Stable models of Lubin-Tate curves with level three
aa r X i v : . [ m a t h . N T ] D ec Stable models of Lubin-Tate curves with level three
Naoki Imai and Takahiro Tsushima
Abstract
We construct a stable formal model of a Lubin-Tate curve with level three, and study theaction of a Weil group and a division algebra on its stable reduction. Further, we study astructure of cohomology of the Lubin-Tate curve. Our study is purely local and includes thecase where the characteristic of the residue field of a local field is two.
Introduction
Let K be a non-archimedean local field with a finite residue field k of characteristic p . Let p be themaximal ideal of the ring of integers O K of K . Let n be a natural number. We write LT( p n ) for theLubin-Tate curve with full level n as a deformation space of formal O K -modules by quasi-isogenies.Let D be the central division algebra over K of invariant 1 /
2. Let ℓ be a prime number differentfrom p . We write C for the completion of an algebraic closure of K . Then the groups W K , GL ( K )and D × act on lim −→ m H (LT( p m ) C , Q ℓ ) , and these actions partially realize the local Langlands correspondence and the local Jacquet-Langlands correspondence for GL . The realization of the local Langlands correspondence wasproved by global automorphic methods in [Ca]. Since Lubin-Tate curves are purely local objects,it is desirable to have a purely local proof which only makes use of the geometry of Lubin-Tatecurves.We put K ( p n ) = (cid:26)(cid:18) a bc d (cid:19) ∈ GL ( O K ) (cid:12)(cid:12)(cid:12)(cid:12) c ≡ , d ≡ p n (cid:27) . Let LT ( p n ) be the Lubin-Tate curve with level K ( p n ) as a deformation space of formal O K -modules by quasi-isogenies. Then the cohomology group H (LT ( p n ) C , Q ℓ ) = (cid:16) lim −→ m H (LT( p m ) C , Q ℓ ) (cid:17) K ( p n ) will give representations of W K and D × that correspond to smooth irreducible representationsof GL ( K ) with conductor less than or equal to n . The purpose of this paper is to study thiscohomology in the case n = 3. We note that 3 is the smallest conductor of a two-dimensionalrepresentation of W K which can not be written as an induction of a character. Such a representationis called a primitive representation.Our method is purely local and geometric. In fact, we construct a stable model of the connectedLubin-Tate curve X ( p ) with level K ( p ) by using the theory of semi-stable coverings (cf. [CM, Mathematics Subject Classification . Primary: 11G20; Secondary: 11F80.Key words: Lubin-Tate curve, stable model, Galois representation p = 2, and in this case, primitive Galois represen-tations of conductor 3 appear in the cohomology of X ( p ). It gives a geometric understanding ofa realization of the primitive Galois representations.Our method of the calculation of the stable reduction is similar to that in [CM]. In [CM],Coleman-McMurdy calculate the stable reduction of the modular curve X ( p ) under the assump-tion p ≥
13. The calculation of the stable reductions in the modular curve setting is equivalent tothat in the Lubin-Tate setting where K = Q p . As for the calculation of the stable reduction of themodular curve X ( p n ), it is given in [DR] if n = 1.We explain the contents of this paper. In Section 1, we recall a definition of the connectedLubin-Tate curve, and study the action of a division algebra in a general setting. In Section 2, westudy the cohomology of Lubin-Tate curves as representation of GL ( K ) by purely local methods.By this result, we can calculate the genus of some Lubin-Tate curves. In Section 3, we constructa stable covering of the connected Lubin-Tate curve with level K ( p ), which is used to study acovering of X ( p ).In Section 4, we define several affinoid subspaces Y , , Y , and Z , of X ( p ), and calculatetheir reductions. Let k ac be the residue field of C . We put q = | k | and S = ( µ q − ( k ac ) if q is odd, µ q − ( k ac ) if q is even.The reductions of Y , and Y , are isomorphic to the affine curve defined by x q y − xy q = 1.This affine curve has genus q ( q − /
2, and is called the Deligne-Lusztig curve for SL ( F q ) or theDrinfeld curve. Here, the genus of a curve means the genus of the smooth compactification of thenormalization of the curve. The reduction Z , of Z , is isomorphic to the affine curve defined by Z q + X q − + X − ( q − = 0. This affine curve has genus 0 and singularities at X ∈ S .Next, we analyze tubular neighbourhoods {D ζ } ζ ∈S of the singular points of Z , . If q is odd, D ζ is a basic wide open space with the underlying affinoid X ζ . See [CM, 2B] for the precisedefinition of a basic wide open space. Roughly speaking, it is a smooth geometrically connectedone-dimensional rigid space which contains an affinoid such that the reduction of the affinoid isirreducible and has at worst ordinary double points as singularities, and the complement of theaffinoid is a disjoint union of open annuli. The reduction of X ζ is isomorphic to the Artin-Schreieraffine curve of degree 2 defined by z q − z = w . This affine curve has genus ( q − / q is even, it is harder to analyze D ζ , because the space D ζ is not a basicwide open space. First, we find an affinoid P ζ . The reduction P ζ of P ζ has genus 0 and singularpoints parametrized by ζ ′ ∈ k × . Secondly, we analyze the tubular neighborhoods of singularpoints of P ζ . As a result, we find an affinoid X ζ,ζ ′ , whose reduction X ζ,ζ ′ is isomorphic to theaffine curve defined by z + z = w . The smooth compactification of this curve is the uniquesupersingular elliptic curve over k ac , whose j -invariant is 0, and its cohomology gives a primitiveGalois representation. By using these affinoid spaces, we construct a covering C ( p ) of X ( p ).In Section 5, we calculate the action of O × D on the reductions of the affinoid spaces in X ( p ),where O D is the ring of integers of D . In Section 6, we calculate an action of a Weil group on thereductions. In the case where q is even, we construct an SL ( F )-Galois extension of K ur , and showthat the Weil action on X ζ,ζ ′ up to translations factors through the Weil group of the constructedextension. For such a Galois extension, see also [Weil, 31].In Section 7, we show that the covering C ( p ) is semi-stable. To show this, we calculate thesummation of the genera of the reductions of the affinoid spaces in X ( p ), and compare it withthe genus of X ( p ). Using the constructed semi-stable model, we study a structure of cohomologyof X ( p ).The dual graph of the semi-stable reduction of X ( p ) in the case where q is even is the following:2 Y c1 , ◦ Z , c1 , ◦ Y c2 , ◦ ⑧⑧⑧⑧⑧⑧⑧ P , c ζ · · ·· · ·· · ·· · ·◦ ⑧⑧⑧⑧⑧⑧⑧ X c ζ ,ζ ′ ◦· · · X c ζ ,ζ ′ q − ◦ ❄❄❄❄❄❄❄ P , c ζ q − ◦ X c ζ q − ,ζ ′ ◦ ❄❄❄❄❄❄❄ · · · X c ζ q − ,ζ ′ q − where µ q − ( k ac ) = { ζ , . . . , ζ q − } , k × = { ζ ′ , . . . , ζ ′ q − } and X c denotes the smooth compactificationof the normalization of X for a curve X over k ac . The constructed semi-stable model is in factstable, except in the case where q = 2. If q = 2, we get the stable model by blowing down some P -components.The realization of the local Jacquet-Langlands correspondence in cohomology of Lubin-Tatecurves was proved in [Mi] by a purely local method. Therefore, the remaining essential part of thestudy of the realization of the local Langlands correspondence is to study actions of Weil groupsand division algebras. In the paper [IT3], we give a purely local proof of the realization of the localLanglands correspondence for representations of conductor three using the result of this paper.At last, we mention some recent progress on related topics according to a suggestion of a referee.In [Wein], Weinstein constructs semi-stable models of Lubin-Tate curves for arbitrary level in thecase where the residue characteristic is not equal to two using Lubin-Tate perfectoid spaces. In[IT4] and [IT5], some of our results in this paper are generalized to arbitrary dimensional cases forLubin-Tate perfectoid spaces. In [IT6], we construct an affinoid in the two-dimensional Lubin-Tatespace such that the cohomology of the reduction of the affinoid realizes representations which area bit more ramified than the epipelagic representations. Acknowledgment
The authors thank Seidai Yasuda for helpful discussion on the subject in Paragraph 6.2.2. Theyare grateful to a referee for suggestions for improvements.
Notation
In this paper, we use the following notation. Let K be a non-archimedean local field. Let O K denote the ring of integers of K , and k denote the residue field of K . Let p be the characteristicof k . We fix a uniformizer ̟ of K . Let q = | k | . We fix an algebraic closure K ac of K . For anyfinite extension F of K in K ac , let G F denote the absolute Galois group of F , W F denote the Weilgroup of F and I F denote the inertia subgroup of W F . The completion of K ac is denoted by C .Let O C be the ring of integers of C and k ac the residue field of C . For an element a ∈ O C , wewrite ¯ a for the image of a by the reduction map O C → k ac . Let v ( · ) denote the valuation of C suchthat v ( ̟ ) = 1. Let K ur denote the maximal unramified extension of K in K ac . The completion of K ur is denoted by b K ur . For a, b ∈ C and a rational number α ∈ Q ≥ , we write a ≡ b (mod α ) ifwe have v ( a − b ) ≥ α , and a ≡ b (mod α +) if we have v ( a − b ) > α . For a curve X over k ac , wedenote by X c the smooth compactification of the normalization of X , and the genus of X meansthe genus of X c . For an affinoid X , we write X for its reduction. The category of sets is denotedby Set . For a representation τ of a group, the dual representation of τ is denoted by τ ∗ . We takerational powers of ̟ compatibly as needed. 3 Preliminaries
Let Σ denote a formal O K -module of dimension 1 and height 2 over k ac , which is unique up toisomorphism. Let n be a natural number. We define K ( p n ) as in the introduction. In the following,we define the connected Lubin-Tate curve X ( p n ) with level K ( p n ).Let C be the category of Noetherian complete local O b K ur -algebras with residue field k ac . For A ∈ C , a formal O K -module F = Spf A [[ X ]] over A and an A -valued point P of F , the correspondingelement of the maximal ideal of A is denoted by x ( P ). We consider the functor A ( p n ) : C →
Set ; A [( F , ι, P )] , where F is a formal O K -module over A with an isomorphism ι : Σ ≃ F ⊗ A k ac and P is a ̟ n -torsionpoint of F such that Y a ∈O K /̟ n O K (cid:0) X − x ([ a ] F ( P )) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) [ ̟ n ] F ( X )in A [[ X ]]. This functor is represented by a regular local ring R ( p n ) by [Dr, §
4. B) Lemma]. Wewrite X ( p n ) for Spf R ( p n ). Its generic fiber is denoted by X ( p n ), which we call the connectedLubin-Tate curve with level K ( p n ). The space X ( p n ) is a rigid analytic curve over b K ur . We candefine the Lubin-Tate curve LT ( p n ) with level n by changing C to be the category of O b K ur -algebraswhere ̟ is nilpotent, and ι to be a quasi-isogeny Σ ⊗ k ac A/̟A → F ⊗ A A/̟A . We consider LT ( p n )as a rigid analytic curve over b K ur .The ring R (1) is isomorphic to the ring of formal power series O b K ur [[ u ]]. We simply write B (1)for Spf O b K ur [[ u ]]. Let B (1) denote an open unit ball such that B (1)( C ) = { u ∈ C | v ( u ) > } . Thegeneric fiber of B (1) is equal to B (1). Then, the space X (1) is identified with B (1). Let F univ denote the universal formal O K -module over X (1).In this subsection, we choose a parametrization of X (1) ≃ B (1) such that the universal formal O K -module has a simple form. Let F be a formal O K -module of dimension 1 over a flat O K -algebra R . For a nontrivial invariant differential ω on F , a logarithm of F means a unique isomorphism F : F ∼ → G a over R ⊗ K with dF = ω (cf. [GH, 3]). In the sequel, we always take an invariantdifferential ω on F so that a logarithm F has the following form: F ( X ) = X + X i ≥ f i X q i with f i ∈ R ⊗ K. Let F ( X ) = P i ≥ f i X q i ∈ K [[ u, X ]] be the universal logarithm over O K [[ u ]]. By [GH, (5.5),(12.3), Proposition 12.10], the coefficients { f i } i ≥ satisfy f = 1 and ̟f i = P ≤ j ≤ i − f j v q j i − j for i ≥
1, where v = u , v = 1 and v i = 0 for i ≥
3. Hence, we have the following: f = 1 , f = u̟ , f = 1 ̟ (cid:18) u q +1 ̟ (cid:19) , f = 1 ̟ (cid:18) u + u q + u q + q +1 ̟ (cid:19) , · · · . (1.1)By [GH, Proposition 5.7] or [Ha, 21.5], if we set F univ ( X, Y ) = F − ( F ( X ) + F ( Y )) , [ a ] F univ ( X ) = F − ( aF ( X )) (1.2)for a ∈ O K , it is known that these power series have coefficients in O K [[ u ]] and define the universalformal O K -module F univ over O b K ur [[ u ]] of dimension 1 and height 2 with logarithm F ( X ). We havethe following approximation formula for [ ̟ ] u ( X ).4 emma 1.1. We have the following congruence: [ ̟ ] F univ ( X ) ≡ ̟X + uX q + X q − u̟ { ( uX q + X q ) q − u q X q − X q } mod ( ̟ X q , u̟X q , ̟X q , X q +1 ) . Proof.
