Curves and vectorial fibers in \(p\)-adic Hodge theory. With a preface by Pierre Colmez. (Courbes et fibrés vectoriels en théorie de Hodge \(p\)-adique.) (French) Zbl 1470.14001

Fargues, Laurent; Fontaine, Jean-Marc

Astérisque 406. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-896-1/pbk). xiii, 382 p. (2018).

This is a voluminous and comprehensive book on the famous ‘Curve of \(p\)-adic Hodge Theory’ which, since its invention about 10 years ago by Laurent Fargues and Jean-Marc Fontaine, has become perhaps the arguably most exciting structure found in \(p\)-adic Hodge Theory. It allows the unification of various by-now-classical theorems in \(p\)-adic Hodge Theory – most prominently, in catchwords: ‘weak admissible implies admissible’ and ‘de Rham implies potentially semistable’ – and in fact sheds an entirely new light on the category of \(p\)-adic Galois representations, by interpreting them equivalently as (semistable of slope \(0\)) fibre bundles on ‘The Curve’.
Let \(k\) be a perfect field of characteristic \(p\), put \(K_0=W(k)[\frac{1}{p}]\), let \(K/K_0\) be a finite totally ramified extension field with algebraic closure \(\overline{K}\), and put \(G_K=\mathrm{Gal}(\overline{K}/K)\). Write \(C\) for the completion of \(\overline{K}\) and put\[C^*=\{x=(x^{(n)})_{n\in{\mathbb N}};\, x^{(n)}\in C, (x^{(n+1)})^p=x^{(n)}\mbox{ for all }n\}.\]For \(x=(x^{(n)})_n\in C^*\) write \(x^{\sharp}=x^{(0)}\). Conversely, for \(x\in C\) let \(x^*\in C^*\) be an arbitrary element with \((x^*)^{\sharp}=x\). Let \(\epsilon=(1,\zeta_p,\ldots)\in C^*\) where \(\zeta_p\) is a primitive \(p\)-th root of unity.
Now \(C^*\) can in a natural way be endowed with the structure of an algebraically closed field of characteristic \(p\), complete for the valuation \(v_{C^*}\) given by \(v_{C^*}(x)=v_p(x^{{\sharp}})\), and having the same residue field as \(C\). Put \(\mathbf{A}_{\mathrm{inf}}=W({\mathcal O}_{C^*})\). Each element in \(\mathbf{A}_{\mathrm{inf}}\) can in a unique way be written as \[\sum_{k\ge0}p^k[x_k]\quad\quad\mbox{ with }x_k\in {\mathcal O}_{C^*}.\]By functoriality, \(\mathbf{A}_{\mathrm{inf}}\) is endowed with a Frobenius endomorphism \(\varphi\) given by \[\varphi(\sum_{k\ge0}p^k[x_k])=\sum_{k\ge0}p^k[x_k^p]\] and an action by \(G_K\) commuting with \(\varphi\). There is a surjective ring morphism\[\theta:\mathbf{A}_{\mathrm{inf}}\longrightarrow{\mathcal O}_C,\quad\quad \sum_{k\ge0}p^k[x_k]\mapsto \sum_{k\ge0}p^kx_k^{{\sharp}}\]whose kernel is generated by \(p-[p]^*\). Next, \(\mathbf{B}_{\mathrm{dR}}^+={\lim}(\mathbf{A}_{\mathrm{inf}}[\frac{1}{p}]/(p-[p]^*)^k)\) is a discrete valuation ring with residue field \(C\), containing the \(p\)-adic completion \(\mathbf{A}_{\mathrm{cris}}\) of \(\mathbf{A}_{\mathrm{inf}}[\frac{(p-[p]^*)^k}{k!}\,;\,k\ge1]\). The element \[t=\log[\epsilon]=-\sum_{k\ge1}\frac{(1-[\epsilon])^k}{k}\]of \(\mathbf{A}_{\mathrm{cris}}\) is a uniformizer in \(\mathbf{B}_{\mathrm{dR}}^+\). Put \(\mathbf{B}_{\mathrm{cris}}=\mathbf{A}_{\mathrm{cris}}[\frac{1}{t}]\) and \(\mathbf{B}_{\mathrm{dR}}=\mathbf{B}_{\mathrm{dR}}^+[\frac{1}{t}]\). To \(\mathbf{B}_{\mathrm{cris}}\) the action by Frobenius \(\varphi\) extends by continuity and the formula \(\varphi(t)=pt\). The action of \(G_K\) naturally extends to all these rings (with \(\sigma(t)=\chi(t)t\) for \(\sigma\in G_K\), where \(\chi\) denotes the cyclotomic character). Putting\[\mathbf{B}_e=\mathbf{B}_{\mathrm{cris}}^{\varphi=1}\]we have an exact sequence\[0\longrightarrow{\mathbb Q}_p\longrightarrow\mathbf{B}_e\longrightarrow\mathbf{B}_{\mathrm{dR}}/\mathbf{B}_{\mathrm{dR}}^+\longrightarrow0.\]
All these constructions and their eminent meaning for the theory of \(p\)-adic representations of \(G_K\) had been established more than thirty years ago by Fontaine.
The starting point for the invention of ‘The Curve’ now seems to have been discussions – among Fargues, Fontaine, Colmez – on the structure of \(\mathbf{B}_e\), and related discussions on the notion of ‘B-pairs’, as introduced by Laurent Berger, which generalizes that of \(p\)-adic \(G_K\)-representations. The observation was made that the category of \(B\)-pairs shares features of that of fibres bundles on a projective curve (over a field), as it allows similar Harder-Narasimhan filtrations. Boldly speculating, this invited the search for some geometric object, resembling a projective curve (over a field) but instead carrying B-pairs. And right away, a candidate for an ‘open affine piece’ of that curve-to-be-found was identified: the spectrum of \(\mathbf{B}_e\), as \(\mathbf{B}_e\) was recognized to be a principal ideal domain. Yet, how to ‘compactify’ this ‘curve’ ? Faithful to the concept of \(B\)-pairs, the completion of the structure sheaf (of the ‘compactified curve’ to be found) at the missing point complementary to \(\mathrm{Spec}(\mathbf{B}_e)\) was identified as being, out of necessity, \(\mathbf{B}_{\mathrm{dR}}^+\). And indeed, these data can in a purely formal way be ‘geometrically glued’ along \[{\mathbb Q}_p=\mathbf{B}_e\bigcap \mathbf{B}_{\mathrm{dR}}^+.