Gal(Q p /Q p ) as a geometric fundamental group

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1 Gal(Q p /Q p ) as a geometric fundamental group Jared Weinstein April 28, Introduction Let p be a prime number. In this article we present a theorem, suggested by Peter Scholze, which states that Gal(Q p /Q p ) is the étale fundamental group of certain object Z which is defined over an algebraically closed field. As a consequence, p-ic representations of Gal(Q p /Q p ) correspond to Q p -local systems on Z. The precise theorem involves perfectoid spaces, [Sch12]. Let C/Q p be complete and algebraically closed. Let D be the open unit disk centered at 1, considered as a rigid space over C, and given the structure of a Z p -module where the composition law is multiplication, and a Z p acts by x x a. Let D = lim D. x x p Then D is no longer a classical rigid space, but it does exist in Huber s category of ic spaces, and is in fact a perfectoid space. Note that D has the structure of a Q p -vector space. Let D = D\ {0}; this mits an action of Q p. Theorem The category of Q p -equivariant finite étale covers of D is equivalent to the category of finite étale Q p -algebras. The object Z of the first paragraph is then the quotient Z = D /Q p. This quotient doesn t belong to the category of ic spaces. Rather, one has a Yoneda-style construction. The category Perf C of perfectoid spaces over C has a pro-étale topology, [Sch13], and one has a sheaf of sets Z on Perf C, namely the sheafification of X Hom(X, D )/Q p. Thus Z belongs to the category of sheaves of sets on Perf C which mit a surjective map from a representable sheaf. In this category, a morphism F F is called finite 1

2 étale if the pull-back to any representable sheaf is finite étale in the usual sense. Then one may define the fundamental group for such objects, and Thm asserts that πét 1 (Z) = Gal(Q p/q p ). Theorem can be generalized to a finite extension E/Q p. Let π E be a uniformizer, and let H be the corresponding Lubin-Tate formal O E - module. Then D gets replaced by the generic fiber HC ; this is the unit disc centered at 0, considered as an ic space over C. HC is endowed with the O E -module structure coming from H. Form the universal cover H C = lim H C, where the inverse limit is taken with respect to multiplication by π in H. Then H C is an E-vector space object in the category of perfectoid spaces over C. Form the quotient Z E = ( H C \ {0})/E. Then the main theorem (Thm ) states that the categories of finite étale covers of Z E and Spec E are equivalent, so that πét 1 (Z E) = Gal(E/E). The proof hinges on a combination of two themes: the fundamental curve X of p-ic Hodge theory, due to Fargues-Fontaine, and the tilting equivalence, due to Scholze. Let us sketch the proof in the case E = Q p. Let C be the tilt of C, a perfectoid field in characteristic p. Consider the punctured open disc D (with parameter t) and its universal cover D. C C Then D is simultaneously a perfectoid space over two fields: C D C Spa F p ((t 1/p )) Spa C Considered as a perfectoid space over C, D has an obvious un-tilt, C namely D C. The other field F p((t 1/q )) is the tilt of ˆQ p (µ p ). It turns out that there is a perfectoid space over ˆQ p (µ p ) whose tilt is also D C, and here is where the Fargues-Fontaine curve comes in. The construction of the Fargues-Fontaine curve X is reviewed in 2. X is an integral noetherian scheme of dimension 1 over Q p, whose closed points parametrize un-tilts of C modulo Frobenius. For our purposes we need the ic version X, which is the quotient of another ic space Y by 2

3 a Frobenius automorphism φ. The extension of scalars Y ˆ ˆQ p (µ p ) is a perfectoid space; by a direct calculation (Prop ) we show that its tilt is isomorphic to D. In this isomorphism, the action of Gal(Q C p (µ p )/Q p ) = Z p on the field of scalars ˆQ p (µ p ) corresponds to the geometric action of Z p on D, and the automorpism φ corresponds (up to absolute Frobenius) C to the action of p on D. C Therefore under the tilting equivalence, finite étale covers of D C and Y ˆ ˆQ p (µ p ) are identified. The same goes for D C /pz and X ˆ ˆQ p (µ p ). Now we apply the key fact that X is geometrically simply connected. The same statement is proved in [FF11] for the algebraic curve X; we have apted the proof for X in Prop Thus finite étale covers of D C /pz are equivalent to finite étale Q p (µ p )-algebras. Now we can descend to Q p : Z p -equivariant finite étale covers of D C /p Z are equivalent to finite étale Q p -algebras, which is Thm This entire article is an elaboration of comments me to me by Peter Scholze. I also thank Laurent Fargues for many helpful remarks. The author is supported by NSF Award DMS The Fargues-Fontaine curve 2.1 The rings B and B + Here we review the construction of the Fargues-Fontaine curve. The construction requires the following two inputs: A finite extension E/Q p, with uniformizer π and residue field F q /F p, A perfectoid field F of characteristic p containing F q. Note that a perfectoid field F of characteristic p is the same as a perfect field of characteristic p which is complete with respect to a nontrivial (rank 1) valuation. Let x x denote one such valuation. Let W (F ) be the ring of Witt vectors of the perfect field F, and let W OE (F ) = W (F ) W (Fq) O E. A typical element of W OE (F ) is a series x = n [x n ]π n, where x n F. Let B b W OE (F )[1/π] denote the subalgebra defined by the condition that x n is bounded as n. Equivalently, B b = 3