This follows from a direct computation using the relation F ([ ̟ ] F univ ( X )) = ̟F ( X ) and(1.1).In the sequel, F univ means the universal formal O K -module with the identification X (1) ≃ B (1)given by (1.2), and we simply write [ a ] u for [ a ] F univ . The reduction of (1.2) gives a simple model ofΣ such that X + Σ Y = X + Y, [ ζ ] Σ ( X ) = ¯ ζX for ζ ∈ µ q − ( O K ) , [ ̟ ] Σ ( X ) = X q . (1.3)We put A n = O b K ur [[ u, X n ]] / (cid:0) [ ̟ n ] u ( X n ) / [ ̟ n − ] u ( X n ) (cid:1) . Then there is a natural identification X ( p n ) ≃ Spf A n (1.4)that is compatible with the identification X (1) ≃ B (1). The Lubin-Tate curve X ( p n ) is identifiedwith the generic fiber of the right hand side of (1.4). We set X i = [ ̟ n − i ] u ( X n ) for 1 ≤ i ≤ n − X (1) for X (1). X ( p n ) Let D be the central division algebra over K of invariant 1 /
2. We write O D for the ring of integersof D . In this subsection, we recall the left action of O × D on the space X ( p n ).Let K be the unramified quadratic extension of K . Let k be the residue field of K , and σ ∈ Gal( K /K ) be the non-trivial element. The ring O D has the following description: O D = O K ⊕ ϕ O K with ϕ = ̟ and aϕ = ϕa σ for a ∈ O K . We define an action of O D on Σ by ζ ( X ) = ¯ ζ X for ζ ∈ µ q − ( O K ) and ϕ ( X ) = X q . Then this give an isomorphism O D ≃ End(Σ) by[GH, Proposition 13.10].Let d = d + ϕd ∈ O × D , where d ∈ O × K and d ∈ O K . By the definition of the action of O D on Σ, we have d ( X ) ≡ ¯ d X + ( ¯ d X ) q mod ( X q ) . (1.5)We take a lifting ˜ d ( X ) ∈ O K [[ X ]] of d ( X ) ∈ k [[ X ]]. Let F ˜ d be the formal O K -module defined by F ˜ d ( X, Y ) = ˜ d (cid:0) F univ ( ˜ d − ( X ) , ˜ d − ( Y )) (cid:1) , [ a ] F ˜ d ( X ) = ˜ d (cid:0) [ a ] u ( ˜ d − ( X )) (cid:1) for a ∈ O K . Then, we have an isomorphism˜ d : F univ ∼ −→ F ˜ d ; ( u, X ) ( u, ˜ d ( X )) . By [GH, Proposition 14.7], the formal O K -module F ˜ d withΣ d − −→ Σ ι −→ F univ ⊗ k ac ˜ d ⊗ k ac −−−→ F ˜ d ⊗ k ac gives an isomorphism d : X (1) → X (1) , (1.6)5hich is independent of a choice of a lifting ˜ d , such that there is the unique isomorphism j : d ∗ F univ ∼ −→ F ˜ d ; ( u, X ) ( u, j ( X ))satisfying j ( X ) ≡ X mod ( ̟, u ), where d ∗ F univ denotes the pull-back of F univ over X (1) by themap (1.6). Hence, we have [ ̟ ] d ∗ F univ ( j − ( X )) = j − ([ ̟ ] F ˜ d ( X )) . (1.7)On the other hand, we have the following isomorphism: d ∗ F univ ∼ → F univ ; ( u, X ′ ) ( d ( u ) , X ′ ) . Furthermore, we consider the following isomorphism under the identification (1.4): ψ d : X ( p n ) −→ X ( p n ); ( u, X n ) (cid:0) d ( u ) , j − ( ˜ d ( X n )) (cid:1) , (1.8)which depends only on d as in [GH, Proposition 14.7]. We put d ∗ ( X ) = j − ( ˜ d ( X )) . We define a left action of d on X ( p n ) by[( F , ι, P )] [( F , ι ◦ d − , P )] . Then this action coincides with ψ d by the definition.By (1.5), we have ˜ d − ( X ) = d − X − d − ( q +1)1 d q X q mod ( ̟, X q ) (1.9)in O K [[ X ]]. We use the following lemma later to compute the O × D -action on the stable reductionof X ( p ). Lemma 1.2.
We assume v ( u ) = 1 / (2 q ) . Let d = d + ϕd ∈ O × D . We set u ′ = d ( u ) . We changevariables as u = ̟ / (2 q ) ˜ u and u ′ = ̟ / (2 q ) ˜ u ′ . Then, we have the following: u ′ ≡ d − ( q − u (1 + d − q d u ) mod ( ̟, u ) , (1.10) j − ( X ) ≡ X + d − q d uX mod ( ̟, u X, uX ) . (1.11) Proof.
We set d − = d ′ + ϕd ′ . Then d ′ ≡ d − , d ′ ≡ − d − ( q +1)1 d (mod 1). First, we prove (1.10). If v ( u ) = 1 / (2 q ), a function w ( u ) in [GH, (25.11)] is well-approximated by a function ̟u ( ̟ + u q +1 ) − .By [GH, (25.13)], we have ̟u ′ ̟ + u ′ q +1 ≡ d ′ q ̟u ( ̟ + u q +1 ) − + ̟d ′ q d ′ ̟u ( ̟ + u q +1 ) − + d ′ ≡ ̟u ( d − d q u q ) d q ( ̟ + u q +1 ) − d ̟u (mod 1+) . Hence, we acquire the following by u = ̟ / (2 q ) ˜ u and u ′ = ̟ / (2 q ) ˜ u ′ :˜ u ′ ˜ u ′ q +1 + ̟ q − q ≡ ˜ u ( d − ̟ d q ˜ u q ) d q ˜ u q +1 + ̟ q − q d q − ̟ d ˜ u (mod 12 +) . (1.12)By taking an inverse of the congruence (1.12), we obtain(˜ u ′ − d − ( q − ˜ u ) q ≡ ̟ q − q (cid:18) ˜ u ′ − d − ( q − ˜ ud − ( q − ˜ u ˜ u ′ (cid:19) + ̟ ( d q − d q ˜ u q − d − d ) (mod 12 +) . (1.13)6ow, we set ˜ u ′ − d − ( q − ˜ u = ̟ / (2 q ) x . By substituting this to (1.13) and dividing it by ̟ / , weobtain ( x − d − q d ˜ u ) q ≡ d q − ˜ u − ( x − d − q d ˜ u ) (mod 0+) . Since x is an analytic function of ˜ u , a congruence x ≡ d − q d ˜ u (mod 0+) must hold. Hence wehave ˜ u ′ ≡ d − ( q − ˜ u (1 + ̟ q d − q d ˜ u ) (mod 12 q +)using ˜ u ′ − d q − ˜ u = ̟ / (2 q ) x . This implies (1.10), because u ′ is an analytic function of u .By Lemma 1.1, (1.7) and (1.9), we have u ′ j − ( X ) q ≡ j − ( ud − ( q − X q ) mod ( ̟, X q ). Hence,the assertion (1.11) follows from (1.10) and j − ( X ) ≡ X mod ( ̟, u ). Let ℓ be a prime number different from p . We take an algebraic closure Q ℓ of Q ℓ . Let LT( p n ) bethe Lubin-Tate curve with full level n over b K ur (cf. [Da, 3.2]). We put H i LT ,̟ = lim −→ n H i c (cid:0) (LT( p n ) /̟ Z ) C , Q ℓ (cid:1) for any non-negative integer i , where LT( p n ) /̟ Z denote the quotient of LT( p n ) by the action of ̟ Z ⊂ D × . Then we can define an action of GL ( K ) × D × × W K on H i LT ,̟ for a non-negativeinteger i (cf. [Da, 3.2, 3.3]).We write Irr( D × , Q ℓ ) for the set of isomorphism classes of irreducible smooth representations of D × over Q ℓ , and Disc(GL ( K ) , Q ℓ ) for the set of isomorphism classes of irreducible discrete seriesrepresentations of GL ( K ) over Q ℓ . LetJL : Irr( D × , Q ℓ ) → Disc(GL ( K ) , Q ℓ )be the local Jacquet-Langlands correspondence. We denote by LJ the inverse of JL. For anirreducible smooth representation π of GL ( K ), let ω π denote the central character of π . We writeSt for the Steinberg representation of GL ( K ).The following fact is well known as a corollary of the Deligne-Carayol conjecture. Here, we givea purely local proof of this fact. Proposition 2.1.
We have isomorphisms H ,̟ ≃ M π π ⊕ π ) ⊕ M χ (St ⊗ ( χ ◦ det)) ,H ,̟ ≃ M χ ( χ ◦ det) as representations of GL ( K ) , where π runs through irreducible cuspidal representations of GL ( K ) such that ω π ( ̟ ) = 1 , and χ runs through characters of K × satisfying χ ( ̟ ) = 1 .Proof. First, we show the second isomorphism. Let X ( p n ) be the connected Lubin-Tate curve withfull level n over b K ur (cf. [St2, 2.1]). We put H X = lim −→ n H (cid:0) X ( p n ) C , Q ℓ (cid:1) , GL ( K ) = { g ∈ GL ( K ) | det g ∈ O × K } . ( K ) acts on H X . By [St2, Theorem 4.4 (i)], we have H X ≃ M χ ( χ ◦ det) (2.1)as representations of GL ( O K ), where χ runs through characters of O × K . Let H be the kernel ofGL ( K ) → Aut( H X ). Then H = SL ( K ), because a normal subgroup of GL ( K ) containingSL ( O K ) is SL ( K ) by [De, Lemme 2.2.5 (iii)]. Hence, we see that (2.1) is an isomorphism asrepresentations of GL ( K ) . The second isomorphism follows from this, because we have H ,̟ ≃ c-Ind GL ( K ) /̟ Z GL ( K ) H X . Next, we show the first isomorphism. By [Mi, Definition 6.2 and Theorem 6.6], the cuspidalpart of H ,̟ is M π π ⊕ π ) . Here, we note that the characteristic of a local field is assumed to be zero in [Mi], but the sameproof works in the equal characteristic case. By [Far2, Th´eor`eme 4.3] and the Faltings-Farguesisomorphism (cf. [Fal] and [FGL]), we see that the non-cuspidal part of H ,̟ is the Zelevinskydual of H ,̟ . Therefore, we have the first isomorphism. In this section, we construct a stable covering of X ( p ). Let ( u, X ) be the parameter of X ( p )given by the identification (1.4).Let Y , , W , W k × , W ∞ , , W ∞ , and W ∞ , be the subspaces of X ( p ) defined by the followingconditions respectively: Y , : v ( u ) = 1 q + 1 , v ( X ) = qq − , v ( X ) = 1 q ( q − . W : 0 < v ( u ) < q + 1 , v ( X ) = 1 − v ( u ) q − , v ( X ) = 1 − qv ( u ) q ( q − . W k × : 0 < v ( u ) < q + 1 , v ( X ) = 1 − v ( u ) q − , v ( X ) = v ( u ) q ( q − . W ∞ , : 0 < v ( u ) < qq + 1 , v ( X ) = v ( u ) q ( q − , v ( X ) = v ( u ) q ( q − . W ∞ , : v ( u ) ≥ qq + 1 , v ( X ) = 1 q − , v ( X ) = 1 q ( q − . W ∞ , : 1 q + 1 < v ( u ) < qq + 1 , v ( X ) = 1 − v ( u ) q − , v ( X ) = 1 − v ( u ) q ( q − . We put W ∞ = W ∞ , ∪ W ∞ , ∪ W ∞ , . Note that we have X ( p ) = Y , ∪ W ∪ W k × ∪ W ∞ . Proposition 3.1.
The Lubin-Tate curve X ( p ) is a basic wide open space with underlying affinoid Y , . Further, W and W ∞ are open annuli, and W k × is a disjoint union of q − open annuli. roof. This is proved in [IT1] by direct calculations without cohomological arguments. Here, wesketch another proof based on arguments in this paper.First, we note that X (1) is a good formal model of X (1). Then, we can show that X ( p ) isisomorphic to an open annulus by a cohomological argument as in the proof of Theorem 7.14 usingthe natural level-lowering map X ( p ) → X (1).Next, we can see that the reduction of Y , is isomorphic to the affine curve defined by x q y − xy q = 1 by a calculation as in the proof of Proposition 4.2 (cf. [IT1, § X ( p ) → X ( p ). ( p ) ( p ) In this subsection, we define several subspaces of X ( p ). Let ( u, X ) be the parameter of X ( p )given by the identification (1.4).Let Y , , Y , and Z , be the subspaces of X ( p ) defined by the following conditions respec-tively: Y , : v ( u ) = 1 q + 1 , v ( X ) = qq − , v ( X ) = 1 q ( q − , v ( X ) = 1 q ( q − . Y , : v ( u ) = 1 q ( q + 1) , v ( X ) = q + q − q ( q − , v ( X ) = 1 q − , v ( X ) = 1 q ( q − . Z , : v ( u ) = 12 q , v ( X ) = 2 q − q ( q − , v ( X ) = 12 q ( q − , v ( X ) = 12 q ( q − . We write down the following possible cases for ( u, X , X ):1 . < v ( u ) < q + 1 , v ( X ) = 1 − v ( u ) q − , v ( X ) = 1 − qv ( u ) q ( q − , . < v ( u ) < q + 1 , v ( X ) = 1 − v ( u ) q − , v ( X ) = v ( u ) q ( q − , . v ( u ) = 1 q + 1 , v ( X ) = qq − , v ( X ) = 1 q ( q − , . < v ( u ) < qq + 1 , v ( X ) = v ( u ) q ( q − , v ( X ) = v ( u ) q ( q − , . v ( u ) ≥ qq + 1 , v ( X ) = 1 q − , v ( X ) = 1 q ( q − , . q + 1 < v ( u ) < qq + 1 , v ( X ) = 1 − v ( u ) q − , v ( X ) = 1 − v ( u ) q ( q − . (4.1)Next, we consider the following possible cases for ( X , X ):1 ′ . v ( X q ) = v ( X ) < v ( uX q ) , ′ . v ( uX q ) = v ( X ) < v ( X q ) , ′ . v ( X ) > v ( X q ) = v ( uX q ) , ′ . v ( X ) = v ( X q ) = v ( uX q ) . (4.2) Lemma 4.1.
