On trivial zeros of Perrin-Riou's LL-functions

In the previous paper we generalized Greenberg's construction of the \Cal L-invariant to semistable representations. Here we prove that this construction is compatible with Perrin-Riou's theory of $p$-adic $L$-functionsComment: 37 page

Th\'eorie d'Iwasawa des repr\'esentations 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 $L \subset \cup_{n=1}^\infty K(\zeta_{p^n})$. 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 $C_{\mathrm{EP}}(L/K,V)$.Comment: 58 page