From e914a9d7727319784285639840e4914a76c7d569 Mon Sep 17 00:00:00 2001 From: Alexander Luzgarev Date: Tue, 19 Jul 2016 20:28:51 +0200 Subject: [PATCH] Add Lecture 10 --- motives.pdf | Bin 675641 -> 713834 bytes motives.tex | 621 +++++++++++++++++++++++++++++++++++++++++++++++++++- 2 files changed, 619 insertions(+), 2 deletions(-) diff --git a/motives.pdf b/motives.pdf index a292276e496a1c1bc21608ab2fac10161bce66fc..517ca70b1783b371b03ce8aae225aad4afedbca8 100644 GIT binary patch delta 223476 zcmZs?RaBlm7cGjryF+nzheC08r?|U&fd{9!ytum*cX!v~?p|Dq+v)f3arU`6W8@+k z$(qU4nsX&HZ8>zOqYP+N%HontY|LB;R5J@hD+p|<&SvnKbrj`D;Cu~JNN1?2WXTX% z4U}Gg>w##CTrOBC2weQ9Dotiu)%{ zP>3$7wPaXXPEaZAa4gP^l3umyONfrnNH%-K;=Sak=-Mcxq0Tu_)<2 zF$T#)X@jsd8)9X(<+1wq@HU|st0`wdJ(^R?baa?zL-d`2wB0(LF#Dy zUiD-Q5YE@oL^?d$(Dl1e{twedf^Mh5Ju4yyQ0d`XMi5{SMd81rRux zcH^3nA@QoL)|5|*bw(~s8fG1~mHWl2Kf1?W@Wyyn0cRKy>m?dHdjqB_6i^|BN>1c$ z_)umjZ4a(ewdHX|#ZafR(ooB`VooQ=!Hv>YQR7w7Qf6UITs8lioBX}X(sC|0{$}U) ztKuC1msW4FC3CemBTE%qw5Jk07QtR{`* 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+\usetikzlibrary{decorations.pathreplacing} \DeclareFontFamily{OT1}{pzc}{} \DeclareFontShape{OT1}{pzc}{m}{it}{<-> s * [1.1] pzcmi7t}{} @@ -65,6 +68,7 @@ \DeclareMathOperator{\Trd}{Trd} \DeclareMathOperator{\sing}{sing} \DeclareMathOperator{\Bl}{Bl} +\DeclareMathOperator{\Frac}{Frac} %\DeclareFontFamily{OT1}{pzc}{} %\DeclareFontShape{OT1}{pzc}{m}{it}{<-> s * [1.2] pzcmi7t}{} @@ -3767,13 +3771,626 @@ $Y_1(F)\neq\varnothing$ тогда и только тогда, откуда ${}_{\xi}(\mathsf{F}_4/P_4)(F)\neq\varnothing$. Мы получили картинку -$$ +\[ \begin{tikzcd} \Bl_{\Spin_9/P_1}({}_{\xi}(\mathsf{F}_4/P_4)) \arrow{r}{\isom}\arrow{d} & \Bl_{\Spin_9/P_4}Q\arrow{d} \\ {}_{\xi}(\mathsf{F}_4/P_4) & Q \end{tikzcd} -$$ +\] + +% 23.04.2012 + +\subsection{Многообразия, клеточные над общей точкой} + +Вернемся к общей ситуации: по коциклу $\xi\in H^1(F,G)$, где +$G$~--- расщепимая группа, и простому числу $p$ мы построили +набор чисел $J_p(\xi) = (j_1,\dots,j_r)$. +Теорема Зайнуллина--Петрова--Семенова~\ref{thm:ZPS} утверждает, +что если $X$~--- ${}_{\xi}G$-однородное многообразие, клеточное +над $F(X)$, то $M(X)\otimes\mathbb{Z}/p = \bigoplus R_p(G)\{\dots\}$, +и +\[ +P(R_p(G),t) = \prod\frac{1-t^{p^{j_i}d_i}}{1-t^{d_i}}. +\] +\begin{remark} +При помощи знания $J_p(\xi)$ для всех $p$ можно описать все такие $X$. +\end{remark} +Выше мы встречали точную последовательность вида +\[ +\CH^*(BB) \to \CH^*(G/B) \to CH^*(G) \to 0. +\] +Пусть $X = {}_{\xi}(G/P)$. +Попробуем нарисовать аналогичную последовательность для $P$: +\[ +\begin{tikzcd} +\CH^*(BP) \arrow{d} \\ +\CH^*(G/P) \arrow{r} & \CH^*(G)\arrow{r} & \CH^*(P)\arrow{r} & 0 +\end{tikzcd} +\] +Нижняя строка этой диаграммы является точною последовательностью +градуированных колец. +Однако, точность всей последовательности в члене $\CH^*(G/P)$ +неизвестна. +При этом $\CH^*(P) \isom \CH^*(L)$ (поскольку $P/U\isom L$, +а $U$ аффинно). +Более того, мы утверждаем, что $\CH^*(L) \isom \CH^*([L,L])$. +Почему это так? +Рассмотрим коммутативную диаграмму +\[ +\begin{tikzcd} +\CH^*(BB_L) \arrow{r}\arrow{d} +& \CH^*(L/B) \arrow{r}\arrow{d}{\isom} +& \CH^*(L) \arrow{d}\arrow{r} +& 0 \\ +\CH^*(BB_{[L,L]}) \arrow{r} +& \CH^*(L/B) \arrow{r} +& \CH^*([L,L]) \arrow{r} +& 0 +\end{tikzcd} +\] +Здесь $B_L$ обозначает борелевскую подгруппу в $L$, а +$B_{[L,L]}$~--- борелевскую в $[L,L]$. +Левая вертикальная стрелка не обязана быть изоморфизмом, +однако образы $\CH^*(BB_L)$ и $\CH^*(BB_{[L,L]})$ в +$\CH^*(L/B)$ совпадают. +Это утверждение достаточно проверить на $\CH^1$. + +Обозначим $H = [L,L]$. +Мы получили точную последовательность +\[ +\CH^*(G/P)/p \to \CH^*(G)/p \xrightarrow{\ph} \CH^*(H)/p \to 0. +\] +Нам известно, что +$\CH^*(G)/p = (\mathbb{Z}/p)[x_1,\dots,x_r]/(x_i)^{p^{k_i}}$ +и +$\CH^*(H)/p = (\mathbb{Z}/p)[y_1,\dots,y_s]/(y_j)^{p^{l_j}}$. + +Оказывается, есть отображение $\sigma\colon\{1,\dots,s\}\to\{1,\dots,r\}$ +такое, что +\[ +\ph(x_{\sigma(m)}) = c\cdot y_m + \mbox{члены меньшего порядка}, +\] +где $c\in (\mathbb{Z}/p)^*$~--- некоторая константа. + +\begin{theorem}\label{thm:cellularity} +Пусть $G$~--- простая расщепимая группа, +$\xi\in H^1(F,G)$, $X = {}_{\xi}(G/P)$, $Y = {}_{\xi}(G/B)$. +Тогда следующие условия эквивалентны. +\begin{enumerate} +\item $X$~--- клеточное над общей точкой. +\item Композиция отображений +\[ +\CH^*({}_{\xi}(G/B)) \xrightarrow{\res} \CH^*(G/B) \to \CH^*(G) \to +\CH^*(P) +\] +сюръективна. +\item Для любого простого $p$ выполняются условия +\begin{enumerate} +\item $j_{\sigma(m)}(\xi) = 0$, если $m$ такое, что $\deg(y_m) > 1$; +\item\label{item:thm-mod-p} композиция отображений +\[ +\CH^1({}_{\xi}(G/B))/p \xrightarrow{\res} \CH^1(G/B)/p \to \CH^1(G)/p \to +\CH^1(P)/p +\] +сюръективна. +\end{enumerate} +\end{enumerate} +\end{theorem} +\begin{example} +Пусть $\xi\in H^1(F,\mathsf{E}_7)$, $X = {}_{\xi}(\mathsf{E}_7/P_7)$. +Когда $X$ расщепимо над общей точкой? +\begin{itemize} +\item $p=2$. В этом случае +\[ +\begin{tikzpicture}[remember picture] +\node (a1) {$\CH^*(\mathsf{E}_7)/2 = (\underset{\deg:}{\mathbb{Z}/2}) +[\underset{1}{x_1}, \subnode{x1}{$\underset{3}{x_2}$}, \underset{5}{x_3}, +\underset{9}{x_4}]/(\dots)$}; +\node (b1) [below=of a1] {$\CH^*(\mathsf{E}_6)/2 = +(\underset{\deg:}{\mathbb{Z}/2})[\subnode{y1}{$\underset{3}{y_1}$}]/(y_1^2)$}; +\draw[|->] (x1.south) -- (y1.north); +\end{tikzpicture} +\] +Условие~(\ref{item:thm-mod-p}) теоремы~\ref{thm:cellularity} +превращается в $j_2 = 0$. +Элементы $x_3$, $x_4$ получаются операцией Стинрода, и потому +$j_3 = j_4 = 0$ автоматически. +\item $p=3$. В этом случае +\end{itemize} +\[ +\begin{tikzpicture}[remember picture] +\node (a2) {$\CH^*(\mathsf{E}_7)/3 = (\underset{\deg:}{\mathbb{Z}/3}) +[\subnode{x2}{$\underset{4}{x_1}$}]/(x_1^3)$}; +\node (b2) [below=of a2] {$\CH^*(\mathsf{E}_6)/3 = +(\underset{\deg:}{\mathbb{Z}/3})[\subnode{y2}{$\underset{4}{y_1}$}]/(y_1^3)$}; +\draw[|->] (x2.south) -- (y2.north); +\end{tikzpicture} +\] +Условие состоит в том, что $j_1=0$; и тогда весь $j$-инвариант для модуля $3$ +равен нулю. +\end{example} +Задача: посчитать $j_2$ для компактной формы $\mathsf{E}_7$ над $\mathbb{R}$. +Эквивалентно: +\begin{equation}\label{eqn:condition-e7} +\begin{array}{p{0.8\textwidth}} +верно ли, что над полем $\mathbb{R}(\SB(\mathbb{H}))$ +(которое изоморфно $\Frac(\mathbb{R}[x,y]/(x^2+y^2-1))$) +компактная форма $\mathsf{E}_7$ расщепляется? +\end{array} +\end{equation} +Еще одна эквивалентная формулировка: пусть $R_2(\xi)$~--- мотив Роста, +отвечающий кватернионам. +Сравним ${}_{\xi}(\mathsf{E}_7/P_7)$ и $\SB(\mathbb{H})$. +Над $F({}_{\xi}(\mathsf{E}_7/P_7))$ алгебра Титса ($\mathbb{H}$) +тривиальна, и потому $\SB(\mathbb{H})$ приобретает рациональную точку. +Если условие~\ref{eqn:condition-e7} выполняется, то верно и обратное: +над $F(\SB(\mathbb{H}))$ многообразие ${}_{\xi}(\mathsf{E}_7/P_7)$ +имеет рациональную точку. Почему это так? +\begin{theorem} +Пусть $X,Y$~--- проективные однородные многообразия (возможно, +относительно разных групп), $p$~--- простое число. +Предположим, что $X_{F(Y)}$ и $Y_{F(X)}$ имеют рациональные точки. +Тогда $M(X)\otimes(\mathbb{Z}/p)$ и $M(Y)\otimes(\mathbb{Z}/p)$ +имеют общее слагаемое, <<задевающее>> точку +(то есть, $q\in\CH^{\dim X}(X\times X)$~--- проектор, +и $\im(q)$ над замыканием содержит $\pt\in\CH^{\dim X}(X)$). +\end{theorem} + +\begin{proposition} +Если есть изотропная ${}_{\xi}G$, и анизотропное ядро типа $H$, +то $\xi$ приходит из $\zeta\in H^1(F,H)$. +Тогда $J_p(\xi)$ выражается через $J_p(\zeta)$: +\[ +j_i(\xi) = \begin{cases} j_m(\zeta), & \text{если $i=\sigma(m)$ +для некоторого $m$};\\ +0, & \text{иначе}. +\end{cases} +\] +Поэтому имеет смысл считать $J_p(\xi)$ только для анизотропных +групп (то есть, над $\mathbb{R}$~--- только для компактных форм). +\end{proposition} +\begin{itemize} +\item $\mathsf{G}_2$, $\mathsf{F}_4$: +$(\underset{\deg:}{\mathbb{Z}/2})[\underset{3}{x_1}]/(x_1^2)$, +$J_2(\xi) = (1)$~--- а не $0$, ибо у $\mathbb{R}$ нет расширений нечетной +степени. При этом $R_2(\xi)$~--- мотив Роста $\lAngle -1,-1,-1\rAngle$. +\item $\mathsf{E}_6$: компактная форма есть, но она внешняя. +\item $\mathsf{E}_7$: $(\underset{\deg:}{\mathbb{Z}/2}) +[\underset{1}{x_1}, \underset{3}{x_2}, \underset{5}{x_3}, \underset{9}{x_4}]$. +\end{itemize} +\begin{proposition}\label{prop:j-inv-for-e7} +В этом случае $J_2(\xi) = (1, 0, 0, 0)$, $R_2(\xi)$~--- это мотив Роста +квадрики $\lAngle -1, -1\rAngle$. +\end{proposition} +Для доказательства этого утверждения нужен \term{инвариант Роста}. + +\subsection{Инвариант Роста} +Пусть $G$~--- простая, односвязная, но не обязательно расщепимая группа. +Тогда есть инвариант +\[ +r\colon H^1(F,G) \to H^3(F, (\mathbb{Q}/\mathbb{Z})(2)) += \varinjlim_{N} H^3(F, \mu_N^{\otimes 