For ≤ i ≤ in (4.1) and ′ ≤ j ′ ≤ ′ in (4.2) , the case i and j ′ does not happen.Proof. This is an easy exercise.Let W i,j ′ be the subspace of X ( p ) defined by the conditions 1 ≤ i ≤ ′ ≤ j ′ ≤ ′ in (4.2). We note that W , ′ = Y , and W , ′ = Y , . Let W +1 , ′ and W − , ′ be the subspaces of W , ′ defined by 1 / (2 q ) < v ( u ) < / ( q + 1) and 1 / ( q ( q + 1)) < v ( u ) < / (2 q ) respectively.9 .2 Reductions of the affinoid spaces Y , and Y , In this subsection, we compute the reductions of the affinoid spaces Y , and Y , . The reductionsof Y , and Y , are isomorphic to the affine curve defined by x q y − xy q = 1. These curves havegenus q ( q − / Proposition 4.2.
The reduction of Y , is isomorphic to the affine curve defined by x q y − xy q = 1 .Proof. We change variables as u = ̟ / ( q +1) ˜ u , X = ̟ q/ ( q − x , X = ̟ / ( q ( q − x and X = ̟ / ( q ( q − x . By Lemma 1.1, we have˜ u ≡ − x − ( q − , x ≡ ˜ ux q + x q , x ≡ x q (mod 0+) . (4.3)Then we have ˜ u = − x − ( q − + F (˜ u, x ) for some function F (˜ u, x ) satisfying v ( F (˜ u, x )) > v (˜ u ).Substituting ˜ u = − x − ( q − + F (˜ u, x ) to F (˜ u, x ) and repeating it, we see that ˜ u is written as afunction of x . Similarly, by x ≡ x q (mod 0+), we can see that x is written as a function of x and x . By (4.3), we acquire 1 ≡ x q x − x q x q (mod 0+) . (4.4)By setting 1 + x − x q = x q t − and substituting this to (4.4), we obtain t q ≡ x (mod 0+) andhence (1 + x q t − ) q ≡ x q t − (mod 0+). By setting 1 + x q t − = x q t − , we obtain t q ≡ t (mod 0+).Hence (1 + x t − ) q ≡ x q t − (mod 0+). Finally, by setting x = x and 1 + x t − = x q y , we acquire y q ≡ t − (mod 0+). Hence we have x q y − xy q ≡ x = x , y = x (1 + x q ( q − + x ( q +1)( q − ) + x q x x q + q − , (4.5)which we will use later.We put γ i = ̟ q − qi for 1 ≤ i ≤
4. We choose an element c such that c q − γ c + 1 = 0. Note that we have c ≡ − q -th root c /q of c . Proposition 4.3.
The reduction of the space Y , is isomorphic to the affine curve defined by x q y − xy q = 1 .Proof. We change variables as u = ̟ / ( q ( q +1)) ˜ u , X = ̟ ( q + q − / ( q ( q − x , X = ̟ / ( q − x , and X = ̟ / ( q ( q − x . By Lemma 1.1, we have˜ u ≡ − x − ( q − (mod q − q +) , (4.6) x ≡ ˜ ux q + γ ( x q + x ) (mod q − q +) , (4.7) x ≡ x q + ˜ ux q (mod q − q +) . (4.8)By (4.6) and (4.8), we can see that ˜ u is written as a function of x , and that x is written as afunction of x and x . We define a parameter t by x x = c + γ x q t . (4.9)10e note that v ( t ) = 0. By considering x − × (4.7), we have (cid:18) x x (cid:19) q + 1 − γ x x ≡ γ x q x (mod q − q +) . (4.10)By substituting (4.9) to the left hand side of the congruence (4.10), and dividing it by γ x q , weacquire x ≡ t q (cid:18) − γ t q − x q ( q − (cid:19) − (mod q − q +) . (4.11)By this congruence, we can see that x is written as a function of t and x . By considering x − × (4.8), we acquire c + γ x q t ≡ x q x − (cid:18) x x (cid:19) q (mod q − q +) (4.12)by (4.9). Substituting (4.11) to (4.12), we have (cid:18) c /q − x q t + x x (cid:19) q ≡ − γ ( x + x ) q tx q ( q − (mod q − q +) . (4.13)By (4.9) and c ≡ − x ≡ − x (mod 0+). Therefore, we acquire( x + x ) q ≡ x q − x q (mod 0+)by (4.6) and (4.8). In particular, we obtain v ( x + x ) = 0. We introducing a new parameter t as c /q − x q t + x x = − γ q ( x + x ) q t x q − . (4.14)Substituting this to the left hand side of the congruence (4.13), and dividing it by − γ x − q ( q − ( x + x ) q , we acquire t ≡ t q (mod 0+). By this congruence, we can see that t is written as a functionof t and x . By (4.14), we obtain x ≡ t q (1 + x t − ) q (mod 0+)using t ≡ t q (mod 0+) and x ≡ t q (mod 0+). Hence, by setting x = t − and y = t q (1 + x t − ),we acquire x q y − yx q ≡ , In this subsection, we calculate the reduction of the affinoid space Z , . We define S as in theintroduction. The reduction Z , is isomorphic to the affine curve defined by Z q + x q − + x − ( q − =0. This affine curve has genus 0 and singularities at x ∈ S .We put ω i = ̟ qi ( q − , ǫ i = 12 q i for 1 ≤ i ≤
4. We change variables as u = ω q − ˜ u , X = ω q − x , X = ω x and X = ω x . ByLemma 1.1, we have ˜ u ≡ − x − ( q − (mod 12 +) , (4.15) x ≡ ˜ ux q + γ x q + γ x (mod 12 +) , (4.16) x ≡ x q + γ ˜ ux q (mod ǫ +) . (4.17)11ote that we have v ( γ ) > / q = 2. By (4.15) and (4.17), we can see that ˜ u is written as afunction of x , and that x is written as a function of x and x . We define a parameter t by x x = − γ x q t . (4.18)By considering x − × (4.16), we acquire (cid:18) x x + 1 (cid:19) q ≡ γ x q x (cid:18) γ x q − (cid:19) (mod 12 +) (4.19)by (4.15). Substituting (4.18) to (4.19), and dividing it by γ x q , we obtain x ≡ t q (cid:18) γ x q − (cid:19) (mod ǫ +) . (4.20)Therefore we have v ( t ) = 0. By considering x − × (4.17), we acquire (cid:18) x q t (cid:19) q − γ x q t q x q − ≡ γ (cid:18) x q t + (cid:18) x x (cid:19) q (cid:19) (mod ǫ +) (4.21)by (4.15), (4.18) and (4.20). We define a parameter Z by1 + x q t = γ Z . (4.22)We note that v ( Z ) ≥
0. Substituting this to (4.21), and dividing it by γ , we obtain Z q ≡ x q t + (cid:18) x x (cid:19) q + γ q − x q t q x q − (mod ǫ +) . (4.23)By (4.22) and (4.23), we acquire (cid:18) Z + x x − x x (cid:19) q ≡ γ (cid:18) x x (cid:19) q Z + γ q − x q t q x q − (mod ǫ +) . (4.24)We introduce a new parameter Z as Z + x x − x x = γ x x Z. (4.25)We note that v ( Z ) ≥
0. Substituting this to the left hand side of the congruence (4.24), anddividing it by γ ( x /x ) q , we acquire Z q ≡ Z + γ q − q − x q ( q +1)3 t q x q + q − (mod ǫ +) . (4.26)By substituting (4.25) to (4.26), we obtain Z q + x q − (1 − γ Z ) + x − ( q − ≡ − γ q − q − x − q ( q − q +1)3 (mod ǫ +) (4.27)by (4.17), (4.20) and (4.22). Note that we have v ( γ q − q − ) > ǫ , if q = 2.12 roposition 4.4. The reduction of the space Z , is isomorphic to the affine curve defined by Z q + x q − + x − ( q − = 0 . This affine curve has genus and singularities at x ∈ S .Proof. The required assertion follows from the congruence (4.27) modulo 0+.
Definition 4.5.
1. For any ζ ∈ S , we define a subspace D ζ ⊂ Z , × b K ur b K ur ( ω ) by ¯ x = ζ . We call the space D ζ a singular residue class of Z , .2. We define a subspace Z , ⊂ Z , × b K ur b K ur ( ω ) by the complement Z , × b K ur b K ur ( ω ) \ S ζ ∈S D ζ . Proposition 4.6.
The reduction of the space Z , is isomorphic to the affine curve defined by Z q + x q − + x − ( q − = 0 with x / ∈ S .Proof. This follows from Proposition 4.4. , In this subsection, we analyze the singular residue classes {D ζ } ζ ∈S of Z , . If q is odd, the space D ζ is a basic wide open space with an underlying affinoid X ζ , whose reduction X ζ is isomorphicto the affine curve defined by z q − z = w . On the other hand, if q is even, the situation is slightlycomplicated, because the space D ζ is not basic wide open. Hence, we have to cover D ζ by smallerbasic wide open spaces. As a result, in D ζ , we find an affinoid P ζ , whose reduction is isomorphicto the affine curve defined by z f +1 = w ( w q − − . This affine curve has q − w ∈ k × . Then, by analyzing the tubular neighborhoods of these singular points, we find anaffinoid X ζ,ζ ′ ⊂ P ζ for each ζ ′ ∈ k × , whose reduction is isomorphic to the affine curve defined by z + z = w . q : odd We assume that q is odd. For each ζ ∈ µ q − ( k ac ), we define an affinoid X ζ ⊂ D ζ and computeits reduction X ζ .For ι ∈ µ ( k ac ), we choose an element c ′ ,ι ∈ O × K ac such that ¯ c ′ ,ι = − ι and c ′ q ,ι = 4(1 − γ c ′ ,ι ).We take ζ ∈ µ q − ( k ac ). We put c ,ζ = c ′ ,ζ q − , and define c ,ζ ∈ O × K ac by c q − ,ζ = − c − q ,ζ and¯ c ,ζ = ζ . We put a ζ = ω q − c q +12 ,ζ , b ζ = − ζ q − ω q − c − q ,ζ c q +32 ,ζ . Note that we have v ( a ζ ) = 1 / (2 q ) and v ( b ζ ) = 1 / (4 q ).For an element ζ ∈ µ q − ( k ac ), we define an affinoid X ζ by v ( x − c ,ζ ) ≥ / (4 q ). We changevariables as Z = a ζ z + c ,ζ , x = b ζ w + c ,ζ . Then, we acquire a qζ ( z q − z − w ) ≡ ǫ +)by (4.27). Dividing this by a qζ , we have z q − z = w (mod 0+). Hence, the reduction of X ζ isisomorphic to the affine curve defined by z q − z = w . Proposition 4.7.
For each ζ ∈ µ q − ( k ac ) , the reduction X ζ is isomorphic to the affine curvedefined by z q − z = w and the complement D ζ \ X ζ is an open annulus. roof. We have already proved the first assertion. We prove the second assertion. We changevariables as Z = z ′ + c ,ζ , x = w ′ + c ,ζ with 0 < v ( w ′ ) < / (4 q ). Substituting them to (4.27), we obtain z ′ q ≡ w ′ (mod 2 v ( w ′ )+) . Note that we have 0 < v ( z ′ ) < / (2 q ). By setting w ′ = z ′′ z ′ ( q − / , we acquire z ′′ ≡ z ′ (mod v ( z ′ )+) . Hence, we can see that z ′ is written as a function of z ′′ . Then w ′ is also written as a function of z ′′ . Therefore, ( D ζ \ X ζ )( C ) is identified with { z ′′ ∈ C | < v ( z ′′ ) < / (4 q ) } . q : even We assume that q is even. We put Z = x q − . Then, the congruence (4.27) has the following form: Z q + Z (1 − γ Z ) + Z − ≡ − γ q − q − Z − q ( q +1)1 (mod ǫ +) . (4.28)
1. Projective lines
For each ζ ∈ k × , we define subaffinoid P ζ ⊂ D ζ by v ( Z ) ≥ / (4 q ). Wechange variables as Z = ̟ q w , Z = 1 + ̟ q z . Substituting these to (4.28) and dividing it by ̟ / (4 q ) , we acquire( z + w q ) + ̟ q z + ̟ q z + ̟ q − q w + ̟ q − q z w ≡ ̟ q − q (mod 14 q +) . (4.29)We can check that v ( z ) ≥
0. We set q = 2 f and put l i = (2 i − q i , m i = 12 i +2 q for 1 ≤ i ≤ f + 1. Furthermore, we define parameters z i for 2 ≤ i ≤ f + 1 by z i + w l i = ̟ m i +1 z i +1 for 1 ≤ i ≤ f . (4.30) Lemma 4.8.
