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Rigid Geometry and Applications

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Rigid Geometry and Applications Rigid Geometry and Applications Kazuhiro Fujiwara & Fumiharu Kato Talk Plan • Aim: • To give a survey on foundation (based on recent developments). • To show how to use it. • Schedule: • I (Mon. 13): What is Rigid Geometry ? • II (Tue. 14): • Birational Geometry from Zariski’s Viewpoint • Birational Approach to Rigid Geometry • III (Wed. 15): Applications - How to use it. Example: An elliptic curve • E/Q When v = , Qv = R C. !• Base change∞ ⊂ ! Complex analytificationEQ What is Rigidv Geometry ? – v 2When, 3, 5, 7v,=. .p,, : place When v = p, ∈ {• ∞} • – Qv: v-adic completion ofan Q EC = When v = , Qv =QpR C. Qp CpQp = QpQp Cp = Qp • Uniformization∞ ⊂⊂ ⊂ ⊂ ⊂ ! Complex analytification Question: Analysis over Cp ? • • ! ! Q Q an C = Q C p C/Λ "p EC pp p −→⊂ • | |∞⊂ | | Question: Analysis over Cp ? • Λ an= Z= + Z τ ! Real - ComplexEC · Rigid analytic H analCytic geometrp y geometry Uniformizationτ •=| z|∞ | Im| z > 0 : Teichmullar¨ parameter. • ∈ { ∈ | } C C/Λ " Ean −→ C Λ = Z + Z τ · τ H = z C Im z > 0 : Teichmullar¨ parameter. ∈ { ∈ | } 1 2 3 3 2 When v = p, When v = , Qv = R C. • When•v = , Qv∞= R C⊂. • ! Comple∞ x analytification⊂ ! Complex analytification Qp Qp vs. Cp = Qp ⊂ Ean⊂= When v = p,anC Similarities • When v =Ep, = When v = p, Uniformization• C ! •Q Q • C = Q Uniformizationp QC pC/QΛ Ean C p = Qp p Qp Qpp" CQCpp p= QQpp pCp = Qp • ⊂ −→⊂ ⊂ ⊂⊂⊂ ⊂ ⊂ Algebraically closed Algebraicalan ly closed Question: AnalysisQuestionCQuestionΛo:v=C:erZ/+ΛCZ QuestionτE? : ! ! • • "p•· C ! • – −Analysis→ over Cp ?– Analysis over Cp ?! τ CompletH =– Analysise zwith– Analytification respectC oImv teroz CComplet>Epan?0–?Analytification: e Twitheichm respectullar¨E anto ? ∈ { ∈Λ =| Z + ZCp τ} Cp parameterp the absolut–.Analytificatione value norm theE anp-adic? norm p C· p • | |∞ | | • | p|∞ | | τ H = z C •Im| |∞z >| |0 : Teichmullar¨ parameter. ∈ { ∈ | } 3 3 2 3 2 3 When v = , Qv = R C. • ∞ ⊂ When!v = Comple, Qv = Rx analytificationC. • ∞ ⊂ ! Complex analytificationAnalytic Method: Uniformizationan an E = WhenEC v== p, C • UniformizationRecall: Jacobi’s uniformization /C. • • Uniformization • C C/Λ C / C/Λ AA ; −→A ww an AA ww QCAA ww C/QΛ E C = Q pA ww " p Λ√= Z + Zw τ p C p e2π 1 ⊂ ⊂ − · C×−→· τ H = z C Im z >Z0 Z τ an ∈ QuestionZ { ∈ Λ| :=2π √ +1}τ ! ! EC " C•×/q , q = e − . · Jacobi’s uniformizationTake q C suchTatethat’s uniformizationq p < 1, τ H =– Analysisz • C oImv∈erz C×p>p ?0 : T| eichm| ullar¨ Z Z ∈ C× {C×∈/q = |C/Λ an}C C /q parameter−–→.Analytification EC ?