4 W OE (O F )[1/π, 1/[ϖ]], where ϖ F is any element with 0 < ϖ < 1. Then B b clearly contains B b,+ = W (O F ) W (k) O E as a subring. For every 0 < ρ < 1 we define a norm ρ on B b by x ρ = sup x n ρ n. n Z Definition Let B denote the Fréchet completion of B b with respect to the family of norms ρ, ρ (0, 1). B can be expressed as the inverse limit lim I B I of Banach E-algebras, where I ranges over closed subintervals of (0, 1), and B I is the completion of B b with respect to the norm x I = sup x ρ. ρ I (Note that x ρ is continuous in ρ and therefore bounded on I.) It will be useful to give an explicit description of B I. Lemma Let I (0, 1) be a closed subinterval whose endpoints lie in the value group of F : I = [ ϖ 1, ϖ 2 ]. [ Let B b, I ] be subring of elements x B b with x I 1. Then B b, I := B b,+ [ϖ1 ] π, π [ϖ 2 ]. Proof. It is easy to see that [ϖ 1 ] π = π I [ϖ 2 ] = 1, [ ] which proves that B b,+ [ϖ1 ] π, π [ϖ 2 ] B b, I. Conversely, suppose x = n [x n]π n B b, I. Then x n ϖ 1 n for n < 0 and x n ϖ 2 n for n 0. This shows that the tail term n<0 [x n]π n lies in B b,+ [[ϖ 1 ]/π]. It also shows that for n 0, each term [x n ]π n lies in B b,+ [π/[ϖ 2 ]]. Since x B b, there exists C > 0 with x n C for all n. Let N be large enough so that ϖ 2 N C 1; then for n N we have ϖ N 2 x n 1. Then the sum of the terms of x of index N are π N ([x N ] + [x N+1 ]π +... ) = πn ϖ2 N [ ] which lies in B b,+ [ϖ1 ] π, π [ϖ 2 ]. ([ϖ N x N ] + [ϖ N x N+1 ]π +... ), 4

5 Lemma If BI is the closed unit ball in B I, then BI is the π-ic completion of B b, I. Proof. B I is the completion of B b with respect to the norm I. This norm induces the π-ic topology on B b, I, so that the π-ic completion of B b, I is the same as the closed unit ball in B I. The qth power Frobenius automorphism of F induces a map φ: B b,+ B b,+. For ρ (0, 1) we have x φ ρ = x q ρ 1/q. As a result, φ induces an endomorphism of B. We have B φ=1 = E. (See Prop. 7.1 of [FF11].) We put P = d 0 B φ=πd, a gred E-algebra. Definition Let X E,F = Proj P. Theorem [FF11], Théorème X F,E is an integral noetherian scheme which is regular of dimension H 0 (X F,E, O XF,E ) = E. (E is the field of definition of X F,E ) 3. For a finite extension E /E there is a canonical isomorphism X F,E = X F,E E E. 3 The ic curve X F,E 3.1 Generalities on formal schemes and ic spaces The category of ic spaces is introduced in [Hub94]. In brief, an ic space is a topological space X equipped with a sheaf of topological rings O X, which is locally isomorphic to Spa(R, R + ). Here (R, R + ) is an affinoid ring, R + R is an open and integrally closed subring, and Spa(R, R + ) is the space of continuous valuations on R with R + 1. Definition A topological ring R is f-ic if it contains an open subring R 0 whose topology is generated by a finitely generated ideal I R 0. An affinoid ring is a pair (R, R + ), where R is f-ic and R + R is an open and integrally closed subring consisting of power-bounded elements. A morphism of affinoid algebras (R, R + ) (S, S + ) is a continuous homomorphism R S which sends R + S +. R is a Tate ring if it contains a topologically nilpotent unit. 5

6 Note that if (R, R + ) and R is Tate, say with topologically nilpotent π R, then R = R + [1/π] and πr R is open. Note also that if (R, R + ) is an affinoid algebra over (E, O E ), then R is Tate. It is important to note that a given affinoid ring (R, R + ) does not necessarily give rise to an ic space Spa(R, R + ), because the structure sheaf on Spa(R, R + ) is not necessarily a sheaf. Let us say that (R, R + ) is sheafy if the structure presheaf on Spa(R, R + ) is a sheaf. Huber shows (R, R + ) is a sheaf when R is strongly noetherian, meaning that R X 1,..., X n is noetherian for all n. In 2 of [SW13] we constructed a larger category of general ic spaces, whose objects are sheaves on the category of complete affinoid rings (this category can be given the structure of a site in an obvious way). If (R, R + ) is a (not necessarily sheafy) affinoid ring, then Spa(R, R + ) belongs to this larger category. If X is an ic space in the general sense, let us call X an honest ic space if it belongs to the category of ic spaces in the sense of Huber; i.e. if it is locally Spa(R, R + ) for a sheafy (R, R + ). Now suppose R is an O E -algebra which is complete with respect to the topology induced by a finitely generated ideal I R which contains π. Then Spf R is a formal scheme over Spf O E, and (R, R) is an affinoid ring. One can form the (general) ic space Spa(R, R), which is fibered over the two-point space Spa(O E, O E ). By [SW13], Prop , Spf R Spa(R, R) extends to a functor M M from the category of formal schemes over Spf O E locally mitting a finitely generated ideal of definition, to the category of (general) ic spaces over Spa(O E, O E ). Definition Let M be a formal scheme over Spf O E which locally mits a finitely generated ideal of definition. The ic generic fiber Mη is the fiber of M over the generic point η = Spa(E, O E ) of Spa(O E, O E ). We will also notate this as ME. Lemma Let R be a flat O E -algebra which is complete with respect to the topology induced by a finitely generated ideal I containing π. Let f 1,..., f r, π be generators for I. For n 1, let S n = R[f1 n/π,..., f r n /π] (π-ic completion). Let R n = S n [1/π], and let R n + be the integral closure of S n in R n. There are obvious inclusions R n+1 R n and R n+1 + R+ n. Then ME = lim Spa(R n n, R n + ). Proof. This amounts to showing that whenever (T, T + ) is a complete affinoid 6