We assume that v ( Z ) ≥ / (4 q ) . Then we have z f +1 + w q − + w + ̟ q z f +1 w q ≡ ( q/ ̟ q (mod 14 q +) . (4.31) Proof. If q = 2, we can check that z + w + w + ̟ z w ≡ ̟ ( w z + z + w + 1) (mod 164 +) (4.32)by z = − w + ̟ z . We have v ( z + w + w ) >
0. Therefore, we obtain w z + z + w ≡ w ( z + w + w ) ≡ . f ≥
2. For 1 ≤ i ≤ f +1,we put n i = q − i − i +1 q . We prove ( z i + w l i ) + ̟ m i z i w q + ̟ n i w ≡ i +1 q +) (4.33)for 2 ≤ i ≤ f + 1 by induction on i . Eliminating z from (4.29) by (4.30) and dividing it by ̟ / (8 q ) ,we obtain ( z + w q ) + ̟ q z w q + ̟ q − q w + ̟ q w q ( z + w q ) ≡ q +) . This shows v (cid:16) z + w q (cid:17) ≥ q . Hence we have (4.33) for i = 2. Assuming (4.33) for i . Eliminating z i from (4.33) by (4.30) anddividing it by ̟ m i , we obtain (4.33) for i + 1. Hence, we have (4.33) for f + 1, which is equivalentto (4.31). Proposition 4.9.
For each ζ ∈ k × , the reduction P ζ is isomorphic to the affine curve defined by z f +1 = w ( w q − − , which has genus and singularities at w ∈ k × , and the complement D ζ \ P ζ is an open annulus.Proof. The claim on P ζ follows from the congruence (4.31) modulo 0+. We prove the last assertion.We change a variable as Z = 1 + z ′ with 0 < v ( z ′ ) < / (8 q ). Similarly as (4.30), we introduceparameters { z ′ i } ≤ i ≤ f +1 by z ′ i + Z l i = z ′ i +1 for 1 ≤ i ≤ f . Then, by similar computations to thosein the proof of Lemma 4.8, we obtain z ′ f +1 ≡ Z q − (mod 2 v ( z ′ f +1 )+) . By setting z ′ f +2 = Z q /z ′ f +1 , we obtain z ′ f +2 ≡ Z (mod v ( Z )+) . Then we can see that all parameters z ′ i for 1 ≤ i ≤ f + 1 and Z are written as functions of z ′ f +2 .Hence, ( D ζ \ P ζ )( C ) is identified with { z ′ f +2 ∈ C | < v ( z ′ f +2 ) < / (8 q ) } .
2. Elliptic curves
For ζ ′ ∈ k × , we choose c ,ζ ′ ∈ O × C such that ¯ c ,ζ ′ = ζ ′ and c q − ,ζ ′ + 1 + ̟ q c q − ,ζ ′ = 0 , and a square root c / ,ζ ′ of c ,ζ ′ . Further, we choose c ,ζ ′ such that c ,ζ ′ + ̟ q c q ,ζ ′ c ,ζ ′ + c ,ζ ′ ( c q − ,ζ ′ + 1) = q ̟ q , and b ,ζ ′ such that b ,ζ ′ = ̟ / (4 q ) c ,ζ ′ . We put a ,ζ ′ = ̟ q c q ,ζ ′ , b ,ζ ′ = c q − ,ζ ′ b ,ζ ′ . ζ ′ ∈ k × , we define a subspace D ζ,ζ ′ ⊂ P ζ by v ( w − c ,ζ ′ ) >
0. Furthermore, we define X ζ,ζ ′ ⊂ D ζ,ζ ′ by v ( w − c ,ζ ′ ) ≥ / (12 q ). We put P ζ = P ζ \ [ ζ ′ ∈ k × D ζ,ζ ′ . We take ( ζ , ζ ′ ) ∈ k × × k × and compute the reduction of X ζ,ζ ′ . In the sequel, we omit thesubscript ζ ′ of a ,ζ ′ , b ,ζ ′ , b ,ζ ′ c ,ζ ′ and c ,ζ ′ , if there is no confusion. We change variables as z f +1 = a z + b w + c , w = b w + c . By substituting these to (4.31), we acquire a ( z + z + w ) ≡ q +) (4.34)by the definition of a , b , b , c and c . Proposition 4.10.
For each ( ζ , ζ ′ ) ∈ k × × k × , the reduction of X ζ,ζ ′ is isomorphic to the affinecurve defined by z + z = w and the complement D ζ,ζ ′ \ X ζ,ζ ′ is an open annulus.Proof. The first assertion follows from (4.34). We prove the second assertion. We change variablesas z f +1 = z ′ + c q − w ′ + c , w = w ′ + c with 0 < v ( w ′ ) < / (12 q ). Substituting them to (4.31), we acquire z ′ ≡ c q − w ′ (mod 2 v ( z ′ )+)by the choice of c . Note that we have v ( z ′ ) = 3 v ( w ′ ) / < q . By setting z ′′ = z ′ / ( c q − w ′ ), we obtain z ′′ ≡ w ′ (mod v ( w ′ )+) . Then we can see that z ′ and w ′ are written as functions of z ′′ . Hence, ( D ζ,ζ ′ \ X ζ,ζ ′ )( C ) is identifiedwith { z ′′ ∈ C | < v ( z ′′ ) < / (24 q ) } . ( p ) In this subsection, we show the existence of the stable covering of X ( p ) over some finite extensionof the base field b K ur . See [CM, Section 2.3] for the notion of semi-stable coverings. A semi-stablecovering is called stable, if the corresponding semi-stable model is stable. Proposition 4.11.
There exists a stable covering of X ( p ) over a finite extension of the base field.Proof. First, we show that, after taking a finite extension of the base field, X ( p ) is a wide openspace. By [St1, Theorem 2.3.1 (i)], X ( p ) is the Raynaud generic fiber of the formal completion ofan affine scheme over O b K ur at a closed point on the special fiber. Then we can apply [CM, Theorem2.29] to the formal completion of the affine scheme along its special fiber, after shrinking the affinescheme. Hence, X ( p ) is a wide open space over some extension.By [CM, Theorem 2.18], a wide open space can be embedded to a proper algebraic curve so thatits complement is a disjoint union of closed disks. Therefore, X ( p ) has a semi-stable coveringover some finite extension by [CM, Theorem 2.40]. Then a simple modification gives a stablecovering. 16n the following, we construct a candidate of a semi-stable covering of X ( p ) over some finiteextension. We put V = W +1 , ′ ∪ [ ≤ i ≤ W i, ′ , V = W − , ′ ∪ [ ≤ i ≤ W ,i ′ , U = W , ′ \ [ ζ ∈S X ζ . We note that V ⊃ Y , , V ⊃ Y , , U ⊃ Z , , V ∩ V = ∅ , V ∩ U = W +1 , ′ and V ∩ U = W − , ′ .We consider the case where q is even in this paragraph. We set ˆ D ζ = D ζ \ (cid:0)S ζ ′ ∈ k × X ζ,ζ ′ (cid:1) for ζ ∈ k × . Then, ˆ D ζ contains P ζ as the underlying affinoid. On the other hand, for ( ζ , ζ ′ ) ∈ k × × k × the space D ζ,ζ ′ has the underlying affinoid X ζ,ζ ′ .We put S = ( S if q is odd, k × × k × if q is even.Now, we define an admissible covering of X ( p ) as C ( p ) = ( { V , V , U , {D ζ } ζ ∈S } if q is odd, { V , V , U , { ˆ D ζ } ζ ∈ k × , {D ζ,ζ ′ } ( ζ,ζ ′ ) ∈S if q is even.In Subsection 7.2, we will show that C ( p ) is a semi-stable covering of X ( p ) over some finiteextension. In this section, we determine the action of of O × D on the reductions Y , , Y , Z , , { P ζ } ζ ∈ k × and { X ζ } ζ ∈S by using the description of O × D -action in (1.8). We take d = d + ϕd ∈ O × D , where d ∈ O × K and d ∈ O K . We put κ ( d ) = ¯ d , κ ( d ) = − ¯ d − q ¯ d . Lemma 5.1.
The element d induces the following morphisms: Y , → Y , ; ( x, y ) ( κ ( d ) x, κ ( d ) − q y ) , Y , → Y , ; ( x, y ) ( κ ( d ) − x, κ ( d ) q y ) . Proof.
We prove the assertion for Y , . By (1.5), we have d ∗ x ≡ d x , d ∗ x ≡ d x (mod 0+) . Therefore, the required assertion follows from (4.5). The assertion for Y , is proved similarly.Now, let the notation be as in Subsection 4.3. We put x ′ i = d ∗ x i for 1 ≤ i ≤ ,t ′ = d ∗ t, Z ′ = d ∗ Z , Z ′ = d ∗ Z. We have j − ( x ) ≡ x + d − q d ̟ ǫ ˜ ux (mod ǫ +) ,j − ( x ) ≡ x + d − q d ̟ ǫ ˜ ux (mod ǫ +) ,j − ( x ) ≡ x (mod ǫ +)17y (1.11). On the other hand, we have˜ d ( x ) ≡ d x (mod ǫ +) , ˜ d ( x ) ≡ d x + d q ̟ ǫ x q (mod ǫ +) , ˜ d ( x ) ≡ d x + d q ̟ ǫ x q (mod ǫ +)by (1.5). Hence, we obtain x ′ ≡ d x + d − ( q − d ̟ ǫ ˜ ux (mod ǫ +) , (5.1) x ′ ≡ d x + d − ( q − d ̟ ǫ ˜ ux + d q ̟ ǫ x q (mod ǫ +) , (5.2) x ′ ≡ d x + d q ̟ ǫ x q (mod ǫ +) . (5.3)By the definition of t and the equation x ′ /x ′ = − γ ( x ′ q /t ′ ), we acquire t ′ ≡ d q t − d q − d q t − q ̟ ǫ (mod ǫ +) (5.4)using (5.1) and (5.2). We put G = d − q d x q ( q − + d − d q x − q ( q − . By the definition of Z and the equation 1 + ( x ′ q /t ′ ) = γ Z ′ , we obtain Z ′ ≡ Z − ̟ ǫ G (mod ǫ +) (5.5)using (5.3) and (5.4). We put G = G + d − d q ( x x q − + x − x q ) . By the definition of Z and the equation Z ′ + ( x ′ /x ′ ) − ( x ′ /x ′ ) = γ ( x ′ /x ′ ) Z ′ , we obtain Z ′ ≡ Z − x x ̟ ǫ G (mod ǫ +) (5.6)using (5.1), (5.2), (5.3) and (5.5). We have G ≡ d − q d x q ( q − + d − d q x ( q − q +2)3 (mod 0+)by x ≡ − x q , x ≡ x q (mod 0+). We put∆ = d − q d x − ( q − + d − d q x q − . Then the congruence (5.6) has the following form: Z ′ ≡ Z − ̟ ǫ ∆ (mod ǫ +) . (5.7) Proposition 5.2.
The element d acts on Z , by ( Z, x ) ( Z, κ ( d ) x ) .Proof. This follows from (5.3) and (5.7).
Proposition 5.3.
The element d induces the morphism P ζ → P κ ( d ) ζ ; w w . Proof.
This follows from (5.7), Proposition 5.2 and Z = ̟ / (4 q ) w .18 roposition 5.4. We take ζ ∈ S . Further, we take ζ ′ ∈ k × , if q is even. We set as follows: η = ( ζ if q is odd, ( ζ , ζ ′ ) if q is even, dη = ( κ ( d ) ζ if q is odd, ( κ ( d ) ζ , ζ ′ ) if q is even, f d = ( Tr k /k ( ζ − q κ ( d )) if q is odd, Tr k / F ( ζ − q ζ ′− κ ( d )) if q is even,where η, dη ∈ S . Then, the element d induces X η → X dη : ( ( z, w ) ( κ ( d ) − ( q +1) ( z + f d ) , κ ( d ) − ( q +1) / w ) if q is odd, ( z, w ) ( z + f d , w ) if q is even.Proof. First, we assume that q is odd. Recall that Z = a ζ z + c ,ζ and x = b ζ w + c ,ζ . Similarly,we have Z ′ = a ¯ d ζ z ′ + c , ¯ d ζ and x = b ¯ d ζ w ′ + c , ¯ d ζ . Then, the claim follows from (5.7).Next, we assume that q is even. By (5.7) and d ∗ x ≡ d x (mod ( ǫ / d ∗ z f +1 − z f +1 ≡ ̟ ǫ f X i =1 w q − i ∆ i − (mod ǫ v ( Z ) ≥ ǫ /
2. By z f +1 = a ,ζ ′ z + b ,ζ ′ w + c ,ζ ′ and w = b ,ζ ′ w + c ,ζ ′ , we obtain d ∗ z − z ≡ f X i =1 c − i ,ζ ′ ∆ i − (mod 0+) ,d ∗ w ≡ w (mod ǫ X ζ,ζ ′ by (5.7) and (5.8). On the other hand, we have f X i =1 ¯ c − i ,ζ ′ ∆ i − = f d , because ¯ x = ζ and ¯ c ,ζ ′ = ζ ′ . Hence, we have proved the claim. In this section, we compute the actions of the Weil group on the reductions Y , , Y , , Z , , { P ζ } ζ ∈ k × and { X η } η ∈S .Let X be a reduced affinoid over C with an action of W K . For P ∈ X ( C ), the image of P under the natural reduction map X ( C ) → X ( k ac ) is denoted by P . The action of W K on X is ahomomorphism w X : W K → Aut( X )characterized by σ ( P ) = w X ( σ )( P ) for σ ∈ W K and P ∈ X ( C ). For σ ∈ W K , we define r σ ∈ Z sothat σ induces the q − r σ -th power map on the residue field of K ac . Remark 6.1.
