##
**Berkeley lectures on \(p\)-adic geometry.**
*(English)*
Zbl 1475.14002

Annals of Mathematics Studies 207. Princeton, NJ: Princeton University Press (ISBN 978-0-691-20209-9/hbk; 978-0-691-20208-2/pbk; 978-0-691-20215-0/ebook). x, 250 p. (2020).

The following review consists partly of quotations (and points even slightly altered) from the book under discussion, partly of extracts formulated by the reviewer; the distinction is not made explicit.

From the Foreword:

This is a revised version of the lecture notes for the course on \(p\)-adic geometry given by Peter Scholze in the fall of 2014 at UC Berkeley.

In the first half of the course (Lectures 1–10) we construct the category of diamonds, which are quotients of perfectoid spaces by so-called pro-étale equivalence relations. In brief, diamonds are to perfectoid spaces what algebraic spaces are to schemes.

In the second half of the course (Lectures 11–25), we define spaces of mixed-characteristic local shtukas, which live in the category of diamonds. This requires making sense of products like \(\mathrm{Spa}{\mathbb Q}_p \times S\), where \(S\) is an adic space over \({\mathbb F}_p\).

The proper foundations on diamonds can only be found in [P. Scholze, “Étale cohomology of diamonds”, arXiv:1709.07343]; here, we only survey the main ideas in the same way as in the original lectures. In this way, we hope that this manuscript can serve as an informal introduction to these ideas.

From the Introduction:

1.1 Motivation: Drinfeld, L. Lafforgue and V. Lafforgue

The starting point is Drinfeld’s work on the global Langlands correspondence over function fields. Fix \(X/{\mathbb F}_p\) a smooth projective curve, with function field \(K\). The Langlands correspondence for \(\mathrm{GL}_n /K\) is a bijection \(\pi\mapsto\sigma(\pi)\) between the following two sets (considered up to isomorphism):

\({\bullet}\) Cuspidal automorphic representations of \(\mathrm{GL}_n(\mathbf{A}_K)\), where \(\mathbf{A}_K\) is the ring of adeles of \(K\), and

\({\bullet}\) Irreducible representations \(\mathrm{Gal}(\overline{K}/K)\to\mathrm{GL}_n(\overline{\mathbb Q}_{\ell})\).

Whereas the global Langlands correspondence is largely open in the case of number fields \(K\), it is a theorem for function fields, due to Drinfeld (\(n = 2\)) and L. Lafforgue (general \(n\)). The key innovation in this case is Drinfeld’s notion of an \(X\)-shtuka.

A family of stacks \[f:\mathrm{Sht}_{\mathrm{GL}_n,\{\mu_1,\ldots,\mu_m\},N}\longrightarrow (X\backslash N)^m\](depending on a divisor \(N\) on \(X\) and on additional data \(\mu_1,\ldots,\mu_m\)) is considered, and then the cohomology \(R^d(f_N)_!\overline{\mathbb Q}_{\ell}\), where \(d\) is the relative dimension of \(f\).

The analysis of \(R^d(f_N)_!\overline{\mathbb Q}_{\ell}\) runs along the following lines, explained by means of a \(\overline{\mathbb Q}_{\ell}\)-sheaf \({\mathbb L}\) on \(X^m\) which becomes lisse when restricted to \(U^m\) for some dense open subset \(U\) of \(X\). We can think of \({\mathbb L}\) as a representation of the étale fundamental group \(\pi_1(U^m)\) on a \(\overline{\mathbb Q}_{\ell}\)-vector space. Ultimately we want to relate this to \(\pi_1(U)\), because this is a quotient of \(\mathrm{Gal}(\overline{K}/K)\).

For \(i =1,\ldots, m\) we have a partial Frobenius map \(F_i: X^m\to X^m\), which is \(\mathrm{Frob}_X\) on the \(i\)-th factor, and the identity on each other factor. For an étale morphism \(V\to X^m\), let us say that a system of partial Frobenii on \(V\) is a commuting collection of isomorphisms \(F_i^*V \cong V\) over \(X^m\) (and whose product is the relative Frobenius of \(V\to X^m\)). Finite étale covers of \(U^m\) equipped with partial Frobenii form a Galois category, and thus they are classified by continuous actions of a profinite group \(\pi_1(U^m/\mathrm{ partial Frob})\) on a finite set.

Lemma: (Drinfeld) The natural map\[\pi_1(U^m/\mathrm{partial Frob})\longrightarrow \longrightarrow\pi_1(U)\times\cdots\times\pi_1(U)\quad\quad(m\,\, \mathrm{copies})\] is an isomorphism.

