\(p\)-adic representations of a local field
Let \(K\) be a finite extension of the field \(\mathbb Q_p\) of \(p\)-adic numbers and let \(\mathcal G_K= \text{Gal}(\overline{K}/K)\). Let \(V\) be a finite-dimensional vector space over \(\mathbb Q_p\) that is a representation space for \(\mathcal G_K\). Associated to \(V\) is a vector space \(D(V)\) over a field \textbf{B}\(_K\) constructed by Fontaine. The space \(D(V)\), called the \((\phi,\Gamma)\)-module of \(V\), is much easier to work with than the original \(V\), and it is possible to deduce properties of \(V\) from those of \(D(V)\). The present article surveys some applications of this fact. \textit{R. Coleman} [Invent. Math. 53, 91-116 (1979; Zbl 0429.12010)] constructed a map from the inverse limit of units in the cyclotomic tower over \(K\) to measures on \(\mathbb Z_p^*\), when \(K/\mathbb Q_p\) is unramified. \textit{B. Perrin-Riou} [Invent. Math. 115, 81-149 (1994; Zbl 0838.11071)] generalized this construction to crystalline representations of \(\mathcal G_K\), which allowed her to give a conjectural definition of the \(p\)-adic \(L\)-function attached to a motive having good reduction at \(p\). The author describes extensions of these maps to more general representations and describes resulting explicit reciprocity laws. More details are contained in the author's papers [Ann. Math. (2) 148, No. 2, 485-571 (1998; Zbl 0928.11045)] and [with \textit{F. Cherbonnier}, J. Am. Math. Soc. 12, No. 1, 241-268 (1999; Zbl 0933.11056)].
scientific article; zbMATH DE number 1184272