桔梗, 宏孝

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TitleACFAとManin-Mumford予想 (幾何学的モデル理論の研究)Author(s)桔梗, 宏孝Citation数理解析研究所講究録 (2002), 1283: 21-26Issue Date2002-09URL http://hdl.handle.net/2433/42406RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityACFA Manin-Mumfordkikyo ss.u-tokai.ac.jp1Hrushovski Manin-Mumford (Raynaud ).Ll $A$ $X$ $K$ (i.e. $\mathbb{Q}\subseteq K\subseteq \mathbb{Q}^{\mathrm{a}}$ ). $X\cap \mathrm{T}\mathrm{o}\mathrm{r}(A)$ $\mathrm{T}\mathrm{o}\mathrm{r}(B)+b$. , $B$ $A$ , $b\in A$ . ,$A$ $X$ ., Mordell-Lang ,, $\mathrm{T}\mathrm{o}\mathrm{r}(A)$ () . , $\mathrm{T}\mathrm{o}\mathrm{r}(A)$, LMS(locally modular, stable) . LMS. ,.Hrushovski , generic 1. ACFA (ACFA). $A$ $\mathrm{i}$ generic $\sigma$ ACFA, $\mathrm{T}\mathrm{o}\mathrm{r}(A)$ $\mathrm{K}\mathrm{e}\mathrm{r}(P(\sigma))$ $P[T]\in \mathbb{Z}[T]$ ., Weil , $P[T]$. , $\mathrm{K}\mathrm{e}\mathrm{r}(P(\sigma))$ locally modular stable .. Mordell-LangHrushovski , ., ACFA $\mathrm{T}\mathrm{o}\mathrm{r}(A)$ ,$p$ , $p$ .., . $K$ ,$K^{\mathrm{a}}$ . $\varphi$ $L$ , $A$1283 2002 21-2621$\varphiarrow L(A)$ . , $\varphi$ $\forall$ $\exists$ ,$\varphi \mathrm{C}L^{\ovalbox{\tt\small REJECT}}(A)$ $<$ ( $\mathrm{q}\mathrm{f}$ quanfifier-free ).2ACFA &Psf, $T$ (existentially closedmodel, $\mathrm{e}\mathrm{c}$ ) .2.1 $T$ $\mathcal{L}$ . $M$ $T$ ,2 .(1) $M\models T(M$ $T$ )(2) $M\subset N\models T,$ $\varphi(x_{1}, \ldots,x_{n})\in \mathcal{L}^{\mathrm{q}\mathrm{f}}(M),$ $N\models\exists x_{1},$$\ldots,$$x_{n}\varphi(x_{1}, \ldots, x_{n})$, $M\models\exists x_{1},$$\ldots,$$x_{n}\varphi(x_{1}, \ldots, x_{n})$ .$\mathcal{L}$$\{+, -, \cdot, 0,1\}$ , $T$ , $\varphi(x_{1}, \ldots,x_{n})$$M$ , $M$ $T$, $M$ Hilbert .$\cdot$,. $\mathcal{L}$. Hrushovski Manin-Mumford , $K$ , $\sigma$1 , $(K, +, -, \cdot, 0,1, \sigma)$.2.2 $\mathcal{L}$ ACF , $\mathcal{L}_{\sigma}=\mathcal{L}\cup\{\sigma\}$ , ACF, $=\mathrm{A}\mathrm{C}\mathrm{F}\cup${\sigma } (L ) . ACF, differenoe field.2.3 $(K, \tau)\models \mathrm{A}\mathrm{C}\mathrm{F}_{\sigma}$ , $(K,\tau)$ $(\tilde{K},\tilde{\tau})$ ACF,. ACF, $\mathcal{L}_{\sigma}$$[1, 7]$ . ACFA .$(K, \sigma)$ $\mathrm{A}\mathrm{C}\mathrm{F}_{\sigma}$ , $\sigma$ generic . $\sigma$( ). ACF, , $\forall x,$ $\ldots\exists y,$ $\ldots\varphi(x, \ldots, y, \ldots)(\varphi\in \mathcal{L}_{\sigma}^{\mathrm{q}\mathrm{f}})$. -# , ,. , . ACFA,$\mathcal{L}$ .$T$ ACFA. $T$ ACF [6].222.4 $\mathcal{L}$ . $\mathcal{L}$ , “” Psf ., , Psf() ., $M$ ACFA Psf , $M$ ACFA $M$Psf .25(Ax) $K$ Psf 3.(1) $K$ .(2) $K$ $K^{\mathrm{a}}$ ( $K$ ) PAC ( $I\dot{\mathrm{t}}^{r}$ $K$).