Now \(R^d(f_N)_!\overline{\mathbb Q}_{\ell}\) is not exactly like an \({\mathbb L}\) as above, but only approximately so, yet the Lemma can be suitably adapted so that one gets a big representation of \[\mathrm{GL}_n(\mathrm{A}_K)\times\mathrm{ Gal}(\overline{K}/K)\times\cdots\times\mathrm{Gal}(\overline{K}/K)\] on \(\mathrm{lim}_{\to N}R^d(f_N)_!\overline{\mathbb Q}_{\ell}\). One then expects the latter to decompose under the said action to yield the desired Langlands correspondence.

1.2 The possibility of Shtukas in mixed characteristic

It would be desirable to have moduli spaces of shtukas over number fields, but the first immediate problem is that such a space of shtukas would live over something like \(\mathrm{Spec}({\mathbb Z})\times\mathrm{Spec}({\mathbb Z})\), where the product is over \({\mathbb F}_1\) somehow.

In this course we will give a rigorous definition of \(\mathrm{Spec}({\mathbb Z}_p)\times\mathrm{Spec}({\mathbb Z}_p)\), the completion of \(\mathrm{Spec}({\mathbb Z})\times\mathrm{Spec}({\mathbb Z})\) at \((p, p)\). It lives in the world of nonarchimedean analytic geometry, so it should properly be called \(\mathrm{Spa}({\mathbb Z}_p)\times\mathrm{Spa}({\mathbb Z}_p)\). (The notation \(\mathrm{Spa}\) refers to the adic spectrum.)

Whatever it is, it should contain \(\mathrm{Spa}({\mathbb Q}_p)\times\mathrm{Spa}({\mathbb Q}_p)\) as a dense open subset.

We present a model for the product \(\mathrm{Spa}({\mathbb Q}_p)\times\mathrm{Spa}({\mathbb Q}_p)\), specified by picking one of the factors: one copy of \({\mathbb Q}_p\) appears as the field of scalars, but the other copy appears geometrically. Consider the open unit disc \(D_{{\mathbb Q}_p} = \{x\,;\, |x| < 1\}\) as a subgroup of (the adic version of) \({\mathbb G}_m\), via \(x\mapsto 1 + x\). Then \(D_{{\mathbb Q}_p}\) is in fact a \({\mathbb Z}_p\)-module with multiplication by \(p\) given by \(x\mapsto (1 + x)^p-1\), and we consider\[\widetilde{D}_{{\mathbb Q}_p}=\projlim_{x\mapsto (1 + x)^p-1} D_{{\mathbb Q}_p}.\] After base extension to a perfectoid field, this is a perfectoid space, which carries the structure of a \({\mathbb Q}_p\)-vector space. Thus its punctured version \(\widetilde{D}^*_{{\mathbb Q}_p}=\widetilde{D}_{{\mathbb Q}_p}-\{0\}\) has an action of \({\mathbb Q}_p^{\times}\), and we consider the quotient \[\mathrm{Spa}({\mathbb Q}_p)\times \mathrm{Spa}({\mathbb Q}_p):=\widetilde{D}^*_{{\mathbb Q}_p}/ {\mathbb Z}_p^{\times}.\]Note that this quotient does not exist in the category of adic spaces (the quotient being taken in a formal sense). On \(\widetilde{D}^*_{{\mathbb Q}_p}/ {\mathbb Z}_p^{\times}\), we have an operator \(\varphi\), corresponding to \(p\in{\mathbb Q}_p^{\times}\). Let \[X =(\widetilde{D}^*_{{\mathbb Q}_p}/ {\mathbb Z}_p^{\times})/\varphi^{\mathbb Z}=\widetilde{D}^*_{{\mathbb Q}_p}/ {\mathbb Q}_p^{\times}.\]One can define a finite étale cover of \(X\) simply as a \({\mathbb Q}_p^{\times}\)-equivariant finite étale cover of \(\widetilde{D}^*_{{\mathbb Q}_p}\). There is a corresponding profinite group \(\pi_1(X)\) which classifies such covers. We have the following theorem, which is a local version of Drinfeld’s lemma in the case \(m = 2\).

Theorem: \[\pi_1(X) \cong \mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\times\mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p).\]

This theorem suggests that if one could define a moduli space of \({\mathbb Q}_p\)-shtukas which is fibered over products such as \(\mathrm{Spa}({\mathbb Q}_p)\times\mathrm{Spa}({\mathbb Q}_p)\), then its cohomology would produce representations of \(\mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\times \mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\).