7 algebra over (E, O E ), then M E (T, T + ) = lim n Hom((R n, R + n ), (T, T + )). An element of the left hand side is a continuous O E -linear homomorphism g : R T + ; we need to produce a corresponding homomorphism R + n T +. We have that g(f i ) T + is topologically nilpotent for i = 1,..., r, since f i is. Since πt + is open in T, there exists n so that g(f i ) n πt +, i = 1,..., r. Thus g extends to a map R + [f n 1 /π,..., f n r /π] T +. Passing to the integral closure of the completion gives a map R n T + as required. Example The ic generic fiber of R = Spf O E t is the rigid open disc over E. 3.2 Formal schemes with perfectoid generic fiber In this section K is a perfectoid field of characteristic 0, and ϖ K satisfies 0 < ϖ p. Assume that ϖ = ϖ for some ϖ O K, so that ϖ 1/pn O K for all n 1. Definition Let S be a ring in characteristic p. S is semiperfect if the Frobenius map S S is surjective. If S is semiperfect, let S = lim S, x x p a perfect topological ring. If R is a topological O K -algebra with R/ϖ semiperfect, then write R = (R/ϖ), a perfect topological O K -algebra. If R is complete with respect to a finitely generated ideal of definition, then so is R. Proposition Let R be an O K -algebra which is complete with respect to a finitely generated ideal of definition. Assume that R/π is a semiperfect ring. Then (Spf R) K and (Spf R ) are a perfectoid spaces over K and K K respectively, so that in particular they are honest ic spaces. There is a natural isomorphism of perfectoid spaces over K : (Spf R), K = (Spf R ) K Proof. First we show that (Spf R ) is a perfectoid space over K. Let η f 1,..., f r be generators for an ideal of definition of R. Then (Spf R ) η 7

8 is the direct limit of subspaces Spa(R n, R,+ n ) as in Lemma Here R n = R f n i /ϖ [1/ϖ ]. Let ( ) f Sn, = R n 1/p ( ) 1 f n 1/p ϖ,, r ϖ Rn Then Sn, is a ϖ -ically complete flat O K -algebra. We claim that Sn, a is a perfectoid O a -algebra, which is to say that the Frobenius map S, K n /ϖ 1/p n /ϖ is an almost isomorphism. For this we refer to [Sch12], proof of Lemma 6.4(i) (case of characteristic p). This shows that Rn = Sn, [1/ϖ ] is a perfectoid K -algebra, and thus Spf(Rn, Rn,+ ) is a perfectoid affinoid, and in particular it is an honest ic space. To conclude that (Spf R ) is a perfectoid space, one needs to show that K its structure presheaf is a sheaf. (I thank Kevin Buzzard for pointing out this subtlety.) But since (Spf R ) is the direct limit of the Spa(R K n, R,+ n ), its structure presheaf is the inverse limit of the pushforward of the structure presheaves of the Spa(Rn, Rn,+ ), which are all sheaves. Now we can use the fact that an arbitrary inverse limit of sheaves is again a sheaf. We now turn to characteristic 0. Recall the sharp map f f, which is a map of multiplicative monoids R R. The elements f 1,..., f r generate an ideal of definition of R. Then (Spf R) η is the direct limit of subspaces Spa(R n, R n + ), where R n = R fi n /ϖ [1/ϖ]. The proof of [Sch12], Lemma 6.4(i) (case of characteristic 0) shows that Spa(R n, R n + ) is a perfectoid affinoid, and Lemma 6.4(iii) shows that the tilt of Spa(R n, R n + ) is Spa(Rn, Rn,+ ). S, 3.3 The ic spaces Y F,E and X F,E Once again, E/Q p is a finite extension. A scheme X/E of finite type mits a canonical analytification X : this is a rigid space together with a morphism X X of locally ringed spaces satisfying the appropriate universal property. GAGA holds as well: If X is a projective variety, then the categories of coherent sheaves on X and X are equivalent. The Fargues-Fontaine curve X = X F,E, however, is not of finite type, and it is not obvious that such an analytification exists. However, in [Far13] there appears an ic space X defined over E, such that X and X satisfy a suitable formulation of GAGA. 8