In the usual sense, W K does not act on X ( p ) , because the action of W K does notpreserve the connected components of LT ( p ) . Precisely, w X is the action of { ( σ, ϕ − r σ ) ∈ W K × D × } , which preserves the connected components of LT ( p ) . .1 Actions of the Weil group on Y , , Y , and Z , For σ ∈ W K , we put λ ( σ ) = σ ( ̟ / ( q − ) /̟ / ( q − ∈ k × . We note that λ is not a group homomorphism in general. Lemma 6.2.
Let σ ∈ W K . Then, the element σ induces the automorphisms Y , → Y , ; ( x, y ) ( λ ( σ ) q x q − rσ , λ ( σ ) − y q − rσ ) , Y , → Y , ; ( x, y ) ( λ ( σ ) − x q − rσ , λ ( σ ) q y q − rσ ) as schemes over k .Proof. We prove the claim for Y , . We set σ ( ̟ q q − ) = ξ̟ q q − with ξ ∈ µ q ( q − ( K ac ). Let P ∈ Y , ( C ). We have X ( σ ( P )) = σ ( X ( P )). By applying σ to X ( P ) = ̟ / ( q ( q − x ( P ), we obtain x ( σ ( P )) = ξσ ( x ( P )) ≡ ξx ( P ) q − rσ (mod 0+) . In the same way, we have x ( σ ( P )) ≡ ξ q x ( P ) q − rσ (mod 0+) . Therefore, we acquire x σ = ¯ ξx q − rσ and y σ = ¯ ξ − q y q − rσ by (4.5). Hence, the claim follows from¯ ξ = λ ( σ ) q . We can prove the claim for Y , similarly.For σ ∈ W K , we put ξ σ = σ ( ω ) ω ∈ µ q ( q − ( K ac ) . Lemma 6.3.
Let σ ∈ W K . Then, σ acts on Z , by ( Z, x ) ( Z q − rσ , ¯ ξ σ x q − rσ ) .Proof. We use the notation in Subsection 4.3. Let P ∈ Z , ( C ). Since we set X = ω q − x , X = ω x and X = ω x , we have x ( σ ( P )) = ξ q (2 q − σ σ ( x ( P )) ,x ( σ ( P )) = ξ q σ σ ( x ( P )) ,x ( σ ( P )) = ξ σ σ ( x ( P )) . Hence, we obtain x ( σ ( P )) x ( σ ( P )) = ξ − q ( q − σ σ (cid:18) x ( P ) x ( P ) (cid:19) ≡ σ (cid:18) x ( P ) x ( P ) (cid:19) (mod ǫ +) . Since we set x /x = − γ ( x q /t ), we acquire t ( σ ( P )) ≡ ξ q σ σ ( t ( P )) (mod ǫ +) . Therefore, we obtain x ( σ ( P )) q t ( σ ( P )) = ξ − q ( q − σ σ (cid:18) x ( P ) q t ( P ) (cid:19) ≡ σ (cid:18) x ( P ) q t ( P ) (cid:19) (mod ǫ +) . x q /t ) = γ Z , we obtain Z ( σ ( P )) ≡ σ ( Z ( P )) (mod ǫ +) . Therefore we acquire Z ( σ ( P )) ≡ σ ( Z ( P )) (mod ǫ +) (6.1)by Z + ( x /x ) − ( x /x ) = γ ( x /x ) Z .The assertion follows from x ( σ ( P )) = ξ σ σ ( x ( P )) ≡ ξ σ x ( P ) q − rσ (mod 0+)and (6.1). η In this subsection, let ζ ∈ µ q − ( k ac ). Until Lemma 6.8, let σ ∈ W K . q : odd We assume that q is odd. We use the notation in Paragraph 4.4.1. By (6.1) and x ( σ ( P )) = ξ σ σ ( x ( P )), we have a ¯ ξ σ ζ q − rσ z ( σ ( P )) + c , ¯ ξ σ ζ q − rσ = Z ( σ ( P )) ≡ σ ( Z ( P ))= σ ( a ζ ) σ ( z ( P )) + σ ( c ,ζ ) (mod ǫ +) (6.2)and b ¯ ξ σ ζ q − rσ w ( σ ( P )) + c , ¯ ξ σ ζ q − rσ = x ( σ ( P )) = ξ σ σ ( x ( P ))= ξ σ σ ( b ζ ) σ ( w ( P )) + ξ σ σ ( c ,ζ ) (6.3)for P ∈ X ζ ( C ). Note that c , ¯ ξ σ ζ q − rσ = c ,ζ and c , ¯ ξ σ ζ q − rσ = ξ q σ ζ q − rσ − c ,ζ . We have v ( σ ( c ,ζ ) − c ,ζ ) ≥ ǫ by (6.2). We put a σ,ζ = σ ( a ζ ) ζ r σ ( q − ξ q +1 σ a ζ , b σ,ζ = σ ( c ,ζ ) − c ,ζ ζ r σ ( q − ξ q +1 σ a ζ , c σ,ζ = σ ( b ζ ) ζ ( q − rσ − q +32 ξ q +12 σ b ζ . Then we have a σ,ζ , b σ,ζ , c σ,ζ ∈ O K ac . In the sequel, we omit the subscript ζ of a σ,ζ , b σ,ζ and c σ,ζ . Proposition 6.4.
We have ¯ a σ ∈ k × , ¯ b σ ∈ k and ¯ a σ = ¯ c σ . Further, σ induces the morphism X ζ → X ¯ ξ σ ζ q − rσ ; ( z, w ) (¯ a σ z q − rσ + ¯ b σ , ¯ c σ w q − rσ ) . Proof.
We have v ( ξ σ σ ( c ,ζ ) − ξ q σ ζ q − rσ − c ,ζ ) ≥ ǫ by v ( σ ( c ,ζ ) − c ,ζ ) ≥ ǫ . Hence we have the last assertion by (6.2) and (6.3). By the definition of a ζ , b ζ and c ,ζ , we can check that ¯ a q − σ = 1 , ¯ b qσ = ¯ b σ , ¯ a σ = ¯ c σ using c q ,ζ ≡ − ι (2 − γ c ,ζ ) (mod ( q − /q ). 21e put L = K ( ̟ / ) and L = K ( ̟ / ) in K ac . Let LT L be the formal O L -module over O L ur of dimension 1 such that[ ̟ ] LT L ( X ) = ̟ X − X q , [ ζ ] LT L ( X ) = ζ X for ζ ∈ µ q − ( L ) ∪ { } . We put ̟ ,L = ̟ / (2( q − and take ̟ ,L ∈ O K ac such that [ ̟ / ] LT L ( ̟ ,L ) = ̟ ,L . LetArt L : L × ∼ → W ab L be the Artin reciprocity map normalized so that the image by Art L of auniformizer is a lift of the geometric Frobenius. We consider the following homomorphism: I L → k × × k ; σ (cid:0) λ ( σ ) , λ ( σ ) (cid:1) = σ ( ̟ ,L ) ̟ ,L , ̟ ,L σ ( ̟ ,L ) − σ ( ̟ ,L ) ̟ ,L σ ( ̟ ,L ) ̟ ,L ! . This map is equal to the composite I L → O × L → k × × k , where the first homomorphism is induced from the inverse of Art L , and the second homomorphismis given by a + b̟ / (¯ a, ¯ b/ ¯ a ) for a ∈ µ q − ( L ) and b ∈ O L . Then, we rewrite Proposition 6.4as follows: Corollary 6.5.
Let σ ∈ I L . We put g = 2 ζ − ( q +1) ( λ ( σ ) q + ζ q − λ ( σ )) ∈ k. Then, σ induces the morphism X ζ → X λ ( σ ) q +1 ζ ; ( z, w ) ( λ ( σ ) − q +1) ( z + g ) , λ ( σ ) − ( q +1) w ) . Proof.
We can check that ¯ a σ = λ ( σ ) − q +1) and ¯ c σ = λ ( σ ) − ( q +1) easily. We prove that¯ b σ = λ ( σ ) − q +1) g . We simply write ̟ i for ̟ i,L . We put ι = ζ q − and C = ̟ q − q (cid:26)(cid:18) ̟ ̟ (cid:19) q + ι (cid:18) ̟ ̟ (cid:19)(cid:27) . Then, we have C q − ιγ C ≡ − ̟ q − ̟ / ̟ = − ̟ . We can easily check the equality σ ( C ) − C ≡ ̟ ǫ ( λ ( σ ) q + ιλ ( σ )) (mod ǫ +) . On the other hand, we can check c q ,ζ ≡ − ι (2 − γ c ,ζ ) (mod q − q +)by the definition of c ,ζ . Therefore, the elements C and c q ,ζ / (2 ι ) satisfy x q − ιγ x ≡ − . Hence, we obtain C ≡ c q ,ζ / (2 ι ) (mod ǫ +). This implies( σ ( c ,ζ ) − c ,ζ ) q ≡ ι ( σ ( C ) − C ) (mod ǫ +) . Therefore, we obtain ¯ b σ ≡ ¯ b q σ ≡ λ ( σ ) − q +1) g (mod 0+)by ξ σ = λ ( σ ) q +1 (mod 0+). 22 .2.2 q : even We assume that q is even. We use the notation in Paragraph 4.4.2. For P ∈ P ( C ), we have w ( σ ( P )) ≡ σ ( w ( P )) (mod 14 q +) (6.4)by (6.1). We can see that z f +1 ( σ ( P )) ≡ σ ( z f +1 ( P )) (mod 18 q +) (6.5)using (4.31) and (6.4). Lemma 6.6.
The element σ induces the morphism P ζ → P ¯ ξ σ ζ q − rσ ; w w q − rσ . Proof.
This follows from Lemma 6.3 and (6.4).We take ζ ′ ∈ k × . By (6.4) and (6.5), we have a ,ζ ′ z ( σ ( P )) + b ,ζ ′ w ( σ ( P )) + c ,ζ ′ ≡ a ,ζ ′ σ ( z ( P )) + σ ( b ,ζ ′ ) σ ( w ( P )) + σ ( c ,ζ ′ ) (mod 18 q +) (6.6)and b ,ζ ′ w ( σ ( P )) + c ,ζ ′ ≡ σ ( b ,ζ ′ ) σ ( w ( P )) + σ ( c ,ζ ′ ) (mod 14 q +) (6.7)using σ ( a ,ζ ′ ) ≡ a ,ζ ′ (mod 1 / (8 q )+). We put a σ,ζ ′ = σ ( b ,ζ ′ ) b ,ζ ′ , b σ,ζ ′ = σ ( b ,ζ ′ ) b ,ζ ′ − b ,ζ ′ σ ( b ,ζ ′ ) a ,ζ ′ b ,ζ ′ ,b ′ σ,ζ ′ = σ ( c ,ζ ′ ) − c ,ζ ′ b ,ζ ′ , c σ,ζ ′ = σ ( c ,ζ ′ ) − c ,ζ ′ − b ,ζ ′ b − ,ζ ′ ( σ ( c ,ζ ′ ) − c ,ζ ′ ) a ,ζ ′ . In the sequel, we omit the subscript ζ ′ of a σ,ζ ′ , b σ,ζ ′ , b ′ σ,ζ ′ and c σ,ζ ′ . We note that v ( a σ ) = 0. Wehave v ( b ′ σ ) ≥ v ( b σ ) ≥
0. By (6.6) and (6.7), we obtain v ( c σ ) ≥ v ( b σ ) ≥ Proposition 6.7.
The element σ induces the morphism X ζ,ζ ′ → X ¯ ξ σ ζ q − rσ ,ζ ′ ; ( z, w ) ( z q − rσ + ¯ b σ w q − rσ + ¯ c σ , ¯ a σ w q − rσ + ¯ b ′ σ ) . Proof.
This follows from (6.6) and (6.7).In the following, we simplify the description of ¯ a σ , ¯ b σ , ¯ b ′ σ and ¯ c σ . Let ˜ ζ ′ ∈ µ q − ( K ) be the liftof ζ ′ . We put h ζ ′ ( x ) = x − ̟ ˜ ζ ′ x − ˜ ζ ′ . Lemma 6.8.