(3) $n\geq 1$ , Il\nearrow $n$ $I\mathrm{t}^{\nearrow \mathrm{a}}$ 1 .2.6 $(I\mathrm{t}^{\nearrow}, \sigma)$ ACFA , $k=\mathrm{f}\mathrm{i}\mathrm{x}(\sigma)$ $=\{x\in K : \sigma(x)=x\}$.(1) $n\geq 1$ , $(I\iota^{\nearrow}, \sigma^{n})$ ACFA.(2) $k=\mathrm{f}\mathrm{i}\mathrm{x}(\sigma)$ [ Psf.(3) fix(\sigma ) $n$ $\mathrm{f}\mathrm{i}\mathrm{x}(\sigma^{n})$ .(4) $k=\mathrm{f}\mathrm{i}\mathrm{x}(\sigma)$ $K$ stably embedded. , $\mathcal{L}_{\sigma}(K)$$k^{n}=k\cross\cdots\cross k$ ( $n$ ) , $\mathcal{L}_{\sigma}(k)$.$\mathrm{G}\mathrm{a}1(\mathrm{f}\mathrm{i}\mathrm{x}(\sigma)^{\mathrm{a}}/\mathrm{f}\mathrm{i}\mathrm{x}(\sigma))$ $\langle\sigma\rangle$ ,.3LMS, $P(X)$ , ACFA$(K, \sigma)$ $\mathrm{K}\mathrm{e}\mathrm{r}(P(\sigma))$ LMS (locally modular stable set).31ACFA . $D$. $D$ $a=(a_{1}, \ldots, a_{m})$ $E$ , $\mathrm{t}\mathrm{p}(a/E)$non-forking ( ) , $D$23stable . , $D$ $A$ . $D^{\mathrm{q}}$$B,$ $C$ , $B$ $C$ $\mathrm{a}\mathrm{c}1(BA)\cap \mathrm{a}\mathrm{c}1(CA)$ , $D$ locallymodular .[1].32($\mathrm{A}\mathrm{C}\mathrm{F}\mathrm{A}_{0}$ Dichotomy Theorem) 0 ACFA , $\sigma$ generic, $X$ . $SU(X)=1$ , $X$ LMS \searrow$X\angle \mathrm{f}\mathrm{i}\mathrm{x}(\sigma)$ .Chatzidakis, Hrushovski [1] . ACFAunstable , 0 , unstable fix(\sigma ) (Psf) .33 $B$ $\mathrm{c}$ , $B$ $H$$[B:H]$ .3.4 $(K, \sigma)$ 0 ACFA, $k=\mathrm{f}\mathrm{i}\mathrm{x}(\sigma)$ , $A$ $k$, $B$ $A$ $\mathrm{c}$ , $B$ LMS ,$B\subset \mathrm{K}\mathrm{e}\mathrm{r}(\sigma^{N}-.1)$ $N$ . A=G ., .3.5 $A$ $k$ , $P(X)\in \mathbb{Z}[X]$ , $\mathrm{k}^{r}\mathrm{e}\mathrm{r}(P(\sigma))$LMS , $P(X)$ .3.4 ., $C\subset K$ . $B$ $C$ $\sigma$ , $X\subset B$ , $X$ $C$$SU(X)=1$ ., SU Zilber , $n$, $B’=(XX^{-1})^{n}$ $B$ .$B$ $\mathrm{c}$ , $B’= \bigcap_{:}H.\cdot,$ $[B : H.\cdot]$ , . , $H_{-}$.$B$ , $B’$ , $X$LMS $B’$ LMS , , $B$ LMS ., $X$ LMS .$\mathrm{A}\mathrm{C}\mathrm{F}\mathrm{A}_{0}$ Dichotomy Theorem , $X\angle \mathrm{f}\mathrm{i}\mathrm{x}(\sigma)=k$ . ,$h:Xarrow k$ 1 ., $B’\subset \mathrm{a}\mathrm{c}1(k\cup C)$ . $B/B’$, $C$ $B\subset \mathrm{a}\mathrm{c}1(k\cup C)$ .$k$ Psf , Psf Hrushovski Pilay[5], , $h’$ : $B\mapsto H(k)$24( , $h’$ $\mathrm{K}\mathrm{e}\mathrm{r}(h’)$ ). , $H$ $k$. $k$ stably embedded ,$k$ $k_{1}$ . , $k_{1}=\mathrm{f}\mathrm{i}\mathrm{x}(\sigma^{n})$$n$ . $k_{1}$ $B/\mathrm{K}\mathrm{e}\mathrm{r}(h’)$, $n|n’$ , $B/\mathrm{K}\mathrm{e}\mathrm{r}(h’)$ [ $\sigma^{n’}$ .$\mathrm{K}\mathrm{e}\mathrm{r}(h’)$$\sigma^{n}$ , , $n|l$$l$ [ $\sigma^{l}$ [ . $\tau=\sigma^{l}$ , $a\in \mathrm{K}\mathrm{e}\mathrm{r}(h’)$ $\tau(a)=a$. $\mathrm{K}\mathrm{e}\mathrm{r}(h’)$ $m$ . $x\in B$ , $\tau$ $x$, $\tau(x)=x+a(a\in \mathrm{K}\mathrm{e}\mathrm{r}(h’))$ . , $\tau^{2}(x)=\tau(x+a)=$$\tau(x)+\tau(a)=(x+a)+a=x+2a,$ $\tau^{3}(x)=x+3a,$ $\ldots,$ $\tau^{m}(x)=x+ma=x(m$$\mathrm{K}\mathrm{e}\mathrm{r}(h’)$ $a\in \mathrm{K}\mathrm{e}\mathrm{r}(h’)$ ma $=0$). , $B$ [ $N$$\sigma^{N}$ . , $B\subset \mathrm{K}\mathrm{e}\mathrm{r}(\sigma^{N}-1)$ ..4Manin-MumfordACFA, , p$n$ t $n$ .. .