9 Definition Give B b,+ = W OE (O F ) the I-ic topology, where I = ([ϖ], π) and ϖ O F is an element with 0 < ϖ < 1. Note that B b,+ is I-ically complete. Let Y = (Spf B b,+ ) E \ {0}, where 0 refers to the valuation on B b,+ = W OE (O F ) pulled back from the π-ic valuation on W OE (k F ). Proposition As (general) ic spaces we have Y = lim I Spa(B I, B I ). This shows that our definition of Y agrees with the definition in [Far13], Définition 2.5. Proof. By Prop we have (Spf B b,+ ) E = lim ϖ1 Spa(R ϖ1, R + ϖ 1 ), where ϖ 1 runs over elements ] of O F with 0 < ϖ < 1, R ϖ + 1 is the π-ic, and R ϖ1 = R ϖ + 1 [1/π]. [ completion of B b,+ [ϖ1 ] π By definition, Y is the complement in (Spf B b,+ ) E of the single valuation pulled back from the π-ic valuation on W OE (k). Thus if x Y we have [ϖ](x) = 0 for all ϖ O F \ {0}. Thus there exists ϖ 2 O F with ϖ 2 (x) π(x). On the other hand we have x Spa(R ϖ1, R + ϖ 1 ) for some ϖ 1. Thus x belongs to the rational subset Spa(B I, B I ) Spa(R ϖ 1, R + ϖ 1 ), where I = [ ϖ 1, ϖ 2 ]. This shows that the Spa(B I, B I ) cover Y. The following conjecture is due to Fargues. Conjecture Let I (0, 1) be a closed interval. The Banach algebra B I is strongly noetherian. That is, for every n 1, B I T 1,..., T n is noetherian. Despite not being able to prove Conj , we have the following proposition (Théorème 2.1 of [Far13]), which is proved by extending scalars to a perfectoid field and appealing to results of [Sch12]. Proposition Y is an honest ic space. Definition The ic space X is the quotient of Y by the automorphism φ. 9

10 3.4 The ic Fargues-Fontaine curve in characteristic p The entire story of the Fargues-Fontaine curve can be retold when E is replaced by a local field of residue field F q. The construction is very similar, except that B b,+ = W (O F ) W (Fq) O E must be replaced by O F ˆ Fq O E. One arrives at a curve X F,E defined over E satisfying the same properties as Thm One also gets spaces X F,E and Y F,E. Namely, Y F,E = Spf ( O F ˆ Fq O E ) E \ {0}, where 0 refers to the pullback of the valuation on O F through the quotient map O F ˆ Fq O E O F. As before, XF,E is defined as the quotient of Y F,E by the automorphism φ coming from the Frobenius on F. Since E = F q ((t)), we have Y F,E = Spf O F t F \ {0}. This is nothing but the punctured rigid open disc DF. We have the quotient XF,E = Y F,E /φz. Note that since φ does not act F -linearly, XF,E does not make sense as a rigid space over F. 3.5 Tilts Suppose once again that E has characteristic 0. Let K be a perfectoid field containing E. Let YF,E ˆ K be the strong completion of the base change to K of Y. Then Y F,E ˆ K = Spf ( W OE (O F ) ˆ W (Fq)O K ) K \ {0}. Note that (W OE (O F ) ˆ W (Fq)O K )/π = O F Fq O K /π is semiperfect, and (W OE (O F ) ˆ W (Fq)O K ) = O F ˆ Fq O K. By Prop , YF,E ˆ K is a perfectoid space, and (YF,E ˆ K) ( ) = Spf OF ˆ Fq O K \ {0}. (3.5.1) K As a special case, let E n be the field obtained by joining the π n -torsion in a Lubin-Tate formal group over E, and let E = n 1 E n. Then Ê is a perfectoid field. Let L(E) be the imperfect field of norms for the extension E /E. As a multiplicative monoid we have L(E) = lim E n, 10

11 where the inverse limit is taken with respect to the norm maps E n+1 E n. L(E) = F q ((t)) is a local field, and Ê = F q ((t 1/q )) is the completed perfection of L(E). The following proposition follows immediately from Eq. (3.5.1). Proposition This is Thm. 2.7(2) of [Far13]. (Y F,E ˆ Ê ) = Y F,L(E) ˆ Ê. 3.6 Classification of vector bundles on X F,E In this section we review the results of [FF11] concerning the classification of vector bundles on X F,E. Recall that X F,E = Proj P, where P is the gred ring d 0 (B+ ) φ=πd. For d Z, let P [d] be the gred P -module obtained from P by shifting degrees by d, and let O XE (d) be the corresponding line bundle on X E. For h 1, let E h /E be the unramified extension of degree h, let π h : X F,Eh = X F,E E h X F,E be the projection. If (d, h) = 1, define O XF,E (d/h) = π h O XF,Eh (d), a vector bundle on O XF,E of rank h. One thus obtains a vector bundle O XF,E (λ) for any λ Q, which satisfy O XF,E (λ) O XF,E (λ ) = O XF,E (λ + λ ). Proposition Let λ Q. Then H 0 (λ) 0 if and only if λ 0. Theorem Every vector bundle F on X F,E is isomorphic to one of the form s i=1 O X F,E (λ i ), with λ i Q. Furthermore, F determines the λ i up to permutation. 3.7 Classification of vector bundles on X F,E In the absence of the noetherian condition of Conj , one does not have a good theory of coherent sheaves on XF,E when E has characteristic 0. This frustrates attempts to prove an analogue of Thm for XF,E. Nonetheless, in [Far13], Fargues gives an hoc notion of vector bundle on XF,E, and proves a GAGA theorem relating vector bundles on X F,E to those on XF,E. When E has characteristic p, however, YF,E is isomorphic to the punctured disc D = (Spf O F t ) F \ {0} over F. In particular it is a rigid space, which does have a well-behaved theory of coherent sheaves. In this case we 11