There is a root δ of h ζ ′ ( x ) = 0 such that δ ≡ c q ,ζ ′ + q ̟ ˜ ζ ′ (mod 14 +) . roof. We put h ( x ) = x q − + 1 + ̟ x q − . By the definition of c ,ζ ′ , we have h ( c q ,ζ ′ ) ≡ c ′ of h such that c ′ ≡ c q ,ζ ′ (mod 3 /
4) by Newton’s method. We can check that c ′ ≡ ˜ ζ ′ + ̟ ˜ ζ ′ (mod 116 +) . We define a parameter s with v ( s ) ≥ /
16 by x = ˜ ζ ′ + s . Then we have h ( ˜ ζ ′ + s ) ≡ ˜ ζ ′− s + (cid:18) q − (cid:19) ˜ ζ ′− s + ̟ ( ˜ ζ ′ + s ) ≡ ˜ ζ ′− h ζ ′ ( x ) + (cid:18) q − (cid:19) ˜ ζ ′− s + ̟ s ˜ ζ ′− (mod 12 +) . This implies h ζ ′ ( c ′ ) ≡ q ̟ ˜ ζ ′ (mod 12 +) . Therefore, we have a root δ of h ζ ′ ( x ) = 0 such that δ ≡ c ′ + q ̟ ˜ ζ ′ (mod 14 +)by Newton’s method.By the definition of b ,ζ ′ , we have b q ,ζ ′ ̟ − ≡ ˜ ζ ′ (mod 0+) . Let ζ ′′ be the element of µ q − ( K ur ) satisfying ζ ′′ ≡ b q ,ζ ′ ̟ − / (mod 0+). Note that ζ ′′ = ˜ ζ ′ .We take δ as in Lemma 6.8 and put δ = δ / ( ζ ′′ ̟ / ). Then we have δ − δ = 1 ζ ′′ ̟ . Note that v ( δ ) = − /
12. We take ζ ∈ µ ( K ur ) such that ζ = 1, and put h δ ( x ) = x − (1 + 2 ζ ) ̟ δ q x − ̟ δ q − (1 + 2 ̟ δ ) . Lemma 6.9.
There is a root θ of h δ ( x ) = 0 such that θ ≡ c q ,ζ ′ (mod 1 / .Proof. By the definition of c ,ζ ′ and c ,ζ ′ , we have h δ ( c q ,ζ ′ ) ≡ / θ as in Lemma 6.9 and put θ = θ ̟ δ q − ζ . Then we have θ − θ = δ . Note that v ( θ ) = − /
8. Let σ ∈ W K in this paragraph. We put ζ ,σ = σ ( ζ ′′ ̟ ) ζ ′′ ̟ .
24e take ν σ ∈ µ ( K ur ) ∪ { } such that σ ( δ ) ≡ ζ − ,σ ( δ + ν σ ) (mod 5 / (cid:0) σ ( θ ) − θ + ν σ δ (cid:1) ≡ σ ( θ ) − θ + ν σ δ + ν σ , (cid:0) σ ( θ ) − θ + ν σ δ + ν σ (cid:1) ≡ σ ( θ ) − θ + ν σ δ (mod 0+) . (6.8)By these equations, we can take µ σ ∈ µ ( K ur ) ∪ { } such that µ σ ≡ σ ( θ ) − θ + ν σ δ + ν σ + σ ( ζ ) − ζ (mod 0+) . Then we have µ σ + µ σ ≡ ν σ (mod 1) by (6.8) and ν σ , µ σ ∈ µ ( K ur ) ∪ { } . Lemma 6.10.
1. Let σ ∈ W K . Then we have a σ ≡ ζ ,σ , b σ ≡ ζ ,σ ν σ , b ′ σ ≡ ν σ , c σ ≡ µ σ (mod 0+) .
2. Let σ ∈ W K . Then we have ¯ a σ ∈ F × and ¯ b σ , ¯ b ′ σ , ¯ c σ ∈ F . Further, ¯ a σ ¯ b σ = ¯ b ′ σ and ¯ b σ = ¯ c σ + ¯ c σ hold.Proof. By the definition of b ,ζ ′ , we have a q σ ≡ σ ( ζ ′′ ̟ ) ζ ′′ ̟ (mod 0+) . Hence we have ¯ a q σ = ¯ ζ ,σ ∈ F × . This implies ¯ a σ = ¯ ζ ,σ ∈ F × .By the definition of a ,ζ ′ and b ,ζ ′ , we have b q σ ≡ σ ( b q ,ζ ′ ) (cid:0) σ ( c q (2 q − ,ζ ′ ) − c q (2 q − ,ζ ′ (cid:1) ̟ c q ,ζ ′ ≡ a q σ b q ,ζ ′ (cid:0) σ ( δ q − ) − δ q − (cid:1) ̟ δ q ≡ ζ ,σ ζ ′′ (cid:0) σ ( δ ) − δ (cid:1) ̟ δ ≡ ζ ,σ (cid:18) σ ( ζ ′′ ̟ ) ζ ′′ ̟ σ ( δ ) − δ (cid:19) ≡ ζ ,σ ν σ (mod 0+) , where we use Lemma 6.8 in the second congruence, b q ,ζ ′ /̟ / ≡ ζ ′′ (mod 0+) in the third congru-ence, δ = ˜ ζ ′ (mod 1 /
4) and ζ ′′ = ˜ ζ ′ in the fourth congruence and σ ( ζ ′′ ̟ / ) / ( ζ ′′ ̟ / ) ≡ ζ ,σ (mod 0+) in the last congruence. Hence, we obtain ¯ b σ = ¯ ζ ,σ ¯ ν σ ∈ F .By Lemma 6.8 and b q ,ζ ′ /̟ / ≡ ζ ′′ (mod 0+), we have b ′ q σ ≡ σ ( c q ,ζ ′ ) − c q ,ζ ′ b q ,ζ ′ ≡ σ ( δ ) − δ ζ ′′ ̟ = σ ( ζ ′′ ̟ ) ζ ′′ ̟ σ ( δ ) − δ ≡ ν σ (mod 0+) . Hence, we obtain ¯ b ′ σ = ¯ ν σ ∈ F .By Lemma 6.8, Lemma 6.9 and the definition of a ,ζ ′ , we have c q σ ≡ σ ( θ ) − θ − δ q − ( σ ( δ ) − δ ) ̟ δ q ≡ δ − q σ (cid:0) δ q ̟ ( θ + ζ ) (cid:1) − ̟ ( θ + ζ ) − δ − ( σ ( δ ) − δ ) ̟ ≡ σ (cid:0) ̟ ( θ + ζ ) (cid:1) − ̟ ( θ + ζ ) − ̟ δ (cid:0) σ ( ̟ δ ) − ̟ δ (cid:1) ̟ ≡ σ ( θ ) − θ + ν σ δ + σ ( ζ ) − ζ (mod 0+) , where we use σ ( δ ) ≡ δ (mod 1 /
4) in the second congruence, δ = ˜ ζ ′ (mod 1 /
4) in the thirdcongruence. Then we have ¯ c q σ ∈ F by (6.8). Hence we have ¯ c σ ∈ F and c σ ≡ µ σ (mod 0+) againby (6.8).By the above calculations, we can easily check ¯ a σ ¯ b σ = ¯ b ′ σ and ¯ b σ = ¯ c σ + ¯ c σ .25 emma 6.11. The field K ( ζ , ζ ′′ ̟ / , θ ) is a Galois extension over K .Proof. Let σ ∈ W K . It suffices to show σ ( θ ) ∈ K ( ζ , ζ ′′ ̟ / , θ ). We put θ σ = θ + ν σ δ + ν σ + µ σ + σ ( ζ ) − ζ . Then we have θ σ − θ σ ≡ σ ( δ ) (mod 2 / θ ′ such that θ ′ − θ ′ = σ ( δ ) and θ ′ ≡ θ σ (mod 2 / µ σ , we have θ ′ = σ ( θ ) (mod 0+). Hence, we obtain θ ′ = σ ( θ ).We take σ ′ ∈ W K such that σ ′ ( θ ) = σ ( θ ). We can define θ σ ′ as above, and have σ ′ ( θ ) ≡ θ σ ′ (mod 2 / ν σ = ν σ ′ , then we have ζ ,σ σ ( δ ) ≡ ζ ,σ ′ σ ′ ( δ ) (mod 5 / ζ ,σ σ ( δ ) = ζ ,σ ′ σ ′ ( δ ) because both sides are roots of x − x − ζ ′′ ̟ = 0 . Hence, if σ ( δ ) = σ ′ ( δ ) , we have ν σ = ν σ ′ , which implies σ ( θ ) ≡ θ σ θ σ ′ ≡ σ ′ ( θ ) (mod 0+) . If σ ( δ ) = σ ′ ( δ ) , we have σ ( θ ) σ ′ ( θ ) (mod 0+). Therefore we have v ( σ ( θ ) − θ σ ) > v ( σ ′ ( θ ) − θ σ ) . Then, we obtain σ ( θ ) ∈ K ( θ σ ) ⊂ K ( ζ , ζ ′′ ̟ / , θ )by Krasner’s lemma.Let E be the elliptic curve over k ac defined by z + z = w . We put Q = ( g ( α, β, γ ) = α β γα β α ∈ GL ( F ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) αγ + α γ = β ) . We note that | Q | = 24 and Q is isomorphic to SL ( F ) (cf. [Se, 8.5. Exercices 2]). Let Q ⋊ Z bea semidirect product, where r ∈ Z acts on Q by g ( α, β, γ ) g ( α q r , β q r , γ q r ). Then Q ⋊ Z actsfaithfully on E as a scheme over k , where ( g ( α, β, γ ) , r ) ∈ Q ⋊ Z acts on E by( z, w ) (cid:0) z q − r + α − ( βw q − r + γ ) , α ( w q − r + ( α − β ) ) (cid:1) for k ac valued points. Proposition 6.12.
The element σ ∈ W K sends X ζ,ζ ′ to X ¯ ξ σ ζ q − rσ ,ζ ′ . We identify X ζ,ζ ′ with X ¯ ξ σ ζ q − rσ ,ζ ′ by ( z, w ) ( z, w ) . Then the action of W K gives a homomorphism Θ ζ ′ : W K → Q ⋊ Z ⊂ Aut k ( X ζ,ζ ′ ); σ (cid:0) g ( ¯ ζ ,σ , ¯ ζ ,σ ¯ ν σ , ¯ ζ ,σ ¯ µ σ ) , r σ (cid:1) . Proof.
This follows from Proposition 6.7 and Lemma 6.10.
Proposition 6.13.
The homomorphism Θ ζ ′ factors through W ( K ur ( ̟ / , θ ) /K ) and gives an iso-morphism W ( K ur ( ̟ / , θ ) /K ) ≃ Q ⋊ Z .Proof. By Lemma 6.10.1, the homomorphism Θ ζ ′ factors through W ( K ur ( ̟ / , θ ) /K ) and inducesan injective homomorphism W ( K ur ( ̟ / , θ ) /K ) → Q ⋊ Z .To prove the surjectivity, it suffices to show that Θ ζ ′ sends I K onto Q . Let g = g ( α, β, γ ) ∈ Q .We take ζ α ∈ µ ( K ur ), ν β , µ γ ∈ µ ( K ur ) ∪ { } such that ¯ ζ α = α , ¯ ν β = α − β and ¯ µ γ = α − γ . Weput δ g = ζ − α ( δ + ν β ) and θ g = θ + ν β δ + ν β + µ γ . Then we have δ g − δ g ≡ ζ α ζ ′′ ̟ (mod 56 ) . Hence, we can find δ ′ g such that δ ′ g − δ ′ g = 1 / ( ζ α ζ ′′ ̟ / ) and δ ′ g ≡ δ g (mod 5 / θ g − θ g ≡ δ ′ g (mod 2 / θ ′ g such that θ ′ g − θ ′ g = δ ′ g and θ ′ g ≡ θ g (mod 2 / ̟ / ζ α ̟ / and θ θ ′ g gives an element of I K , whose image by Θ ζ ′ is g .26 Cohomology of X ( p ) In this section we show that the covering C ( p ) is semi-stable, and study a structure of ℓ -adiccohomology of X ( p ). In the sequel, for a projective smooth curve X over k , we simply write H ( X, Q ℓ ) for H ( X k ac , Q ℓ ). For a finite abelian group A , the character group Hom Z ( A, Q × ℓ ) isdenoted by A ∨ . Let X DL be the smooth compactification of the affine curve over k defined by X q − X = Y q +1 .The curve X DL is also the smooth compactification of the Deligne-Lusztig curve x q y − xy q = 1 forSL ( F q ). Then, a ∈ k acts on X DL by α a : ( X, Y ) ( X + a, Y ) . On the other hand, ζ ∈ k × acts on X DL by β ζ : ( X, Y ) ( ζ q +1 X, ζ Y ) . By these actions, we consider H ( X DL , Q ℓ ) as a Q ℓ [ k × k × ]-module. Lemma 7.1.