What would a \({\mathbb Q}_p\)-shtuka over \(S\) look like? It should be a vector bundle \({\mathcal E}\) over \(\mathrm{Spa} {\mathbb Q}_p\times S\), together with a meromorphic isomorphism \(\mathrm{Frob}^*_S{\mathcal E}- - \to{\mathcal E}\). In order for this to make any sense, we would need to give a geometric meaning to \(\mathrm{Spa} {\mathbb Q}_p\times S\) (and to \(\mathrm{Frob}_S\)) just as we gave one to \(\mathrm{Spa}({\mathbb Q}_p)\times \mathrm{Spa}({\mathbb Q}_p)\).

We will give a meaning to \(\mathrm{Spa} {\mathbb Q}_p\times S\) whenever \(S\) is a perfectoid space of characteristic \(p\), which lets us define moduli spaces of \(p\)-adic shtukas. In general, these are not representable by perfectoid spaces or classical rigid spaces, but instead they are diamonds: That is, quotients of perfectoid spaces by pro-étale equivalence relations. A large part of this course is about the definition of perfectoid spaces and diamonds.

Table of contents:

Lecture 1: Introduction

Lecture 2: Adic spaces

Lecture 3: Adic spaces II

Lecture 4: Examples of adic spaces

Lecture 5: Complements on adic spaces

Lecture 6: Perfectoid rings

Lecture 7: Perfectoid spaces

Lecture 8: Diamonds

Lecture 9: Diamonds II

Lecture 10: Diamonds associated with adic spaces

Lecture 11: Mixed characteristic shtukas

Lecture 12: Shtukas with one leg

Lecture 13: Shtukas with one leg II

Lecture 14: Shtukas with one leg III

Lecture 15: Examples of diamonds

Lecture 16: Drinfel’s lemma for diamonds

Lecture 17: The v-topolgy

Lecture 18: v-sheaves associated with perfect and formal schemes

Lecture 19: The \(B_{dR}^+\)-affine Grassmannian

Lecture 20: Families of affine Grassmannians

Lecture 21: Affine flag varieties

Lecture 22: Vector bundles and \(G\)-torsors

Lecture 23: Moduli spaces of shtukas

Lecture 24: Local Shimura varieties

Lecture 25: Integral models of local Shimura varieties

From the Foreword:

This is a revised version of the lecture notes for the course on \(p\)-adic geometry given by Peter Scholze in the fall of 2014 at UC Berkeley.

In the first half of the course (Lectures 1–10) we construct the category of diamonds, which are quotients of perfectoid spaces by so-called pro-étale equivalence relations. In brief, diamonds are to perfectoid spaces what algebraic spaces are to schemes.

In the second half of the course (Lectures 11–25), we define spaces of mixed-characteristic local shtukas, which live in the category of diamonds. This requires making sense of products like \(\mathrm{Spa}{\mathbb Q}_p \times S\), where \(S\) is an adic space over \({\mathbb F}_p\).

The proper foundations on diamonds can only be found in [P. Scholze, “Étale cohomology of diamonds”, arXiv:1709.07343]; here, we only survey the main ideas in the same way as in the original lectures. In this way, we hope that this manuscript can serve as an informal introduction to these ideas.

From the Introduction:

1.1 Motivation: Drinfeld, L. Lafforgue and V. Lafforgue

The starting point is Drinfeld’s work on the global Langlands correspondence over function fields. Fix \(X/{\mathbb F}_p\) a smooth projective curve, with function field \(K\). The Langlands correspondence for \(\mathrm{GL}_n /K\) is a bijection \(\pi\mapsto\sigma(\pi)\) between the following two sets (considered up to isomorphism):

\({\bullet}\) Cuspidal automorphic representations of \(\mathrm{GL}_n(\mathbf{A}_K)\), where \(\mathbf{A}_K\) is the ring of adeles of \(K\), and

\({\bullet}\) Irreducible representations \(\mathrm{Gal}(\overline{K}/K)\to\mathrm{GL}_n(\overline{\mathbb Q}_{\ell})\).

Whereas the global Langlands correspondence is largely open in the case of number fields \(K\), it is a theorem for function fields, due to Drinfeld (\(n = 2\)) and L. Lafforgue (general \(n\)). The key innovation in this case is Drinfeld’s notion of an \(X\)-shtuka.