×p −→ ×p √ p = 2π 1τ q C×, q p < 1 q e − ∈ p | | Consider • qnt mkqm ℘ (t) = , s = . 1 n 2 k 1 qm n Z (1 q t) m 1 !∈ − !≥ − Define ℘(t) = ℘1(t) + s1. 2 ! automorphic in the sense that ℘(qt) = ℘(t), 2 4 3 5 6 6 Theorem (Tate). Set B = 5s3 and C = (5s3+ / − − Theor7s5) 12.emThen(Tate).weSethavBe the= equality5s3 and C = (5s3+ 2 3 − − 7s5)/12.(℘"Then) + ℘we℘"ha=v(e℘)the+equalityB℘ + C. 2 3 Moreov(℘er",)by+the℘℘"mapping= (℘) + B℘ + C. Moreover, byt the(℘mapping(t) : ℘ (t) : 1), #→ " C /qZ is mappedt (℘bijecti(t) : ℘v(elyt) :onto1), the algebraic ×p #→ " Z P2 C Csubset×p/q inis mapped(Tpat) definedebijecti Curvbyelyvetheontocubicthe equationalgebraic 2 subset in P (Cp) defined by the cubic equation Y2Z + XYZ = X3 + BXZ2 + CZ3. Y2Z + XYZ = X3 + BXZ2 + CZ3. Embedding onto a cubic curve • Embedding onto a cubic curve • (℘:℘":1) 2 C / P (C ) ×p F p FF s9 FF s FF (℘:℘ :1) s FF " s 2 C F" s / P (C ) ×p F p FF Z s9 FF s FFC /q s FF ×p s F" s j(E) p > 1 C /qqZq p < 1 such that E ! • | | ⇐⇒ ×p∃ | | C / Z j(×E)qp. > 1 q q p < 1 such that E ! • | |Z ⇐⇒ ∃ | | C×p/q . Why do we need analytic methods ? Example: An elliptic curve • E/Q 19 Base change 7 ! There should exist an analytic geometry over Complex Analysis •RigidE(C) Anal! C/Λysis EQv C19p by which Z C×p C×p/q – v E(C2), 3,C5/,Λ7, . , : placeE(Q ) Q /q−Z→ ∈ { ! ∞} can bepreg!arded×p as an analytic map. – Q : v-adic completion of QApplication: v Z • E(Qp) ! Q×p/q Nagell-Lutz Theorem. K: algebraic number field, E/K: an elliptic curve. = E(K)tor is a finite set. ⇒ 1 7 There should exist an analytic geometry over • Cp by which C C /qZ ×p −→ ×p can be regardedNasotationan analytic map. Application: • Nagell-Lutz Theorem. (K,K| : |):algebraic completenumber non-archimedeanfield, valued field with non-trivialE/K: an ellipticvaluationcurv | e.|. = E(K)tor is a finite set. ⇒ : K R : multiplicative valuation of ht. 1, • | | → 0 i.e., ≥ (1) x = 0 x = 0. | | ⇐⇒ (2) xy = x y . | | | || | (3) x + y max x , y . | | ≤ {| | | |} : non-trivial K ! 1 . • | | ⇐⇒ | ×| { } Tate’s Rigid Analytic 14 Geometry n k[X1, . , Xn] Ak Algebraic Geometry /k Rigid Geometry /K K X , . , X Dn 1 n KTopologically finitely Function !! Finitely generated"" generated algebra A/K algebra algebra A/k K X , . , Xn (called: Affinoid algebra) !! 