We have an isomorphism H ( X DL , Q ℓ ) ≃ M ψ ∈ k ∨ \{ } M χ ∈ µ q +1 ( k ) ∨ \{ } ψ ⊗ χ as Q ℓ [ k × µ q +1 ( k )] -modules.Proof. As Q ℓ [ k × µ q +1 ( k )]-modules, we have the short exact sequence0 → M ψ ∈ k ∨ ψ → H c ( X DL \ X DL ( k ) , Q ℓ ) → H ( X DL , Q ℓ ) → . (7.1)Let L ψ denote the Artin-Schreier Q ℓ -sheaf associated to ψ ∈ k ∨ . Let K χ denote the Kummer Q ℓ -sheaf associated to χ ∈ µ q +1 ( k ) ∨ . Since X DL \ X DL ( k ) → G m ; ( X, Y ) Y q +1 is a finite etale Galois covering with a Galois group k × µ q +1 ( k ), we have the isomorphism H c ( X DL \ X DL ( k ) , Q ℓ ) ≃ M ψ ∈ k ∨ M χ ∈ µ q +1 ( k ) ∨ H c ( G m , L ψ ⊗ K χ ) (7.2)as Q ℓ [ k × µ q +1 ( k )]-modules. Note that we havedim H c ( G m , L ψ ⊗ K χ ) = 1if ψ = 1 by the Grothendieck-Ogg-Shafarevich formula (cf. [SGA5, Expos´e X Th´eor`eme 7.1]).Clearly, if χ = 1, we have H c ( G m , K χ ) = 0 and H c ( G m , L ψ ) ≃ ψ . Hence, we acquire the isomor-phism M ψ ∈ k ∨ M χ ∈ µ q +1 ( k ) ∨ H c ( G m , L ψ ⊗ K χ ) ≃ M ψ ∈ k ∨ \{ } M χ ∈ µ q +1 ( k ) ∨ \{ } H c ( G m , L ψ ⊗ K χ ) ⊕ M ψ ∈ k ∨ ψ (7.3)as Q ℓ [ k × µ q +1 ( k )]-modules. By (7.1), (7.2) and (7.3), the required assertion follows.27or a character ψ ∈ k ∨ and an element ζ ∈ k × , we denote by ψ ζ the character x ψ ( ζ x ). Weconsider a character group ( k × ) ∨ as a subgroup of ( k × ) ∨ by Nr ∨ k /k . Lemma 7.2.
We have an isomorphism H ( X DL , Q ℓ ) ≃ M ˜ χ ∈ ( k × ) ∨ \ ( k × ) ∨ ˜ χ as Q ℓ [ k × ] -modules.Proof. By Lemma 7.1, we take a basis { e ψ,χ } ψ ∈ k ∨ \{ } , χ ∈ µ q +1 ( k ) ∨ \{ } of H ( X DL , Q ℓ ) over Q ℓ such that k × µ q +1 ( k ) acts on e ψ,χ by ψ ⊗ χ . For ζ ∈ k × and a ∈ k , wehave β ζ ◦ α a ◦ β − ζ = α ζ q +1 a in Aut k ( X DL ). Hence, ζ ∈ k × acts on H ( X DL , Q ℓ ) by e ψ,χ c ψ,χ,ζ e ψ ζ − ( q +1) ,χ with some constant c ψ,χ,ζ ∈ Q × ℓ . Therefore, we acquire an isomorphism H ( X DL , Q ℓ ) ≃ M χ ∈ µ q +1 ( k ) ∨ \{ } Ind k × µ q +1 ( k ) ( χ )as Q ℓ [ k × ]-modules. Hence, the required assertion follows. Proposition 7.3.
We have isomorphisms H ( Y c1 , , Q ℓ ) ≃ M ˜ χ ∈ ( k × ) ∨ \ ( k × ) ∨ ( ˜ χ ◦ λ ) ⊗ ( ˜ χ q ◦ κ ) ,H ( Y c2 , , Q ℓ ) ≃ M ˜ χ ∈ ( k × ) ∨ \ ( k × ) ∨ ( ˜ χ ◦ λ ) ⊗ ( ˜ χ ◦ κ ) as ( I K × O × D ) -representations over Q ℓ .Proof. This follows from Lemma 5.1, Lemma 6.2 and Lemma 7.2.Let X AS be the smooth compactification of the affine curve X ′ AS over k defined by z q − z = w .Let a ∈ k act on X AS by α a : ( z, w ) ( z + a, w ) . By this action, we consider H ( X AS , Q ℓ ) as a Q ℓ [ k ]-module. On the other hand, let b ∈ µ q − ( k ac )act on X AS by β b : ( z, w ) ( b z, bw ) . Lemma 7.4.
We assume that q is odd. Let G be the Galois group of the Galois extension F over k (( s )) defined by z q − z = 1 /s . Let G r be the upper numbering ramification filtration of G . Then G r = G if r ≤ , and G r = 1 if r > . roof. We take a ∈ F such that a q − a = 1 /s . Then sa ( q − / is a uniformizer of F . Let v F be thenormalized valuation of F . For σ ∈ G and an integer i , the condition v F (cid:0) σ ( sa q − ) − sa q − (cid:1) ≥ i is equivalent to the condition v F (cid:0) σ ( a ) − a (cid:1) ≥ i − . Hence, the claim follows.For a character ψ ∈ k ∨ and x ∈ k × , we write ψ x ∈ k ∨ for the character y ψ ( xy ). We set V = M ψ ∈ k ∨ \{ } ψ as Q ℓ [ k ]-modules. Let { e ψ } ψ ∈ k ∨ \{ } be the standard basis of V . Lemma 7.5.
We assume that q is odd.1. Then we have H ( X AS , Q ℓ ) ≃ V as Q ℓ [ k ] -modules.2. For b ∈ µ q − ( k ac ) , the automorphism β b of X AS induces the action e ψ c ψ,b e ψ b − on H ( X AS , Q ℓ ) ≃ V with some constant c ψ,b ∈ Q × ℓ . Furthermore, we have c ψ, − = − .Proof. We have H ( X AS , Q ℓ ) ≃ H c ( X ′ AS , Q ℓ ), because the complement X AS \ X ′ AS consists of onepoint. The curve X ′ AS is a finite etale Galois covering of A with a Galois group k by ( z, w ) w .For ψ ∈ k ∨ , let L ,ψ be the smooth Q ℓ -sheaf on A defined by the covering X ′ AS and ψ . Then wehave H c ( X ′ AS , Q ℓ ) ≃ M ψ ∈ k ∨ \{ } H c ( A , L ,ψ )as Q ℓ [ k ]-modules. By Lemma 7.4 and the Grothendieck-Ogg-Shafarevich formula, we havedim H c ( A , L ,ψ ) = 1and H c ( A , L ,ψ ) ≃ ψ as Q ℓ [ k ]-modules for ψ ∈ k ∨ \{ } . Hence, the first assertion follows.The second assertion follows from that β b α a β − b = α ab for a ∈ k and b ∈ µ q − ( k ac ). Theassertion c ψ, − = − U D = { d ∈ O × D | κ ( d ) ∈ k × } . We take ζ ∈ µ q − ( k ac ) \ k × . Let ∆ ∈ ( k × ) ∨ be the character defined by x x q − ∈ {± } ⊂ Q × ℓ for x ∈ k × . If q is odd, we put τ χ,ψ = Ind I K I L (cid:0) ( χ ◦ λ q +11 ) ⊗ ( ψ ◦ Tr k /k ◦ λ ) (cid:1) ,τ ′ χ,ψ = Ind I K I L (cid:16) ( χ ◦ λ q +11 ) ⊗ (cid:0) ψ ◦ Tr k /k ◦ ( − ζ − ( q +1)0 λ ) (cid:1)(cid:17) ,θ χ,ψ = (∆ χ ◦ κ ) ⊗ ( ψ ◦ Tr k /k ◦ κ ) ,θ ′ χ,ψ = (∆ χ ◦ κ ) ⊗ (cid:0) ψ ◦ Tr k /k ◦ ( ζ − q κ ) (cid:1) ,ρ χ,ψ = Ind O × D U D θ χ,ψ ,ρ ′ χ,ψ = Ind O × D U D θ ′ χ,ψ χ ∈ ( k × ) ∨ and ψ ∈ k ∨ \{ } . We note thatdim ρ χ,ψ = dim ρ ′ χ,ψ = q + 1 . For ψ, ψ ′ ∈ k ∨ \{ } , we can check that τ χ,ψ = τ χ,ψ ′ if and only if ψ ′ = ψ − , and ρ χ,ψ = ρ χ,ψ ′ if andonly if ψ ′ = ψ − . Similar things hold also for τ ′ χ,ψ and ρ ′ χ,ψ . We define an equivalence relation ∼ on k ∨ \{ } by ψ ∼ ψ − . We putΠ χ,ψ = τ χ,ψ ⊗ ρ χ,ψ , Π ′ χ,ψ = τ ′ χ,ψ ⊗ ρ ′ χ,ψ for χ ∈ ( k × ) ∨ and ψ ∈ k ∨ \{ } . Proposition 7.6.
We assume that q is odd. Then we have an isomorphism M ζ ∈ µ q − ( k ac ) H ( X c ζ , Q ℓ ) ≃ M χ ∈ ( k × ) ∨ M ψ ∈ ( k ∨ \{ } ) / ∼ Π χ,ψ ⊕ Π ′ χ,ψ as representations of I K × O × D .Proof. The actions of I L and U D on L ζ ∈ k × H ( X c ζ , Q ℓ ) factor through k × × k by Proposition 5.4and Corollary 6.5. On the other hand, the action of k × × k on L ζ ∈ k × H ( X c ζ , Q ℓ ) is induced fromthe action of { } × k on H ( X c1 , Q ℓ ). Hence, we have M ζ ∈ k × H ( X c ζ , Q ℓ ) ≃ M χ ∈ ( k × ) ∨ M ψ ∈ k ∨ \{ } χ ⊗ ψ as representations of k × × k by Lemma 7.5.1. Therefore, we have an isomorphism M ζ ∈ k × H ( X c ζ , Q ℓ ) ≃ M χ ∈ ( k × ) ∨ M ψ ∈ k ∨ \{ } ( χ ◦ λ q +11 ) ⊗ ( ψ ◦ Tr k /k ◦ λ ) ⊗ θ χ,ψ as representations of I L × U D by Proposition 5.4, Corollary 6.5 and Lemma 7.5.2. Inducing thisrepresentation from U D to O × D , we obtain an isomorphism M ζ ∈ k × H ( X c ζ , Q ℓ ) ≃ M χ ∈ ( k × ) ∨ M ψ ∈ k ∨ \{ } ( χ ◦ λ q +11 ) ⊗ ( ψ ◦ Tr k /k ◦ λ ) ⊗ ρ χ,ψ as representations of I L × O × D . On the left hand side of this isomorphism, we have an action of I K that commutes with the action of O × D . Hence, we have M ζ ∈ k × H ( X c ζ , Q ℓ ) ≃ M χ ∈ ( k × ) ∨ M ψ ∈ ( k ∨ \{ } ) / ∼ τ χ,ψ ⊗ ρ χ,ψ as representations of I K × O × D . By the similar arguments, we have M ζ ∈ µ q − ( k ac ) \ k × H ( X c ζ , Q ℓ ) ≃ M χ ∈ ( k × ) ∨ M ψ ∈ ( k ∨ \{ } ) / ∼ τ ′ χ,ψ ⊗ ρ ′ χ,ψ as representations of I K × O × D . Therefore, we have the isomorphism in the assertion.Let E and Q be as in Paragraph 6.2.2. Let Z ⊂ Q be the subgroup consisting of g (1 , , γ ) with γ + γ = 0, and φ be the unique non-trivial character of Z . By [BH, Lemma 22.2], there exists aunique irreducible two-dimensional representation τ of Q such that τ | Z ≃ φ ⊕ , Tr τ ( g ( α, , − α ∈ F × \{ } . Then, it is easily checked that the determinant character of τ is trivial. Note thatevery two-dimensional irreducible representation of Q has a form τ ⊗ χ with χ ∈ ( F × ) ∨ , where weconsider χ as a character of Q by g ( α, β, γ ) χ ( α ).30 emma 7.7. The Q -representation H ( E, Q ℓ ) is isomorphic to τ .Proof. The Q -representation H ( E, Q ℓ ) satisfies (7.4) by Lemma 7.1. Hence, the assertion follows.Let τ ζ ′ be the representation of W K induced from the ( Q ⋊ Z )-representation H ( E, Q ℓ ) by Θ ζ ′ .Then the restriction to I K of τ ζ ′ is isomorphic to the representation induced from τ by Lemma 7.7.We say that a continuous two-dimensional irreducible representation V of W K over Q ℓ is prim-itive, if there is no pair of a quadratic extension K ′ and a continuous character χ of W K ′ such that V ≃ Ind W K W K ′ χ . Lemma 7.8.