12 are in the setting of [HP04], which classifies φ-equivariant vector bundles F on the punctured disc D = YF,E. In [HP04], these are called σ-bundles, and they mit a Dieudonné-Manin classification along the lines of Thm A σ-bundle is one and the same thing as a vector bundle on the quotient YF,E /φz = XF,E. One gets a vector bundle O(λ) on X F,E for every λ Q. For the convenience of the reer we state the main definitions and constructions of [HP04]. A vector bundle on D is by definition a locally free coherent sheaf of O D -modules. The global sections functor is an equivalence between the category of vector bundles on D and the category of finitely generated projective modules over O D (D ). The Frobenius automorphism x x q on F induces an automorphism σ : D D (the arithmetic Frobenius). Note that the F -points of D are {x F 0 < x < 1}, and on this set σ acts as x x q 1. Definition A σ-bundle on D is a pair (F, τ F ), where F is a vector bundle on D and τ F is an isomorphism σ F F. If (F, τ F ) is a σ-bundle, then a global section of F is a global section of F which is invariant under τ F. We will refer to the pair (F, τ F ) simply as F. Then the space of global sections of F will be denoted H 0 (F). The trivial vector bundle O = O D is a σ-bundle whose global sections H 0 (O) are the ring of power series n Z a nt n which converge on D and which satisfy a q n = a n (thus a n F q ). This ring is easily seen to be the field F q ((t)) (see [HP04], Prop. 2.3). Let n Z. The twisting sheaf O(n) is a σ-bundle with underlying sheaf O. The isomorphism τ O(n) is defined as the composition of τ O : σ O D O D followed by multiplication by t n. For r 1, let [r] denote the morphism [r]: D D, t t r. Then for an integer d relatively prime to r we set F d,r = [r] O(d), together with the induced isomorphism τ Fd,r = [r] τ O(d). Then F d,r is a σ-bundle of rank r. In keeping with the notation of [FF11], let us write O(d/r) = F d,r whenever d and r are relatively prime integers with r 1. Then O(λ) makes sense for any λ Q. We have O(λ) O(λ ) = O(λ + λ ), and O(λ) = O( λ). 12

13 Proposition [HP04], Prop. 8.4 Let λ Q. Then H 0 (λ) 0 if and only if λ > 0. Hartl and Pink give a complete description of the category of σ-bundles. In particular we have the following classification theorem of Dieudonné- Manin type. Theorem [HP04], Thm and Cor Every σ-bundle F is isomorphic to one of the form s i=1 O(λ i), with λ i Q. Furthermore, F determines the λ i up to permutation. Since YF,E = D, with φ corresponding to σ, vector bundles on XF,E = YF,E /φz correspond to σ-bundles, and we have a similar classification of them. 3.8 X F,E is geometrically simply connected In [FF11], Thm. 18.1, it is shown that the scheme X = X F,E is geometrically connected, which is to say that every finite étale cover of X E is split. We need a similar result for the ic curve X F,E : Theorem E XF,E E is an equivalence between the category of finite étale E-algebras and the category of finite étale covers of XF,E. Unfortunately, Thm cannot be deduced directly from the corresponding theorem about the scheme X. Following a suggestion of Fargues, we can still mimic the proof of the simple-connectedness of X in this ic context, by first translating the problem into characteristic p, and applying the Hartl-Pink classification of σ-bundles (Thm ). Lemma Let f : Y X be a finite étale morphism of ic spaces of degree d. Then F = f O Y is a locally free O X -module, and ( d ) 2 F = OX. Proof. If A is a ring, and if B is a finite étale A-algebra, then B is flat and of finite presentation, hence locally free. Furthermore, the trace map tr B/A : B B A is perfect, so that B is self-dual as an A-module. Globalizing, we get that F is locally free and F = F = Hom(F, O X ). Taking top exterior powers shows that d F = d F = ( d F), so that the tensor square of d F is trivial. 13