2}), +\] +который порождает всю абелеву группу таких инвариантов. +Вместо предела в правой части можно взять одно достаточно большое $N$. +Заметим, что $r(\xi)$ всегда лежит в кручении. +Посмотрим на наименьшее $N_G$ такое, что всегда $N_G\cdot r(\xi) = 0$ +(для всех расширений $E/F$ и для всех $\xi$). +Тогда $N_G$ зависит только от типа $G$. +А именно, для исключительных групп +$N_{\mathsf{G}_2} = 2$, $N_{\mathsf{F}_4} = N_{\mathsf{E}_6} = 6$, +$N_{\mathsf{E}_7} = 12$, $N_{\mathsf{E}_8} = 60$. +Для $\mathsf{G}_2$ мы это видели: +$\mathsf{G}_2 = \Aut(\mathbb{O})$, и $\mathbb{O}$ определяется формой +$\lAngle a, b, c\rAngle \in H^3(F,\mathbb{Z}/2)$. +Для $\mathsf{F}_4$, $\mathsf{E}_6$: $f_3 = 3r\in H^3(F, \mathbb{Z}/2)$, +$g_3 = 2r \in H^3(F,\mathbb{Z}/3)$, где +$r\in H^3(F,\mathbb{Z}/6)$. + +\begin{exercise}[Исследовательская задача] +Придумать формулу для $r\colon H^1(F,G) \to H^3(f, \mathbb{Z}/4)$, +где $G$~--- односвязная группа типа $\mathsf{D}_6$ +с индексом Титса +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (4) at ($\sm*(4.2, 0)$); +\coordinate (5) at ($\sm*(5.2, 1)$); +\coordinate (6) at ($\sm*(5.2, -1)$); +\draw (1)--(2)--(3)--(4)--(5); +\draw (4)--(6); +\foreach \point in +{1,2,3,4,5,6} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{2,4,5} + { + \draw [black] (\point) circle (5.0pt); + } +\end{tikzpicture} +\] +Такая группа задается кватернионами. +\end{exercise} +\begin{theorem}[Черноусов--Гарибальди] +Если $G$~--- расщепимая группа типа $\mathsf{G}_2$, $\mathsf{F}_4$, +$\mathsf{E}_6$ или $\mathsf{E}_7$, то ядро инварианта Роста тривиально. +\end{theorem} +Пусть $\xi\in H^1(F,G)$. +Хотим узнать, тривиален ли $\xi$. +Теорема Черноусова--Гарибальди говорит, что (оказывается!) +достаточно посчитать $r(\xi)$. +Для $\mathsf{E}_8$ это неверно: можно взять +$\xi\in H^1(\mathbb{R},\mathsf{E}_8)$, задающий компактную форму. + +Воспользуемся этой теоремой для доказательства +утверждения~\ref{prop:j-inv-for-e7}. +Достаточно доказать, что $\xi_{F(SB(\mathbb{H}))} = *$. +Действительно, если это так, то ${}_{\xi}(\mathsf{E}_7/B)$ +и $\SB(\mathbb{H})$ имеют рациональные точки над полями функций +друг друга. +Тогда у них есть общее мотивное слагаемое +$R_2(\xi) = R_2(\lAngle -1, -1\rAngle)$, и потому $J_2(\xi) = (1,0,0,0)$ +из формулы для полинома Пуанкаре. + +Мы должны представить $\xi$ как элемент $H^1(F,G')$, +где $G'$ односвязна. +Расщепимая $G'$ не подойдет: $\xi$ приходит из +$H^1(F, \mathsf{E}_7^{\operatorname{ad}})$, +но не из $H^1(F, \mathsf{E}_7^{\operatorname{sc}})$ +(поскольку алгебра Титса нетривиальна). +Возьмем в качестве $G'$ группу с индексом Титса +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(2.8, -1.4)$); +\coordinate (3) at ($\sm*(1.4, 0)$); +\coordinate (4) at ($\sm*(2.8, 0)$); +\coordinate (5) at ($\sm*(4.2, 0)$); +\coordinate (6) at ($\sm*(5.6, 0)$); +\coordinate (7) at ($\sm*(7.0, 0)$); +\draw (1)--(3)--(4)--(5)--(6)--(7); +\draw (2)--(4); +\foreach \point in +{1,2,3,4,5,6,7} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{1,3,4,6} + { + \draw [black] (\point) circle (5.0pt); + } +\end{tikzpicture} +\] +Тогда $\xi\in H^1(F,G')$. +Что можно сказать про $r(\xi)$? +Мы знаем, что $\xi$ расщепим над $\mathbb{C}$, и потому +$2r(\xi) = 0$. +Стало быть, $r(\xi) \in H^3(\mathbb{R}, \mathbb{Z}/2)$. +Значит, это либо $0$, либо $\lAngle -1, -1, -1\rAngle$. + +Переходим на $F(\SB(\mathbb{H}))$. +Тогда $r(\xi)$ в любом случае становится $0$, +а $G'_{F(\SB(\mathbb{H}))}$ расщепима. +Поэтому можно применить теорему Черноусова--Гарибальди, +и заключить, что $\xi_{F(\SB(\mathbb{H}))} = *$, +чего мы и добивались. + +Посмотрим теперь на компактную форму группы типа $\mathsf{E}_8$. +Мы знаем, что +\[ +\CH^*(\mathsf{E}_8)/2 = (\underset{\deg:}{\mathbb{Z}/2}) +[\underset{3}{x_1},\underset{5}{x_2},\underset{9}{x_3},\underset{15}{x_4}] +/(\dots). +\] +Инвариант Роста равен нулю; $j_1=0$, и потому $j_2 = j_3 = 0$. +Получаем, что $J_2(\xi) = (0, 0, 0, 1)$. +Отсюда следует, что $P(R_2(\xi), t) = 1 + t^{15}$ +и $R_2(\xi) = R_2(\lAngle -1, -1, -1, -1, -1\rAngle)$. + +Мораль: над $\mathbb{R}$ мотивы однородных проективных многообразий +(расщепимых над общей точкой) раскладываются в мотивы Роста +от пфистеровых форм с каким-то количеством $-1$; +количество зависит от типа группы. + +Пусть $\xi\in H^1(F, G)$, где $G$~--- расщепимая группа типа $\mathsf{E}_6$. +Свяжем $J$-инвариант с индексом Титса. +Например, $J_3(\xi)$ определяется индексом Титса над ко-$3$-замыканием +поля $F$. + +Что это значит? +Посмотрим на абсолютную группу Галуа $\operatorname{Gal}(\ol{F}/F)$ +и возьмем в ней $3$-силовскую подгруппу. +Ей соответствует расширение $E/F$, которое называется +\term{ко-$3$-замыканием $F$}. +Неформально говоря, мы игнорируем расширения степеней, не делящихся на $3$. + +\begin{tabular}{|c|c|c|c|c|c|} +\hline +$J_3(\mathsf{E}_6^{\operatorname{ad}})$ & $(0,0)$ & $(1,0)$ & $(0,1)$ & $(1,1)$ & $(2,1)$ \\ +\hline +индекс Титса & +\begin{tikzpicture}[scale=0.5,thin]\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(2.8, -1.4)$); +\coordinate (3) at ($\sm*(1.4, 0)$); +\coordinate (4) at ($\sm*(2.8, 0)$); +\coordinate (5) at ($\sm*(4.2, 0)$); +\coordinate (6) at ($\sm*(5.6, 0)$); +\draw (1)--(3)--(4)--(5)--(6); +\draw (2)--(4); +\foreach \point in +{1,2,3,4,5,6} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{1,2,3,4,5,6} + { + \draw [black] (\point) circle (5.0pt); + } +\end{tikzpicture} +& +\begin{tikzpicture}[scale=0.5,thin]\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(2.8, -1.4)$); +\coordinate (3) at ($\sm*(1.4, 0)$); +\coordinate (4) at ($\sm*(2.8, 0)$); +\coordinate (5) at ($\sm*(4.2, 0)$); +\coordinate (6) at ($\sm*(5.6, 0)$); +\draw (1)--(3)--(4)--(5)--(6); +\draw (2)--(4); +\foreach \point in +{1,2,3,4,5,6} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{2,4} + { + \draw [black] (\point) circle (5.0pt); + } +\end{tikzpicture} +& \multicolumn{3}{|c|}{ + \begin{tikzpicture}[scale=0.5,thin]\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(2.8, -1.4)$); +\coordinate (3) at ($\sm*(1.4, 0)$); +\coordinate (4) at ($\sm*(2.8, 0)$); +\coordinate (5) at ($\sm*(4.2, 0)$); +\coordinate (6) at ($\sm*(5.6, 0)$); +\draw (1)--(3)--(4)--(5)--(6); +\draw (2)--(4); +\foreach \point in +{1,2,3,4,5,6} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\end{tikzpicture} +}\\ +\hline +индекс алгебры Титса & $1$ & $3$ & $1$ & $3$ & $9$ или $27$\\ +\hline +\end{tabular} + +Приведем аналогичную таблицу для $J_2(\mathsf{E}_7^{\operatorname{sc}})$: + +\begin{tabular}{|p{2.9cm}|p{2.5cm}|p{2.6cm}|p{2.9cm}|p{2.5cm}|} +\hline +\begin{center} $J_2(\mathsf{E}_7^{\operatorname{sc}})$ \end{center} +& \begin{center} $(0,0,0)$ \end{center} +& \begin{center} $(1,0,0)$ \end{center} +& \begin{center} $(1,1,0)$ \end{center} +& \begin{center} $(1,1,1)$ \end{center} \\ +\hline +\begin{center}индекс Титса +над ко-$2$-замыканием\end{center} & +\begin{tikzpicture}[scale=0.5,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(2.8, -1.4)$); +\coordinate (3) at ($\sm*(1.4, 0)$); +\coordinate (4) at ($\sm*(2.8, 0)$); +\coordinate (5) at ($\sm*(4.2, 0)$); +\coordinate (6) at ($\sm*(5.6, 0)$); +\coordinate (7) at ($\sm*(7.0, 0)$); +\draw (1)--(3)--(4)--(5)--(6)--(7); +\draw (2)--(4); +\foreach \point in +{1,2,3,4,5,6,7} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{1,2,3,4,5,6,7} + { + \draw [black] (\point) circle (5.0pt); + } +\end{tikzpicture} +& +\begin{tikzpicture}[scale=0.5,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(2.8, -1.4)$); +\coordinate (3) at ($\sm*(1.4, 0)$); +\coordinate (4) at ($\sm*(2.8, 0)$); +\coordinate (5) at ($\sm*(4.2, 0)$); +\coordinate (6) at ($\sm*(5.6, 0)$); +\coordinate (7) at ($\sm*(7.0, 0)$); +\draw (1)--(3)--(4)--(5)--(6)--(7); +\draw (2)--(4); +\foreach \point in +{1,2,3,4,5,6,7} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{1,6,7} + { + \draw [black] (\point) circle (5.0pt); + } +\end{tikzpicture} +& +\begin{tikzpicture}[scale=0.5,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(2.8, -1.4)$); +\coordinate (3) at ($\sm*(1.4, 0)$); +\coordinate (4) at ($\sm*(2.8, 0)$); +\coordinate (5) at ($\sm*(4.2, 0)$); +\coordinate (6) at ($\sm*(5.6, 0)$); +\coordinate (7) at ($\sm*(7.0, 0)$); +\draw (1)--(3)--(4)--(5)--(6)--(7); +\draw (2)--(4); +\foreach \point in +{1,2,3,4,5,6,7} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{1} + { + \draw [black] (\point) circle (5.0pt); + } +\end{tikzpicture} +& +\begin{tikzpicture}[scale=0.5,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(2.8, -1.4)$); +\coordinate (3) at ($\sm*(1.4, 0)$); +\coordinate (4) at ($\sm*(2.8, 0)$); +\coordinate (5) at ($\sm*(4.2, 0)$); +\coordinate (6) at ($\sm*(5.6, 0)$); +\coordinate (7) at ($\sm*(7.0, 0)$); +\draw (1)--(3)--(4)--(5)--(6)--(7); +\draw (2)--(4); +\foreach \point in +{1,2,3,4,5,6,7} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\end{tikzpicture} +\\ +\hline +\begin{center}$r(\xi)$\end{center} +& \begin{center}$0$\end{center} +& \begin{center}чистый символ $\neq 0$ из $H^3(F,\mathbb{Z}/2)$: +$\lAngle a,b,c\rAngle$\end{center} +& \begin{center}сумма двух символов из $H^3(F,\mathbb{Z}/2)$ +с общим слотом: $\lAngle a,b,c\rAngle + \lAngle a,d,e\rAngle$\end{center} +& \begin{center}иначе\end{center} \\ +\hline +\end{tabular} + +Случаи $\mathsf{E}_6\pmod{2}$, $\mathsf{E}_7\pmod{3}$~--- легкие упражнения. +Случаи $\mathsf{E}_8\pmod{2}$, $\mathsf{E}_7^{\operatorname{ad}}\pmod{2}$~--- +исследовательское упражнение. +Случай $\mathsf{E}_8\pmod{5}$~--- легкое упражнение. +Вот ответ для $\mathsf{E}_8\pmod{3}$: + +\begin{tabular}{|p{2.9cm}|p{3.0cm}|p{3.0cm}|p{3.0cm}|} +\hline +\begin{center} $J_3(\mathsf{E}_8)$ \end{center} +& \begin{center} $(0,0)$ \end{center} +& \begin{center} $(1,0)$ \end{center} +& \begin{center} $(1,1)$ \end{center} \\ +\hline +\begin{center}индекс Титса\end{center} +& +\begin{tikzpicture}[scale=0.5,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(2.8, -1.4)$); +\coordinate (3) at ($\sm*(1.4, 0)$); +\coordinate (4) at ($\sm*(2.8, 0)$); +\coordinate (5) at ($\sm*(4.2, 0)$); +\coordinate (6) at ($\sm*(5.6, 0)$); +\coordinate (7) at ($\sm*(7.0, 0)$); +\coordinate (8) at ($\sm*(8.4, 0)$); +\draw (1)--(3)--(4)--(5)--(6)--(7)--(8); +\draw (2)--(4); +\foreach \point in +{1,2,3,4,5,6,7,8} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{1,2,3,4,5,6,7,8} + { + \draw [black] (\point) circle (5.0pt); + } +\end{tikzpicture} +& +\begin{tikzpicture}[scale=0.5,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(2.8, -1.4)$); +\coordinate (3) at ($\sm*(1.4, 0)$); +\coordinate (4) at ($\sm*(2.8, 0)$); +\coordinate (5) at ($\sm*(4.2, 0)$); +\coordinate (6) at ($\sm*(5.6, 0)$); +\coordinate (7) at ($\sm*(7.0, 0)$); +\coordinate (8) at ($\sm*(8.4, 0)$); +\draw (1)--(3)--(4)--(5)--(6)--(7)--(8); +\draw (2)--(4); +\foreach \point in +{1,2,3,4,5,6,7,8} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{7,8} + { + \draw [black] (\point) circle (5.0pt); + } +\end{tikzpicture} +& +\begin{tikzpicture}[scale=0.5,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(2.8, -1.4)$); +\coordinate (3) at ($\sm*(1.4, 0)$); +\coordinate (4) at ($\sm*(2.8, 0)$); +\coordinate (5) at ($\sm*(4.2, 0)$); +\coordinate (6) at ($\sm*(5.6, 0)$); +\coordinate (7) at ($\sm*(7.0, 0)$); +\coordinate (8) at ($\sm*(8.4, 0)$); +\draw (1)--(3)--(4)--(5)--(6)--(7)--(8); +\draw (2)--(4); +\foreach \point in +{1,2,3,4,5,6,7,8} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\end{tikzpicture} +\\ +\hline +\begin{center} $r(\xi)$ \end{center} +& \begin{center} $0$ \end{center} +& \begin{center} чистый символ $\neq 0$ из $H^3(F,\mathbb{Z}/3)$\end{center} +& \begin{center} иначе \end{center} \\ +\hline +\end{tabular} \end{document}