A family of stacks \[f:\mathrm{Sht}_{\mathrm{GL}_n,\{\mu_1,\ldots,\mu_m\},N}\longrightarrow (X\backslash N)^m\](depending on a divisor \(N\) on \(X\) and on additional data \(\mu_1,\ldots,\mu_m\)) is considered, and then the cohomology \(R^d(f_N)_!\overline{\mathbb Q}_{\ell}\), where \(d\) is the relative dimension of \(f\).

The analysis of \(R^d(f_N)_!\overline{\mathbb Q}_{\ell}\) runs along the following lines, explained by means of a \(\overline{\mathbb Q}_{\ell}\)-sheaf \({\mathbb L}\) on \(X^m\) which becomes lisse when restricted to \(U^m\) for some dense open subset \(U\) of \(X\). We can think of \({\mathbb L}\) as a representation of the étale fundamental group \(\pi_1(U^m)\) on a \(\overline{\mathbb Q}_{\ell}\)-vector space. Ultimately we want to relate this to \(\pi_1(U)\), because this is a quotient of \(\mathrm{Gal}(\overline{K}/K)\).

For \(i =1,\ldots, m\) we have a partial Frobenius map \(F_i: X^m\to X^m\), which is \(\mathrm{Frob}_X\) on the \(i\)-th factor, and the identity on each other factor. For an étale morphism \(V\to X^m\), let us say that a system of partial Frobenii on \(V\) is a commuting collection of isomorphisms \(F_i^*V \cong V\) over \(X^m\) (and whose product is the relative Frobenius of \(V\to X^m\)). Finite étale covers of \(U^m\) equipped with partial Frobenii form a Galois category, and thus they are classified by continuous actions of a profinite group \(\pi_1(U^m/\mathrm{ partial Frob})\) on a finite set.

Lemma: (Drinfeld) The natural map\[\pi_1(U^m/\mathrm{partial Frob})\longrightarrow \longrightarrow\pi_1(U)\times\cdots\times\pi_1(U)\quad\quad(m\,\, \mathrm{copies})\] is an isomorphism.

Now \(R^d(f_N)_!\overline{\mathbb Q}_{\ell}\) is not exactly like an \({\mathbb L}\) as above, but only approximately so, yet the Lemma can be suitably adapted so that one gets a big representation of \[\mathrm{GL}_n(\mathrm{A}_K)\times\mathrm{ Gal}(\overline{K}/K)\times\cdots\times\mathrm{Gal}(\overline{K}/K)\] on \(\mathrm{lim}_{\to N}R^d(f_N)_!\overline{\mathbb Q}_{\ell}\). One then expects the latter to decompose under the said action to yield the desired Langlands correspondence.

1.2 The possibility of Shtukas in mixed characteristic

It would be desirable to have moduli spaces of shtukas over number fields, but the first immediate problem is that such a space of shtukas would live over something like \(\mathrm{Spec}({\mathbb Z})\times\mathrm{Spec}({\mathbb Z})\), where the product is over \({\mathbb F}_1\) somehow.

In this course we will give a rigorous definition of \(\mathrm{Spec}({\mathbb Z}_p)\times\mathrm{Spec}({\mathbb Z}_p)\), the completion of \(\mathrm{Spec}({\mathbb Z})\times\mathrm{Spec}({\mathbb Z})\) at \((p, p)\). It lives in the world of nonarchimedean analytic geometry, so it should properly be called \(\mathrm{Spa}({\mathbb Z}_p)\times\mathrm{Spa}({\mathbb Z}_p)\). (The notation \(\mathrm{Spa}\) refers to the adic spectrum.)

Whatever it is, it should contain \(\mathrm{Spa}({\mathbb Q}_p)\times\mathrm{Spa}({\mathbb Q}_p)\) as a dense open subset.