14 The same lemma appears in [FF11], Prop Let H E be the Lubin-Tate formal O E -module corresponding to the uniformizer π, so that [π] HE (T ) T q (mod π). Let K be the completion of the field obtained by joining the torsion points of H E to E. Then K is a perfectoid field and K = F q ((t 1/q )). Proposition Suppose E has characteristic p. Then XF,E is geometrically simply connected. In other words, E XF,E E is an equivalence between the category of finite étale E-algebras and finite étale covers of X F,E. Proof. It suffices to show that if f : Y XF,E is a finite étale cover of degree n with Y geometrically irreducible, then n = 1. Given such a cover, let F = f O Y. This is a sheaf of O X -algebras, so we have a multiplication F,E morphism µ: F F F. Consider F as a σ-bundle. By Thm , F = n i=1 O(λ i), for a collection of slopes λ i Q (possibly with multiplicity). Assume that λ 1 λ n. The proof now follows that of [FF11], Théorème We claim that λ 1 0. Assume otherwise, so that λ 1 > 0. After pulling back F through some [d]: D D, we may assume that λ i Z for i = 1,..., s (see [HP04], Prop. 7.1(a)). The restriction of µ to O(λ 1 ) O(λ 1 ) is the direct sum of morphisms µ 1,1,k : O(λ 1 ) O(λ 1 ) O(λ k ) for k = 1,..., n. The morphism µ 1,1,k is tantamount to a global section of Hom(O(λ 1 ) 2, O(λ k )) = O(λ k 2λ 1 ). But since λ k 2λ 1 < 0, Prop shows that H 0 (O(λ k 2λ 1 )) = 0. Thus the multiplication map O(λ 1 ) O(λ 1 ) 0 is 0. This means that H 0 (Y, O Y ) contains zero divisors, which is a contriction because Y is irreducible. Thus λ 1 0, and thus λ i 0 for all i. By Lemma 3.8.2, ( n F) 2 = O D, from which we deduce n i=1 λ i = 0. This shows that λ i = 0 for all i, and therefore F = OX n. We find that E = H 0 (Y, O Y ) is an étale E-algebra of degree n. Since Y is geometrically irreducible, E E E must be a field for every separable field extension E /E, which implies that E = E and n = 1. is geomet- Proposition Suppose E has characteristic 0. Then X rically simply connected. F,E 14

15 Proof. Let C be a complete algebraically closed field containing E. We want to show that XF,E ˆ C mits no nontrivial finite étale covers. Such covers are equivalent to covers of its tilt, which is (X F,E ˆ C) = X F,L(E) ˆ C. The latter is simply connected, by Prop (note that C is algebraically closed), which shows there are no nontrivial covers. 4 Proof of the main theorem As before, H E is the Lubin-Tate formal O E -module attached to the uniformizer π. Let us recall the construction of H E : choose a power series f(t ) T O E T with f(t ) T q (mod π). Then H E is the unique formal O E -module satisfying [π] HE (T ) = f(t ). Let t = t 1, t 2,... be a compatible family of roots of f(t ), f(f(t )),..., and let E n = E(t n ). For each n 1, H E [π n ] is a free (O E /π n )-module of rank 1, and the action of Galois induces an isomorphism Gal(E n /E) = (O E /π n ). Let H E,0 = H E OE F q. Lemma For all n 2 we have an isomorphism H E,0 [π n 1 ] = Spec O En /t. The inclusion H E,0 [π n 1 ] H E,0 [π n ] corresponds to the qth power Frobenius map Frob q : O En+1 /t O En /t. Proof. We have H E,0 [π n 1 ] = Spec F q [T ]/f (n 1) (T ) = Spec F q [T ]/T qn 1. Meanwhile O En = O E1 [T ]/(f (n 1) (T ) t), so that O En /t = F q [T ]/T q 1. Thus H E,0 [π n 1 ] = Spec O En /t. The second claim in the lemma follows from [π] HE,0 (T ) = T q. Let H E be the universal cover: H E = lim π H E. Then H E is an E-vector space object in the category of formal schemes over O E. We will call such an object a formal E-vector spaces. Let H E,0 = H E OE F q, a formal E-vector space over F q. Since H E,0 = 15

16 Spf F q T and [π] HE,0 (T ) = T q, we have H E,0 = lim Spf F π q T ( = Spf lim F q T T T q = Spf F q T 1/q. In fact we also have H E = Spf O E T 1/q, see [Wei14], Prop (2). Lemma We have an isomorphism of formal E-vector spaces over F q : H E,0 = lim lim H π n E,0 [π n ]. Proof. For each n 1 we have the closed immersion H E,0 [π n ] H E,0. Taking inverse limits gives a map lim H n E,0 [π n ] H E,0, and taking injective limits gives a map lim lim H π n E,0 [π n ] H E,0. The corresponding homomorphism of topological rings is ) F q T 1/q lim lim F q [T 1/qn ]/T = lim F q [T 1/q ]/T, T T q T T q which is an isomorphism. Proposition There exists an isomorphism of formal schemes over F q : H E,0 = Spf OÊ. This isomorphism is E -equivariant, where the action of E on OÊ is defined as follows: O E acts through the isomorphism O E = Gal(E /E) of class field theory, and π E acts as the qth power Frobenius map. Proof. Combining Lemmas and 4.0.6, we get H E,0 = lim lim H π n E,0 [π n ] = lim Spec O E /t Frob q = Spf O E. The compatibility of this isomorphism with the O E follows from the definition of the isomorphism O E = Gal(E /E) of local class field theory. The compatibility of the action of π follows from the second statement in Lemma