\]But then, the definitive observation to be made is that this geometric object can indeed by realized as a true projective curve:\[P=\bigoplus_{n}(\mathbf{B}_{\mathrm{dR}}^+)^{\varphi=p^n}\]is a graded algebra, hence \[X=\mathrm{Proj}(P)\]is a ‘projective variety’ (in fact, a ‘projective curve’ as \(\mathbf{B}_e\) is a principal ideal domain) ! Not surprinsingly, it comes along with bizarre features from the point of view of classical algebraic geometry, e.g. its field of constants is \({\mathbb Q}_p\), whereas its residue field at the point \(t=0\) (whose complement is \(\mathrm{Spec}(\mathbf{B}_e)\)) is \(C\).
The curve \(X\) having been found, the magnificient prospect and project of rewriting the entire theory of \(p\)-adic \(G_K\)-representations and of \(p\)-adic Hodge theory in terms of fibre bundles on \(X\) was opened.
And in fact, this apparently is a very fruitful line of current research, in particular as new ideas for interpreting and finding local Langlands correspondences of various sorts in terms of the curve \(X\) keep springing up.
The book begins with a highly illumination introduction (of 47 pages) by Pierre Colmez. It gives a concise overview into the ramified material. In addition, reconstructing its historical detection it shares interesting and amusing insights into the actual genesis of ’the curve’ (as conceived by Fargues and Fontaine) – by quoting long email passages, by reporting on the restaurants in which the decisive discussions took place, etc. …
The book gathers in a comprehensive and self contained way all the ingredients of the theory of ’The Curve’ in its present state. Besides the very construction of the curve and the more advanced discussion of fibre bundles on it, it also includes many foundational topics, often reinterpreting known constructions from a new and original point of view.
We mention a few topics, without aiming to be exhaustive:
– The rereading of various rings, well known in \(p\)-adic Hodge theory for quite some time, as ‘rings of holomorphis functions in the variable \(p\)’, just as is e.g. suggested by formula. This is the subject matter of the first chapter, in which an intriguing new way of introducing Newton polygons for these rings is explained.
– Finite dimensional Banach spaces (Banach-Colmez spaces)
– Robba rings
– formal \({\mathbb Q}_p\)-vector spaces and \(p\)-divisible groups
– The analytic construction of the curve; this is an alternative description, in terms of adic spaces, as opposed to the algebraic one indicated above.
– Equivariant fibre bundles, Harder Narasimhan filtrations and \(G_K\)-representations
– Lubin-Tate and Drinfeld towers
– weakly admissible implies admissble
– de Rham implies potentially semistable
— GAGA according to Kedlaya-Liu
– Berger’s theory of B-pairs
The chapter titles are as follows:
1. Fonctions holomorphes de la variable \(p\) et anneaux de périodes
2. Zéros des fonctions holomorphes: le cas \(F\) algébriquement clos
3. Zéros des fonctions holomorphes: le cas \(F\) parfait quelconque
4. \({\mathbb Q}_p\)-espaces vectoriels formels et périodes des groupes \(p\)-divisibles
5. Courbes
6. La courbe fondamental lorsque \(F\) est algébriquement clos
7. La courbe fondamental pour \(F\) parfait quelconque
8. Classification des fibrés vectoriels: le cas \(F\) algébriquement clos
9. Classification des fibrés vectoriels: le cas parfait
10. Faiblement admissible implique admissible et la théorème de la monodromie \(p\)-adique
11. \(\varphi\)-module et fibrés

Reviewer: Elmar Große-Klönne (Berlin)

MSC:

14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
14G20	Local ground fields in algebraic geometry
14F30	\(p\)-adic cohomology, crystalline cohomology
11S31	Class field theory; \(p\)-adic formal groups
11S37	Langlands-Weil conjectures, nonabelian class field theory

Keywords:

\(p\)-adic Hodge theory; period rings; Galois representation; crystalline cohomology; B-pair