Iwasawa theory of de Rham representations of a local field.
(Théorie d’Iwasawa des représentations de de Rham d’un corps local.)

*(French)*Zbl 0928.11045In her seminal paper on Iwasawa theory of local \(p\)-adic representations [Invent. Math., 115, No. 1, 81-149 (1994; Zbl 0838.11071)], B. Perrin-Riou considers generalized Coleman maps as machineries to produce generalized \(p\)-adic \(L\)-functions.

More specifically, the usual Coleman map associates to any norm-coherent system \(u\) of local units in \(\bigcup_n \mathbb{Q}_n(\mu_{p^n})\), a measure \(\lambda_u\) on \(\Gamma= \text{Gal} (\mathbb{Q}_p (\mu_{p^\infty})/ \mathbb{Q}_p)\) which is uniquely determined by a certain integration formula involving the logarithm of the well-known Coleman series and choosing \(u\) to be a suitable system of cyclotomic units, the pseudo-measure \((1-\gamma)^{-1}\lambda_u\) (where \(\gamma\) is a topological generator of \(\Gamma)\) gives the Kubota-Leopoldt \(p\)-adic zeta function.

More generally, if \(V\) is a \(p\)-adic crystalline representation of the absolute Galois group of a finite unramified extension \(K\) of \(\mathbb{Q}_p\), B. Perrin-Riou interpolates \(p\)-adically the Bloch-Kato exponential maps, in the sense that she constructs an exponential map \(\text{Exp}_{V,h}\) \((h\gg 0)\) which is uniquely determined by a certain integration formula involving the Bloch-Kato maps \[ \exp_{V(k)}:D_{dR} \bigl(V(k) \bigr)\to H^1\bigl( K,V(k)\bigr),\qquad k\geq 1-h. \] For \(V=\mathbb{Q}_p(1)\), one recovers the inverse of the Coleman map, and it is hoped that \(p\)-adic \(L\)-functions could be attached to crystalline representations \(V\) (i.e. representations with “good reduction”) by using \(\text{Exp}_{V,h}\) and compatible systems of elements in motivic cohomology.

One important conjecture proposed by B. Perrin-Riou in this process was a reciprocity law, \(\text{Rec}(V)\), a far reaching generalization of the usual reciprocity law of local class-field theory (corresponding to \(V=\mathbb{Q}_p(1))\), related to an eventual functional equation for \(p\)-adic \(L\)-functions. In the present paper, P. Colmez extends the construction of the exponential map to the base of “bad reduction”, i.e. to de Rham representations \(V\) over a finite extension \(K\) of \(\mathbb{Q}_p\), using distributions of finite order in \(p\)-adic spaces.

For \(h\geq 1\), he constructs a map \(\text{Exp}_V^{(p)}\) which is uniquely determined by a certain integration formula involving the Bloch-Kato \(\exp_{V(-k)}\), \(0\leq k\leq h-1\). By Fourier transforms, he shows that, in the setting of Perrin-Riou, the two maps \(\text{Exp}^{(h)}_V\) and \(\text{Exp}_{V,h}\) coincide up to a normalization. His construction is explicit enough to give also a logarithm map \(\text{Log}_H^{(h)}\) such that \(\text{Log}_V^{(h)} \circ\text{Exp}_V^{(h)}\) is the identity modulo the kernel of \(\text{Exp}^{(h)}_V\) (which is finite). He then shows a reciprocity law for the logarithm, i.e. a description of \(\text{Log}_V^{(h)}\) in terms of the dual Bloch-Kato exponential \(\exp^*_{V^*(1+k)}\), \(k\notin [0,h-1]\). This yields Perrin-Riou’s conjecture \(\text{Rec}(V)\) if \(V\) is crystalline and \(K\) unramified over \(\mathbb{Q}_p\).

