continuous unitary action of GL_2(Q_p). When V is moreover phi-semi-simple, we use the (phi,Gamma)-module and the Wach module associated to V to show that the representation B(V) is nonzero, topologically irreducible and admissible.

Sur la réduction des représentations cristallines de dimension 2 en poids moyens

We compute the reduction modulo $p$ of 2-dimensional crystalline representations whose Hodge-Tate weights are 0 and $k-1$ with $k \in \{p+2,...,2p-1\}$.

Limites de représentations cristallines

We provide an answer to two questions of Fontaine (in the unramified case). First, we show that a limit of crystalline representations, of bounded Hodge-Tate weights, is itself crystalline. Second, we show that every admissible filtered $ϕ$-module "comes from" a $(ϕ,Γ)$-module of finite q-height.

Théorie d'Iwasawa des représentations cristallines II

Let $K$ be a finite unramified extension of $\Qp$ and let $V$ be a crystalline representation of $\mathrm{Gal}(\Qpbar/K)$. In this article, we give a proof of the $C_{\mathrm{EP}}(L,V)$ conjecture for $L \subset \Qp^{\mathrm{ab}}$ as well as a proof of its equivariant version $C_{\mathrm{EP}}(L/K,V)$ for $L \subset \cup_{n=1}^\infty K(ζ_{p^n})$. The main ingredients are the $δ_{\Zp}(V)$ conjecture about the integrality of Perrin-Riou's exponential, which we prove using the theory of $(ϕ,Γ)$-modules, and Iwasawa-theoretic descent techniques used to show that $δ_{\Zp}(V)$ implies $C_{\mathrm{EP}…

Sur la densité des représentations cristallines du groupe de Galois absolu de Q_p

Let X_d be the p-adic analytic space classifying the d-dimensional (semisimple) p-adic Galois representations of the absolute Galois group of Q_p. We show that the crystalline representations are Zarski-dense in many irreducible components of X_d, including the components made of residually irreducible representations. This extends to any dimension d previous results of Colmez and Kisin for d = 2. For this we construct an analogue of the infinite fern of Gouvêa-Mazur in this context, based on a study of analytic families of trianguline (phi,Gamma)-modules over the Robba ring. We show in partic…

局部伴随点猜想

The Local Companion Points Conjecture

该研究聚焦局部伴随点猜想，在仅假设若干正则性条件的前提下，证明了一般情形下该猜想成立，推进了此前仅在晶态正则伽罗瓦表示假设下的相关证明结果。