Th\'eorie d'Iwasawa des repr\'esentations cristallines II

Abstract

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 $L \subset \cup_{n=1}^\infty K(\zeta_{p^n})$. The main ingredients are the $\delta_{\Zp}(V)$ conjecture about the integrality of Perrin-Riou's exponential, which we prove using the theory of $(\phi,\Gamma)$-modules, and Iwasawa-theoretic descent techniques used to show that $\delta_{\Zp}(V)$ implies $C_{\mathrm{EP}}(L/K,V)$.