NB: The conjecture \(\text{Rec}(V)\) has also been proved independently by Kato-Kurihara-Tsuj (resp. D. Benois) using syntomic cohomology (resp. Galois cohomology via \((\varphi,\Gamma)\)-modules).

More specifically, the usual Coleman map associates to any norm-coherent system \(u\) of local units in \(\bigcup_n \mathbb{Q}_n(\mu_{p^n})\), a measure \(\lambda_u\) on \(\Gamma= \text{Gal} (\mathbb{Q}_p (\mu_{p^\infty})/ \mathbb{Q}_p)\) which is uniquely determined by a certain integration formula involving the logarithm of the well-known Coleman series and choosing \(u\) to be a suitable system of cyclotomic units, the pseudo-measure \((1-\gamma)^{-1}\lambda_u\) (where \(\gamma\) is a topological generator of \(\Gamma)\) gives the Kubota-Leopoldt \(p\)-adic zeta function.

More generally, if \(V\) is a \(p\)-adic crystalline representation of the absolute Galois group of a finite unramified extension \(K\) of \(\mathbb{Q}_p\), B. Perrin-Riou interpolates \(p\)-adically the Bloch-Kato exponential maps, in the sense that she constructs an exponential map \(\text{Exp}_{V,h}\) \((h\gg 0)\) which is uniquely determined by a certain integration formula involving the Bloch-Kato maps \[ \exp_{V(k)}:D_{dR} \bigl(V(k) \bigr)\to H^1\bigl( K,V(k)\bigr),\qquad k\geq 1-h. \] For \(V=\mathbb{Q}_p(1)\), one recovers the inverse of the Coleman map, and it is hoped that \(p\)-adic \(L\)-functions could be attached to crystalline representations \(V\) (i.e. representations with “good reduction”) by using \(\text{Exp}_{V,h}\) and compatible systems of elements in motivic cohomology.

One important conjecture proposed by B. Perrin-Riou in this process was a reciprocity law, \(\text{Rec}(V)\), a far reaching generalization of the usual reciprocity law of local class-field theory (corresponding to \(V=\mathbb{Q}_p(1))\), related to an eventual functional equation for \(p\)-adic \(L\)-functions. In the present paper, P. Colmez extends the construction of the exponential map to the base of “bad reduction”, i.e. to de Rham representations \(V\) over a finite extension \(K\) of \(\mathbb{Q}_p\), using distributions of finite order in \(p\)-adic spaces.

For \(h\geq 1\), he constructs a map \(\text{Exp}_V^{(p)}\) which is uniquely determined by a certain integration formula involving the Bloch-Kato \(\exp_{V(-k)}\), \(0\leq k\leq h-1\). By Fourier transforms, he shows that, in the setting of Perrin-Riou, the two maps \(\text{Exp}^{(h)}_V\) and \(\text{Exp}_{V,h}\) coincide up to a normalization. His construction is explicit enough to give also a logarithm map \(\text{Log}_H^{(h)}\) such that \(\text{Log}_V^{(h)} \circ\text{Exp}_V^{(h)}\) is the identity modulo the kernel of \(\text{Exp}^{(h)}_V\) (which is finite). He then shows a reciprocity law for the logarithm, i.e. a description of \(\text{Log}_V^{(h)}\) in terms of the dual Bloch-Kato exponential \(\exp^*_{V^*(1+k)}\), \(k\notin [0,h-1]\). This yields Perrin-Riou’s conjecture \(\text{Rec}(V)\) if \(V\) is crystalline and \(K\) unramified over \(\mathbb{Q}_p\).

NB: The conjecture \(\text{Rec}(V)\) has also been proved independently by Kato-Kurihara-Tsuj (resp. D. Benois) using syntomic cohomology (resp. Galois cohomology via \((\varphi,\Gamma)\)-modules).

Reviewer: T.Nguyen Quang Do (Besançon)

##### MSC:

11R23 | Iwasawa theory |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

11S20 | Galois theory |

11S25 | Galois cohomology |