We present a model for the product \(\mathrm{Spa}({\mathbb Q}_p)\times\mathrm{Spa}({\mathbb Q}_p)\), specified by picking one of the factors: one copy of \({\mathbb Q}_p\) appears as the field of scalars, but the other copy appears geometrically. Consider the open unit disc \(D_{{\mathbb Q}_p} = \{x\,;\, |x| < 1\}\) as a subgroup of (the adic version of) \({\mathbb G}_m\), via \(x\mapsto 1 + x\). Then \(D_{{\mathbb Q}_p}\) is in fact a \({\mathbb Z}_p\)-module with multiplication by \(p\) given by \(x\mapsto (1 + x)^p-1\), and we consider\[\widetilde{D}_{{\mathbb Q}_p}=\projlim_{x\mapsto (1 + x)^p-1} D_{{\mathbb Q}_p}.\] After base extension to a perfectoid field, this is a perfectoid space, which carries the structure of a \({\mathbb Q}_p\)-vector space. Thus its punctured version \(\widetilde{D}^*_{{\mathbb Q}_p}=\widetilde{D}_{{\mathbb Q}_p}-\{0\}\) has an action of \({\mathbb Q}_p^{\times}\), and we consider the quotient \[\mathrm{Spa}({\mathbb Q}_p)\times \mathrm{Spa}({\mathbb Q}_p):=\widetilde{D}^*_{{\mathbb Q}_p}/ {\mathbb Z}_p^{\times}.\]Note that this quotient does not exist in the category of adic spaces (the quotient being taken in a formal sense). On \(\widetilde{D}^*_{{\mathbb Q}_p}/ {\mathbb Z}_p^{\times}\), we have an operator \(\varphi\), corresponding to \(p\in{\mathbb Q}_p^{\times}\). Let \[X =(\widetilde{D}^*_{{\mathbb Q}_p}/ {\mathbb Z}_p^{\times})/\varphi^{\mathbb Z}=\widetilde{D}^*_{{\mathbb Q}_p}/ {\mathbb Q}_p^{\times}.\]One can define a finite étale cover of \(X\) simply as a \({\mathbb Q}_p^{\times}\)-equivariant finite étale cover of \(\widetilde{D}^*_{{\mathbb Q}_p}\). There is a corresponding profinite group \(\pi_1(X)\) which classifies such covers. We have the following theorem, which is a local version of Drinfeld’s lemma in the case \(m = 2\).

Theorem: \[\pi_1(X) \cong \mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\times\mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p).\]

This theorem suggests that if one could define a moduli space of \({\mathbb Q}_p\)-shtukas which is fibered over products such as \(\mathrm{Spa}({\mathbb Q}_p)\times\mathrm{Spa}({\mathbb Q}_p)\), then its cohomology would produce representations of \(\mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\times \mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\).

What would a \({\mathbb Q}_p\)-shtuka over \(S\) look like? It should be a vector bundle \({\mathcal E}\) over \(\mathrm{Spa} {\mathbb Q}_p\times S\), together with a meromorphic isomorphism \(\mathrm{Frob}^*_S{\mathcal E}- - \to{\mathcal E}\). In order for this to make any sense, we would need to give a geometric meaning to \(\mathrm{Spa} {\mathbb Q}_p\times S\) (and to \(\mathrm{Frob}_S\)) just as we gave one to \(\mathrm{Spa}({\mathbb Q}_p)\times \mathrm{Spa}({\mathbb Q}_p)\).

We will give a meaning to \(\mathrm{Spa} {\mathbb Q}_p\times S\) whenever \(S\) is a perfectoid space of characteristic \(p\), which lets us define moduli spaces of \(p\)-adic shtukas. In general, these are not representable by perfectoid spaces or classical rigid spaces, but instead they are diamonds: That is, quotients of perfectoid spaces by pro-étale equivalence relations. A large part of this course is about the definition of perfectoid spaces and diamonds.

Table of contents:

Lecture 1: Introduction

Lecture 2: Adic spaces

Lecture 3: Adic spaces II

Lecture 4: Examples of adic spaces

Lecture 5: Complements on adic spaces

Lecture 6: Perfectoid rings

Lecture 7: Perfectoid spaces

Lecture 8: Diamonds

Lecture 9: Diamonds II

Lecture 10: Diamonds associated with adic spaces

Lecture 11: Mixed characteristic shtukas

Lecture 12: Shtukas with one leg

Lecture 13: Shtukas with one leg II

Lecture 14: Shtukas with one leg III

Lecture 15: Examples of diamonds

Lecture 16: Drinfel’s lemma for diamonds

Lecture 17: The v-topolgy

Lecture 18: v-sheaves associated with perfect and formal schemes

Lecture 19: The \(B_{dR}^+\)-affine Grassmannian

Lecture 20: Families of affine Grassmannians

Lecture 21: Affine flag varieties

Lecture 22: Vector bundles and \(G\)-torsors

Lecture 23: Moduli spaces of shtukas

Lecture 24: Local Shimura varieties

Lecture 25: Integral models of local Shimura varieties

Reviewer: Elmar Große-Klönne (Berlin)

### MSC:

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14G20 | Local ground fields in algebraic geometry |

14G22 | Rigid analytic geometry |