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

Let $K$ be a finite unramified extension of $\text{Qp}$ and let $V$ be a crystalline representation of $\mathrm{G}\mathrm{a}\mathrm{l}(\text{Qpbar}\mathrm{/}K)$. In this article, we give a proof of the $C_{\mathrm{E}\mathrm{P}}(L,V)$ conjecture for $L \subset \Qp^{$\mathrm{a}\mathrm{b}$}$aswellasaproofofitsequivariantversion$C_{$\mathrm{E}\mathrm{P}$}(L/K,V)$for$L \subset \cup_{n=1}^\infty K(\zeta_{p^n})$. The main ingredients are the ${\delta }_{\text{Zp}}(V)$ conjecture about the integrality of Perrin-Riou’s exponential, which we prove using the theory of $(\varphi ,\mathrm{\Gamma })$-modules, and Iwasawa-theoretic descent techniques used to show that ${\delta }_{\text{Zp}}(V)$ implies $C_{\mathrm{E}\mathrm{P}}(L\mathrm{/}K,V)$.