The representation τ ζ ′ is primitive of Artin conductor .Proof. We use the notations in the proof of Lemma 6.11. The element 1 / ( ̟ / θ ) is a uniformizerof K ur ( ̟ / , θ ). For σ ∈ I K , we can show that v (cid:18) σ (cid:18) ̟ θ (cid:19) − ̟ θ (cid:19) = if ζ ,σ = 1, if ζ ,σ = 1, ν σ = 0, if ζ ,σ = 1, ν σ = 0, µ σ = 0,using σ ( θ ) ≡ θ σ (mod 2 / Q ⋊ Z is Q ⋊ Z , because Q has no index 2 subgroup. Hence,if τ ζ ′ is not primitive, it is induced from a character of W K . However, this is impossible, becausethe restriction of τ ζ ′ to W K is irreducible.We define a character λ ξ : W K → k × by λ ξ ( σ ) = ¯ ξ σ . We put τ ζ ′ ,χ = τ ζ ′ ⊗ ( χ ◦ λ ξ ) ,θ ζ ′ ,χ = ( χ ◦ κ ) ⊗ ( φ ◦ Tr k / F ( ζ ′− κ )) ,ρ ζ ′ ,χ = Ind O × D U D θ ζ ′ ,χ , Π ζ ′ ,χ = τ ζ ′ ,χ ⊗ ρ ζ ′ ,χ for ζ ′ ∈ k × and χ ∈ ( k × ) ∨ . In the sequel, we consider τ ζ ′ ,χ as a representation of I K . Proposition 7.9.
We assume that q is even. Let ζ ′ ∈ k × . Then we have an isomorphism M ζ ∈ k × H ( X c ζ,ζ ′ , Q ℓ ) ≃ M χ ∈ ( k × ) ∨ Π ζ ′ ,χ as representations of I K × O × D .Proof. The actions of I K and U D on L ζ ∈ k × H ( X c ζ,ζ ′ , Q ℓ ) factor through Q × k × by Proposition 5.4and Proposition 6.12. On the other hand, the action of Q × k × on L ζ ∈ k × H ( X c ζ,ζ ′ , Q ℓ ) is inducedfrom the action of Q on H ( X c1 ,ζ ′ , Q ℓ ). Hence, we have an isomorphism M ζ ∈ k × H ( X c ζ,ζ ′ , Q ℓ ) ≃ M χ ∈ ( k × ) ∨ τ ⊗ χ as representations of Q × k × . Therefore, we have an isomorphism M ζ ∈ k × H ( X c ζ,ζ ′ , Q ℓ ) ≃ M χ ∈ ( k × ) ∨ τ ζ ′ ,χ ⊗ θ ζ ′ ,χ as representations of I K × U D by Proposition 5.4 and Proposition 6.12. Inducing this representationfrom U D to O × D , we obtain the isomorphism in the assertion.31 .2 Genus calculation Lemma 7.10.
We have dim H ( X ( p ) C , Q ℓ ) = 2 q − q + 1 .Proof. It suffices to show thatdim H (cid:0) (LT ( p ) /̟ Z ) C , Q ℓ (cid:1) = 4 q − q + 2 , because we have dim H (cid:0) (LT ( p ) /̟ Z ) C , Q ℓ (cid:1) = 2 dim H ( X ( p ) C , Q ℓ ) . For an irreducible smooth representation π of GL ( K ), we write c ( π ) for the conductor of π . ByProposition 2.1, we have H (cid:0) (LT ( p ) /̟ Z ) C , Q ℓ (cid:1) ≃ M π (cid:0) π K ( p ) (cid:1) ⊕ π ) ⊕ M χ (St ⊗ χ ) K ( p ) , where π runs through irreducible cuspidal representations of GL ( K ) such that c ( π ) ≤ ω π ( ̟ ) = 1, and χ runs through characters of K × such that c (St ⊗ χ ) ≤ χ ( ̟ ) = 1. We havethe following list of discrete series representations π of GL ( K ) such that c ( π ) ≤ ω π ( ̟ ) = 1:(1) π ≃ St ⊗ χ for an unramified character χ : K × → Q × ℓ such that χ ( ̟ ) = 1. Then c ( π ) = 1and dim LJ( π ) = 1. There are two such representations.(2) π ≃ St ⊗ χ for a tamely ramified character χ : K × → Q × ℓ that is not unramified and satisfies χ ( ̟ ) = 1. Then c ( π ) = 2 and dim LJ( π ) = 1. There are 2( q −
2) such representations.(3) π ≃ π χ , in the notation of [BH, 19.1], for a character χ : K × → Q × ℓ of level zero such that χ does not factor through Nr K /K and χ ( ̟ ) = 1. Then c ( π ) = 2 and dim LJ( π ) = 2. There are q ( q − / π of GL ( K ) such that c ( π ) = 3 and ω π ( ̟ ) = 1. Thendim LJ( π ) = q + 1 by [Tu, Theorem 3.6]. There are 2( q − such representations by [Tu,Theorem 3.9].We note that dim π K ( p ) = 4 − c ( π ) if π is a discrete series representation of GL ( K ) such that c ( π ) ≤
3. Then we obtain the claim by taking a summation according to the above list.For an affinoid rigid space X , a Zariski subaffinoid of X is the inverse image of a nonemptyopen subscheme of X under the reduction map X → X . Proposition 7.11.
Let W be a wide open rigid curve over a finite extension of b K ur with a stablecovering { ( U i , U u i ) } i ∈ I . Let X be a subaffinoid space of W such that X is a connected smooth curvewith a positive genus. Then there exists i ∈ I such that X is a Zariski subaffinoid of U u i .Proof. Assume that X ∩ U u i is contained in a finite union of residue classes of X for any i ∈ I .Then a Zariski subaffinoid of X appears in an open annulus. This is a contradiction, because X has a positive genus. Hence there exists i ′ ∈ I such that X ∩ U u i ′ is not contained in any finite unionof residue class of X . We fix such i ′ in the sequel.Then some open irreducible subscheme of the reduction of X ∩ U u i ′ does not go to one point in X under the natural map X ∩ U u i ′ → X . Let Y be the inverse image of such an open subschemeunder the reduction map X ∩ U u i ′ → X ∩ U u i ′ . Then we see that Y is a Zariski subaffinoid of X by[CM, Lemma 2.24 (i)]. Each connected component of X \ Y is an open disk, and included in U u i ′ or U u i for i = i ′ or an open annulus outside the underlying affinoids. This can be checked by applying[CM, Corollary 2.39] to every closed disk in a connected component of X \ Y . Hence, X ∩ U u i ′ is aZariski subaffinoid of X . If X ∩ U u i ′ = X , then U u i ′ is connected to an open disk in U u i for i = i ′ orin an open annulus outside the underlying affinoids. This is a contradiction. Therefore, we have X ⊂ U u i . Then we obtain the claim by [CM, Lemma 2.24 (i)].32 emma 7.12. Let W be a wide open rigid curve over a finite extension of b K ur with a stablecovering. Let X be a subaffinoid space of W such that X is a connected smooth curve with genuszero. Then there is a basic wide open subspace of W such that its underlying affinoid is X .Proof. We note that we have the claim if X appears in an open subannulus of W . Let { ( U i , U u i ) } i ∈ I be the stable covering of W .First, we consider the case where X ∩ U u i is contained in a finite union of residue classes of X for any i ∈ I . Then a Zariski subaffinoid of X appears in an open annulus. Further, X itselfappears in the open annulus, because X is connected. Hence, we have the claim in this case.Therefore, we may assume that there exists i ′ ∈ I such that X ∩ U u i ′ is not contained in any finiteunion of residue class of X . We fix such i ′ . By the same argument as in the proof of Proposition7.11, we have X ⊂ U u i . If the image of the induced map X → U u i is one point, we have the claimbecause X appears in an open disk. Otherwise, X is a Zariski subaffinoid of U u i , and we have theclaim.We consider the natural level-lowering map π f : X ( p ) → X ( p ); ( u, X ) ( u, X ) . Lemma 7.13.
The connected components of W , ′ , W , ′ , W , ′ and W , ′ ∪ W , ′ ∪ W , ′ are notopen balls.Proof. Let W ′ be a subannulus of W defined by v ( u ) < / ( q ( q + 1)). Then we have π − f ( W k × ) = W , ′ , π − f ( W ∞ ) = W , ′ ∪ W , ′ ∪ W , ′ and π − f ( W ′ ) = W , ′ ∪ W , ′ . Hence we have the claimby Proposition 3.1 and [Co, Lemma 1.4].The smooth projective curves Y c1 , and Y c2 , have defining equations X q Y − XY q = Z q +1 determined by the equation in Proposition 4.2 and Proposition 4.3. The infinity points of Y , in P k consist of P + a = ( a, ,
0) for a ∈ k and P + ∞ = (1 , , Y , consist of P − a = ( a, ,
0) for a ∈ k and P −∞ = (1 , , W , let e ( W ) be the number of the ends of W , and g ( W ) be the genusof W (cf. [CM, p. 369 and p. 380]). For a proper smooth curve C over k ac , we write g ( C ) for thegenus of C . Theorem 7.14.
The covering C ( p ) is a semi-stable covering of X ( p ) over some finite extension.Proof. We consider the stable covering of X ( p ) C by Proposition 4.11. Then Y c1 , and Y c2 , appearin the stable reduction of X ( p ) C as irreducible components by Proposition 7.11. The point P +0 isthe unique infinity point of Y , whose tube is contained in W +1 , ′ , because v ( X ) > / ( q ( q − W +1 , ′ . Similarly, P − is the unique infinity point of Y , whose tube is contained in W − , ′ . Hence,we have e ( X ( p ) C ) ≥ q by Lemma 7.13. Therefore, we have g ( X ( p ) C ) ≤ q − q + 1 by Lemma7.10. On the other hand, we have g ( X ( p ) C ) ≥ g ( Y c1 , ) + g ( Y c2 , ) + (P ζ ∈ µ q − ( k ac ) g ( X c ζ ) if q is odd, P ζ ∈ k × , ζ ′ ∈ k × g ( X c ζ,ζ ′ ) if q is even,where the summation on the right hand side is q − q + 1 by Proposition 7.3, Proposition 7.6 andProposition 7.9. Then the affinoids Y , , Y , , X ζ for ζ ∈ µ q − ( k ac ) and X ζ,ζ ′ for ζ ∈ k × and ζ ′ ∈ k × are underlying affinoids of basic wide open spaces in the stable covering by Proposition7.11 and Lemma 7.13. Therefore, by the above genus inequalities, we see that e ( X ( p ) C ) = 2 q and the connected components of W , ′ , W , ′ , W , ′ and W , ′ ∪ W , ′ ∪ W , ′ are open annuli.The connected components of X ( p ) \ Z , are two wide open spaces, because each connectedcomponent is connected to Z , at an open subannulus by Lemma 7.12. Then we see that these33wo wide open spaces are basic wide open spaces with underlying affinoids Y , and Y , by theabove genus inequalities. Therefore we have the claim by Proposition 4.7, Proposition 4.9 andProposition 4.10. In this subsection, we study the action of I K × O × D on ℓ -adic cohomology of X ( p ). We put( W K × D × ) = { ( σ, ϕ − r σ ) ∈ W K × D × } . Although it is possible to study the action of ( W K × D × ) using the result of Section 6, here westudy only the inertia action for simplicity. The result in this subsection is essentially used in [IT3].Let X ( p ) be the semi-stable formal scheme constructed from C ( p ) by [IT2, Theorem 3.5].The semi-stable reduction of X ( p ) means the underlying reduced scheme of X ( p ), which isdenoted by X ( p ) k ac . Lemma 7.15.
The smooth projective curves Y c1 , and Y c2 , intersect with Z c1 , at P +0 and P − respectively in the stable reduction X ( p ) k ac .Proof. We see this from the proof of Theorem 7.14.Let Γ be the graph defined by the following: • The set of the vertices of Γ consists of P , P ∞ , P + a and P − a for a ∈ P ( k ) \ { } . • The set of the edges of Γ consists of P P + a , P P − a , P ∞ P + a and P ∞ P − a for a ∈ P ( k ) \ { } .We note that P + a and P − a for a ∈ P ( k ) \ { } are points of Y c1 , and Y c2 , that are not on Z c1 , by Lemma 7.15. Let H (Γ , Q ℓ ) be the cohomology group of Γ with coefficients in Q ℓ (cf. [IT2,Section 2]). The group I K × O × D acts on P + a and P − a for a ∈ P ( k ) \ { } via the action on Y c1 , and Y c2 , . Let I K × O × D act on P and P ∞ trivially. By this action, we consider H (Γ , Q ℓ ) as a Q ℓ [ I K × O × D ]-module. Theorem 7.16.
We have an exact sequence −→ H (Γ , Q ℓ ) −→ H ( X ( p ) C , Q ℓ ) −→ H ( X ( p ) k ac , Q ℓ ) ∗ ( − −→ as representations of ( W K × D × ) . Further, as ( I K × O × D ) -representations, H ( X ( p ) k ac , Q ℓ ) isisomorphic to M ˜ χ ∈ ( k × ) ∨ \ ( k × ) ∨ Π ˜ χ ⊕ (L χ ∈ ( k × ) ∨ L ψ ∈ ( k ∨ \{ } ) / ∼ Π χ,ψ ⊕ Π ′ χ,ψ if q is odd, L ζ ′ ∈ k × L χ ∈ ( k × ) ∨ Π ζ ′ ,χ if q is even,where we put Π ˜ χ = ( ˜ χ ◦ λ ) ⊗ ( ˜ χ ◦ κ ⊕ ˜ χ q ◦ κ ) , and H (Γ , Q ℓ ) is isomorphic to ⊕ M χ ∈ ( k × ) ∨ (cid:0) ( χ ◦ λ q +1 ) ⊗ ( χ ◦ κ q +11 ) (cid:1) ⊕ . Proof.
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