17 Let C be a complete algebraically closed field containing E. Write for the generic fiber of H ˆ O C. By Prop , H E,C H E,C is a perfectoid space. Proposition There exists an isomorphism of ic spaces ( ) ( ) H E,C\ {0} = Y C,E ˆ Ê equivariant for the action of O E (which acts on E by local class field theory). The action of π E on the left corresponds to the action of φ 1 1 on the right, up to composition with the absolute Frobenius morphism on ( Y C,E ˆ Ê ). Proof. By Prop we have an isomorphism H 0 = Spf OÊ. Extending scalars to O C, taking the ic generic fiber and applying tilts gives an isomorphism (cf. Prop ) ) The ic space Spf (O C ˆ Fq OÊ ( ) ( ) H, E,C = Spf OC ˆ Fq O E C ) = Spf (O C ˆ Fq OÊ C C has two Frobenii : one coming from O C and the other coming from OÊ. Their composition is the absolute qth, H C power Frobenius. The action of π on corresponds to the Frobenius on OÊ. Removing the origin from both sides of this isomorphism and using Prop gives the claimed isomorphism. Lemma Let X be a perfectoid space which is fibered over Spec F q, and suppose f : X X is an F q -linear automorphism. Let Frob q : X X be the absolute Frobenius automorphism of X. Then the category of f-equivariant finite étale covers of X is equivalent to the category of f Frob q -equivariant finite étale covers of X. Proof. First observe that a perfectoid algebra in characteristic p is necessarily perfect ([Sch12], Prop. 5.9), which implies that absolute Frobenius is an automorphism of any perfectoid space in characteristic p. Then note that if Y X is a finite étale cover, then Y is also perfectoid ([Sch12], Thm. 7.9(iii)), so that Frob q is an automorphism of Y. The proof of the lemma is now formal: if g : Y X is a finite étale cover and f Y : Y Y lies over f : X X, then f Y Frob q : Y Y lies 17

18 over f Frob q : X X. Since Frob q is invertible on Y, the functor is invertible. We can now prove the main theorem. Theorem There is an equivalence between the category of E - equivariant étale covers of H C \ {0} and the category of finite étale E- algebras. Proof. In the following chain of equivalences, G-cover of X is an abbreviation for G-equivariant finite étale cover of X. { } E -covers of H C { \ {0} } = E, -covers of H C \ {0} [Sch12], Thm { } = O E (φ 1 Frob q ) Z -covers of Y C,L(E) Ê Prop { } = O E φz -covers of Y C,L(E) Ê Lemma { } = O E-covers of X C,L(E) Ê Defn. of X { } C,L(E) = O E -equivariant finite étale Ê -algebras Prop { } = O E-equivariant finite étale Ê -algebras Theory of norm fields = {Finite étale E-algebras} L L O E 4.1 Functoriality in E. Let us write Z E = ( H E,C\ {0})/E. As in the introduction, Z E may be considered as a sheaf on Perf C, although the reer may also interpret the definition of Z E purely formally, keeping in mind that a finite étale cover of Z E means the same thing as an E - equivariant finite étale cover of H E,C. Let us check that Z E really only depends on E. The construction depends on the choice of Lubin-Tate formal O E -module H = H E, which depends in turn on the choice of uniformizer π. If π E is a different uniformizer, with corresponding O E -module H, then H and H become isomorphic after base extension to OÊnr, the ring of integers in the completion of the maximal unramified extension of E. Such an isomorphism is unique up to multiplication by E. Thus the ic spaces 18 H C and ( H ) C are isomorphic,

19 and we get a canonical isomorphism ( H C \ {0})/E ( H C \ {0})/E. Thus Z E only depends on E. It is also natural to ask whether the formation of Z E is functorial in E. That is, given a finite extension E /E of degree d, there ought to be a norm morphism N E /E : Z E Z E, which makes the following diagram commute: πét 1 (Z E ) N E /E Gal(E/E ) πét 1 (Z E) Gal(E/E). (4.1.1) There is indeed such a norm morphism; it is induced from the determinant morphism on the level of π-divisible O E -modules. The existence of exterior powers of such modules is the subject of [Hed10]. Here is the main result we need 1 (Thm of [Hed10]): let G be a π-divisible O E -module of height h (relative to E) and dimension 1 over a noetherian ring R. Then for all r 1 there exists a π-divisible O E -module r O E G of height ( h r) and dimension ( ) h 1 r 1, together with a morphism λ: G r r O E G which satisfies the appropriate universal property. In particular the determinant h O E G has height 1 and dimension 1. Let π be a uniformizer of E, and let H be a Lubin-Tate formal O E - module. Then H [(π ) ] is a π -divisible O E -module over O E of height 1 and dimension 1. By restriction of scalars, it becomes a π-divisible O E - module H [π ] of height d and dimension 1. Then d O E H [π ] is a π- divisible O E -module of height 1 and dimension 1, so that it is the π-power torsion in a Lubin-Tate formal O E -module d H defined over O E. For all n 1 we have an O E /π n -alternating morphism λ: H [π n ] d d H [π n ] of π-divisible O E -modules over O E. Let H 0 = H O E /π. Reducing mod π, taking inverse limits with respect to n and applying Lemma gives a morphism λ 0 : ( H 0) d d H 0 1 This result requires the residue characteristic to be odd, but we strongly suspect this is unnecessary. See [SW13], 6.4 for a construction of the determinant map (on the level of universal covers of formal modules) without any such hypothesis. 19