$A$ $K$ . $K$ $\mathrm{p}$ $A$$A$ , $A$. $\mathrm{p}$ prime of good reduction for $A$ . $\mathrm{p}$ $k$, $p$ . $p$ $n$ $n$ $\mathrm{T}\mathrm{o}\mathrm{r}_{p’}(A)$, $k^{\mathrm{a}}$ .4.1 $X$ $A$ . $X\cap \mathrm{T}\mathrm{o}\mathrm{r}_{p’}(A)$ $A$.( ) $A_{\mathrm{p}}$ $A$ $k$ (in $k^{\mathrm{a}}$). $K_{\mathrm{p}}$ $K$ , $v$ $k$ . $L=K_{\mathrm{p}}^{\mathrm{a}}$$L/\mathrm{A}_{\acute{\mathrm{p}}}$ $v$ $L$ , $k^{\mathrm{a}}$. , $\pi$ : $A(L)arrow A_{\mathrm{p}}(k^{\mathrm{a}})$ . , $\pi$$\mathrm{T}\mathrm{o}\mathrm{r}_{p’}(A)$ $\mathrm{T}\mathrm{o}\mathrm{r}_{p’}(A_{\mathrm{p}})$ ., $q=|k|$ , $\tau(x)=x^{q}$ $\tau\in \mathrm{A}\mathrm{u}\mathrm{t}(k^{\mathrm{a}})$ (). Weil , $P(\tau)$ $A_{\mathrm{p}}(k^{\mathrm{a}})$ 0$P(T)$ 1 . $\rho$ $\tau$ $\mathrm{G}\mathrm{a}1(L/K_{\mathrm{p}})$. , $P(\rho)$ $\mathrm{T}\mathrm{o}\mathrm{r}_{p’}(A)$ 0 . $(L, \rho)$ACFA $(F, \sigma)$ . $\mathrm{T}\mathrm{o}\mathrm{r}_{p’}(A)$ $B=\mathrm{K}\mathrm{e}\mathrm{r}(P(\sigma))$25. 35 , $B$ LMS $\mathrm{S}\mathrm{U}$ . $X$ $A$$X\cap B$ $B$ ,. , $X\cap B$ . ,$X\cap \mathrm{T}\mathrm{o}\mathrm{r}_{p^{\mathrm{t}}}(A)$ .[1] Z. Chatzidakis and E. Hrushovski, The model theory of difference fields,Transactions of AMS, 351 (1999), 2997-3071.[2] Z. Chatzidakis and A. Pilay, Generic structures and simple theories, Annalsof Pure and Applied Logic 95 (1998), 71-92.[3] E. Hrushovski, The Manin-Mumford conjecture and the model theory of dif-ference fields, Annals of Pure and Applied Logic 112 (2001), 43-115.[4] E. Hrushovski, A. Pillay, Weakly normal groups, Logic Colloquium ’85, NorthHoland, Amsterdam, 1987, 233-244.[5] E. Hrushovski, A. Pillay, Definable subgroups of algebraic groups over finitefields, J. Reine Angew. Math. 462 (1995), 69-91.[6] H. Kikyo and A. Pillay, The definable multiplicity property and generic autO-morphism, Annals of Pure and Applied Logic 106 (2000), 263-273.[7] A. Macintyre, Generic automorphisms of fields, Annals of Pure and AppliedLogic 88 (1997), 165-180.[8] A. Pillay, ACFA and the Manin-Mumford conjecture, in Algebraic ModelTheory, B. Hart A. Lachlan, M. Valeriote Eds NATO ASI Series Vol. 496,195-205.[9] A. Pillay, Lecture notes on strongly minimal sets (and fields) with agenericautomorphism, 1999-2000 (UIUC) .26

ACFAとManin-Mumford予想 (幾何学的モデル理論の研究)

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ACFAとManin-Mumford予想 (幾何学的モデル理論の研究)

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