1 "" ν1 νn ν ,...,νn 0 aν1,...,νnT Tn aν1,...,νn 0 as = Points1 ≥ Maximal 1ideals· · · of A| Maximal| → ideals of A K[[T1, . , Tn]] ν1 + + νn ! "(Naive) (with∈ Zariski topology) (with· · · Admissible→ ∞ topology)# kn Building Affine variety Affinoid Block (Spm A, ) OX /k = k /K = K 7. その他 ある。extps を指定すると、本文用の明朝は太ミンに、本文用ゴシックは太ゴになる2）。同時にじゅ んも利用可能になる。 これは、Times Roman フォントが和文に比べると太いために、バランスをとるためである。ただ し、普及型の PS プリンタではモリサワ 5 書体が載っておらず、リュウミンと中ゴシック BBB しか 持っていないことが多いので使用には注意が必要。また、太ミンと太ゴは最近の流行ではない。最近 は見出しミンと見出しゴが多いのかも知れない。 また、平成書体を搭載した PS プリンタでも当然利用できない。注意が必要。Ghostscript では簡 単に追加できる。ただし、TrueType フォントが別途必要。それは Acrobat でも同様。 7.6 その他の命令 るび その他にもいくつか便利な命令を提供している。まず、本文にルビを付けるための命令である Y=ru by{ruby}命令がある。Y=ruby{ 漢字 }{ よみ }として利用する。 ! " ! " また和文ではダッシュは全角ではなく、倍角を用いる。倍角ダッシュを引く命令は Y=dash{dash} である。 難しい内容を記述した段落には「危険な曲がり角」を付けるというブルバキ（Nicolas Bourbaki） 3 流の記述方法がある ）。TEX は標準でそのためのフォントを持っている。ブルバキが用いたフォント There’s no a priori reason why one should • difficult take maximal ideals as points.より は遙かに洗練されていて格好がいい。それを用いることもできる。そ のためには 環 (Depends on approaches; there境を are用いる。 three others.) 最後に便利な命令をお教えしよう。本稿では何ヶ所かで欧文の途中で改行しているものがある。特 In Tate’s approach, one has to introduce the • T X Y=allowdisplaybreaks admissible topology as a Grothendieckに E の 説明を書いている場合によく直面する。たとえば、 なんて命令を topology. 説明していたら悲劇である。常に行をはみ出すかどうかでドキドキできる。和文では別に単語中で改 行したってかまわないのである。 しかし、Y=verb 命令を用いることはできない。今までの Another Manual シリーズでは改行しそう な場所に適当に Y=linebreak[2] を入れて対応しているのだが、Y=footnY=linebreak[2]ote なんて でてきたら思わず笑ってしまう。だから我々には bs2yen パッケージで対応する方法は許されなかっ た。これは面倒だ。そこで、Y=cs 命令を用意した。これは引き数に Y=を付けてタイプライタ体で出力 する。もちろん Y=の直後の改行は禁じられているのでずいぶん安心である。そしてなんといっても、 自動的に改行してくれるのである。四分アキの問題についてはずいぶんトリッキーな対策になってし まった。興味のある人は miscmac.sty ファイルを参照してほしい。もうちょっとだけ命令をみること ができるだろう。 さらに、plate.sty ファイルや moving.sty ファイルの内容も役に立つだろう。 7.7 おわりに このクラスファイルと多くのパッケージファイルは多くの著作の影響を受けている。筆者自身のも のも多数あるが、基本的にこのクラスファイルの使用に関しては著作権問題を気にする必要はない。 クラスファイルを作成したり調整したりしたい人の一助になれば幸いである。 2）ところで、Another Manual シリーズの本文は太ミンで 3）Nicolas Bourbaki 著、EL´ EMENTS´ DE ある。あまり気づいた人はいないようだが……。本文も欧 MATHEMA´ TIQUE 文が 9 ポイント、和文が 12 級強である。 11 n k[X1, . , Xn] A Example of Affinoid Algebrak n K X , . , Xn D !! 1 "" k 8 K X1, . , Xn !Basic! Block:"" Tate algebra. • ν1 νn – (νK,...,, νn):0 completeaν1,...,νnT non-archimedeanTn aν1,...,νn valued0 as = 1 | | ≥ 1 · · · | | → field. K[[T1, . , Tn]] ν1 + + νn ! " n ∈ n · · · → ∞ # – D = z = (z , . , zn) K z 1 . K { 1 ∈ | | i| ≤ } = The algebra of power series converging abso- Dn lutely and uniformly on closed unit disk “ K”. : Banach algebra with Gauss norm • Noetherian and every ideal is closed with re- • spect to the subspace topology. Structure of general affinoid algebras: • A = K X1, . , Xn /I ## 14 $$ Banach algebra by the induced norm ! A = K X , . , Xn /I (I: ideal) is a banach ## 1 $$ algebra by the induced norm . % % n k[X1, . , Xn] Ak n n K X1, . , Xn DK k[X1, . , Xn] Ak !! "" n K X , . , Xn K X1, . .D, Xn !! 