20 of formal vector spaces over O E /π. By the crystalline property of formal vector spaces ([SW13], Prop (ii)), this morphism lifts uniquely to a morphism λ: ( H ) d d H of formal vector space over O E. Since d H and H are both height 1 and dimension 1, they become isomorphic after passing to O C. Let α 1,..., α n be a basis for E /E, and define a morphism of formal schemes H O C d H O C = HOC x λ(α 1 x,..., α d x) After passing to the generic fiber we get a well-defined map N E /E : Z E Z E which does not depend on the choice of basis for E /E. The commutativity of the diagram in Eq. (4.1.1) is equivalent to the following proposition. Proposition The following diagram commutes: } A A E E {Étale E-algebras {Étale E -algebras} {Étale covers of ZE } {Étale covers of Z E }. Here the bottom arrow is pullback via N E /E. Proof. Ultimately, the proposition will follow from the functoriality of the isomorphism in Lemma For n 1, let E n be the field obtained by joining the π n -torsion in H to E. (Note that the π n -torsion is the same as the (π ) en torsion, where e is the ramification degree of E /E.) The existence of λ shows that E n contains the field obtained by joining the π n -torsion in d H to E n. Namely, let x H[π n ](O E n ) be a primitive element, in the sense that x generates H [π n ](O E n ) as an O E /π n -module. If α 1,..., α d is a basis for O E /O E then λ(α 1 x,..., α d x) generates H [π n ](O E n ) as an O E /π n -module. Since d H and H become isomorphic over Ê,nr, we get a compatible family of embeddings Ênr n Ê,nr n. 20

21 By construction, these embeddings are compatible with the isomorphisms in Lemma 4.0.5, so that the following diagram commutes: Spf O E,nr n /t H 0 [πn 1 ] F q Spf O E nr n /t N E /E H0 [π n 1 ] F q Here t O E1 is a uniformizer. From here we get the commutativity of the following diagram: ( Spf OÊ, ˆ OE /π O C ) H, O C ( ) Spf OÊ ˆ Fq O C H OC. N E /E One can now trace this compatibility with the chain of equivalences in the proof of Thm to get the proposition. The details are left to the reer. 4.2 Descent from C to E H The object Z E = ( E,C \ {0})/E is defined over C, but it has an obvious model over E, namely Z E,0 = ( H E \ {0})/E. Since the geometric fundamental group of Z E,0 is Gal(E/E), we have an exact sequence 1 Gal(E/E) πét 1 (Z E,0 ) Gal(E/E) 1. This exact sequence splits as a direct product. In other words: Proposition Let E /E be a finite extension, and let Z E Z E be the corresponding finite étale cover under Thm Then Z E descends to a finite étale cover Z E,0 Z E,0. Proof. Let us write R = H 0 ( H, O H) for the coordinate ring of H, so that R = O E t 1/q. Then R/π = F q t 1/q is a perfect ring and R = W OE (R/π). By Prop we have R/π = OÊ. Thus Z E,0 = Spf W (O ) Ê E \ {0}. Now suppose E /E is a finite extension. Let Ê = E E Ê, a finite étale Ê -algebra. Then Ê is a perfectoid Ê -algebra (in fact it a product of perfectoid fields), so that we can form the tilt Ê,, a perfect ring 21

22 containing Ê. Let OÊ, put be its subring of power-bounded elements. We Z E,0 = Spf W OE (OÊ, ) E \ {0}, a finite étale cover of Z E,0. We claim that Z E = Z E,0 ˆ C. It suffices to show that the tilts are the same. We have ( ZE,0 ˆ C ) = Spf = ( W OE (OÊ, ˆ O C ), ( Spf ( OÊ, ˆ O C which by construction is the tilt of Z E. References ) C \ {0} ) \ {0} /E C ) /E, [Far13] [FF11] Laurent Fargues, Quelque résultats et conjectures concernant la courbe, Submitted to Actes de la conférence en l honneur de Gérard Laumon, Laurent Fargues and Jean-Marc Fontaine, Courbes et fibrés vectoriels en théorie de Hodge p-ique, preprint, [Hed10] Hi Hedayatzeh, Exterior powers of Barsotti-Tate groups, Ph.D. thesis, ETH Zürich, [HP04] U. Hartl and R. Pink, Vector bundles with a frobenius structure on the punctured unit disc, Compositio Mathematica 140 (2004), no. 3, [Hub94] R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), no. 4, [Sch12] [Sch13] [SW13] Peter Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012), , p-ic Hodge theory for rigid-analytic varieties, Forum Math. Pi 1 (2013), e1, 77. Peter Scholze and Jared Weinstein, Moduli of p-divisible groups, Cambridge Journal of Mathematics 1 (2013), no. 2, [Wei14] Jared Weinstein, Semistable models for modular curves of arbitrary level, Preprint,