1 "" !! K "" ν a T 1 T νn a 0 as ν1,...,νn 0 ν1,...,νn 1 n ν1,...,νn K X , . , X = ≥ · · · | | → 1 n K[[T1, . , Tn]] ν1 + + νn !! , . , "" A! "n ∈ · · · → ∞ # k[X1 Xn] ν1 νkn ν ,...,νn 0 aν1,...,νnT Tn aν1,...,νn 0 as n = 1 ≥ 1 · · · | | → k K[[T1, . , Tn]] ν1 + + νn ! " ∈ · · · → ∞ # K X , . , X Dn 1 n n K (Max(A), X) !! "" k O (Max(DictionarA), ) y /k = k OX K X1, . , Xn n !! "" k[X1, . , Xn] A Algebraic Geometry/k = k Rigid Geometry/K = K k ν1 νn ν ,...,νn 0 aν1,...,νnT Tn aν1,...,νn 0 as = 1 ≥ 1 · · · Basic| Block: Tate| →algn ebra. n Kk[[[TX1,,......,,T•Xn]]/]K ν= K+ K+AXν1, . , Xn D 1Basic Block:n– (TKate,1 alg): completeebra!!. knnon-archimedean"" valued field.k ! " ∈ • | | · · · → ∞ # – (K, ): –completeDn = znon-archimedean= (z , . , z ) Kvnaluedz field.1 . n | | K 1 nn i k n { nn ∈ 14| | | ≤ } K kX[X–,1D.,....=,.X,zXn=n](z1, . , zn)withDKA zi 1 . !! 1 K ={The"" algebra of po∈werk |kseries| | ≤ con} verging absolutely = The algebraandofuniformlypower serieson Dconn . verging absolutely 14 K Dn and uniformlyn : Banachon algebraK. with Gaussn norm k[X1, . , Xn] K XA1, . .•, Xn D !!: Banachk algebra"" with Gauss normK Affi• ne space Closed unit polydisk n K X1, . , Xn Dk !! K X1,".". , Xn Noetherian and every ideal is closed with respect to !! "" • Noetheriantheandsubspaceevery idealtopologyis closed. with respect to • ν1 νn ν ,...,νn 0 atheν1subspace,...,νnTStructuretopologyofTn.generalaν1a,...,ffinoidνn algebras:0 as = 1 ≥ • 1 · · · | | → StructureK[[Tof1,general. , Tanffi]]noidA ν=1Kalgebras:+X1, . .+. , Xνnn /I ! " • ∈ ## · · · $$ → ∞ # BanachA =algebraK X ,by. .the, Xninduced/I norm ## 1 $$ Banach algebra! A = byK theX ,induced. , Xn /normI (I: ideal) is a banach alge- ## 1 $$ bra by the induced norm . ! A14= K X , . , Xn /I (I: ideal) is a banach alge- ## 1 $$ % % bra by the induced norm . % % 8 8 14 14 14 14 Basic Block: Tate algebra. • – (K, ): complete non-archimedean valued field. | | Dn Basic Propertiesn – K = z = (z1, . , zn) K zi 1 . { ∈ | | | ≤ } 15 = The algebra of power series converging absolutely K X , . , Xn : and!! 1uniformly""on Dn .
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