This is TeX, Version 3.14159 (preloaded format=LaTeX 1999.7.24) 23 MAY 2000 17:30 **&LaTeX appendix.tex (appendix.tex LaTeX2e <1999/06/01> patch level 1 (:TeX:Inputs:LaTeX:base:latex209.def File: latex209.def 1998/05/13 v0.52 Standard LaTeX file Entering LaTeX 2.09 COMPATIBILITY MODE ************************************************************* !!WARNING!! !!WARNING!! !!WARNING!! !!WARNING!! This mode attempts to provide an emulation of the LaTeX 2.09 author environment so that OLD documents can be successfully processed. It should NOT be used for NEW documents! New documents should use Standard LaTeX conventions and start with the \documentclass command. Compatibility mode is UNLIKELY TO WORK with LaTeX 2.09 style files that change any internal macros, especially not with those that change the FONT SELECTION or OUTPUT ROUTINES. Therefore such style files MUST BE UPDATED to use Current Standard LaTeX: LaTeX2e. If you suspect that you may be using such a style file, which is probably very, very old by now, then you should attempt to get it updated by sending a copy of this error message to the author of that file. ************************************************************* \footheight=\dimen102 \@maxsep=\dimen103 \@dblmaxsep=\dimen104 \@cla=\count79 \@clb=\count80 \mscount=\count81 (:TeX:Inputs:LaTeX:base:tracefnt.sty Package: tracefnt 1997/05/29 v3.0j Standard LaTeX package (font tracing) \tracingfonts=\count82 LaTeX Info: Redefining \selectfont on input line 92. ) \symbold=\mathgroup4 \symsans=\mathgroup5 \symtypewriter=\mathgroup6 \symitalic=\mathgroup7 \symsmallcaps=\mathgroup8 \symslanted=\mathgroup9 LaTeX Font Info: Redeclaring math alphabet \mathbf on input line 284. LaTeX Font Info: Redeclaring math alphabet \mathsf on input line 285. LaTeX Font Info: Redeclaring math alphabet \mathtt on input line 286. LaTeX Font Info: Redeclaring math alphabet \mathit on input line 292. LaTeX Info: Redefining \em on input line 302. (:TeX:Inputs:LaTeX:base:latexsym.sty Package: latexsym 1998/08/17 v2.2e Standard LaTeX package (lasy symbols) \symlasy=\mathgroup10 LaTeX Font Info: Overwriting symbol font `lasy' in version `bold' (Font) U/lasy/m/n --> U/lasy/b/n on input line 38. ) LaTeX Font Info: Redeclaring math delimiter \lgroup on input line 366. LaTeX Font Info: Redeclaring math delimiter \rgroup on input line 368. LaTeX Font Info: Redeclaring math delimiter \bracevert on input line 370. ) (:TeX:Inputs:LaTeX:base:article.cls Document Class: article 1999/01/07 v1.4a Standard LaTeX document class (:TeX:Inputs:LaTeX:base:size12.clo File: size12.clo 1999/01/07 v1.4a Standard LaTeX file (size option) ) \c@part=\count83 \c@section=\count84 \c@subsection=\count85 \c@subsubsection=\count86 \c@paragraph=\count87 \c@subparagraph=\count88 \c@figure=\count89 \c@table=\count90 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 Compatibility mode: definition of \rm ignored. Compatibility mode: definition of \sf ignored. Compatibility mode: definition of \tt ignored. Compatibility mode: definition of \bf ignored. Compatibility mode: definition of \it ignored. Compatibility mode: definition of \sl ignored. Compatibility mode: definition of \sc ignored. LaTeX Info: Redefining \cal on input line 497. LaTeX Info: Redefining \mit on input line 498. \bibindent=\dimen105 ) (bas.sty \msamfam=\mathgroup11 \msbmfam=\mathgroup12 \eulfam=\mathgroup13 \sfifam=\mathgroup14 ) (appendix.aux) LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 136. LaTeX Font Info: ... okay on input line 136. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 136. LaTeX Font Info: ... okay on input line 136. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 136. LaTeX Font Info: ... okay on input line 136. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 136. LaTeX Font Info: ... okay on input line 136. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 136. LaTeX Font Info: ... okay on input line 136. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 136. LaTeX Font Info: ... okay on input line 136. ! Missing number, treated as zero. \def l.136 \begin{document} ? q OK, entering \batchmode... ! Illegal unit of measure (pt inserted). \def l.136 \begin{document} Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) LaTeX Font Warning: Font shape `OT1/cmr/m/n' in size <0> not available (Font) size <5> substituted on input line 136. ! Missing number, treated as zero. \def l.139 \maketitle A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.139 \maketitle Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Missing number, treated as zero. \def l.139 \maketitle A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.139 \maketitle Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) LaTeX Font Info: Calculating math sizes for size <0> on input line 139. LaTeX Font Warning: Font shape `OML/cmm/m/it' in size <0> not available (Font) size <5> substituted on input line 139. LaTeX Font Warning: Font shape `OMS/cmsy/m/n' in size <0> not available (Font) size <5> substituted on input line 139. LaTeX Font Info: External font `cmex10' loaded for size (Font) <0> on input line 139. LaTeX Font Warning: Font shape `OT1/cmr/bx/n' in size <0> not available (Font) size <5> substituted on input line 139. LaTeX Font Warning: Font shape `OT1/cmss/m/n' in size <0> not available (Font) size <5> substituted on input line 139. LaTeX Font Warning: Font shape `OT1/cmtt/m/n' in size <0> not available (Font) size <5> substituted on input line 139. LaTeX Font Warning: Font shape `OT1/cmr/m/it' in size <0> not available (Font) size <5> substituted on input line 139. LaTeX Font Warning: Font shape `OT1/cmr/m/sc' in size <0> not available (Font) size <5> substituted on input line 139. LaTeX Font Warning: Font shape `OT1/cmr/m/sl' in size <0> not available (Font) size <5> substituted on input line 139. LaTeX Font Info: Try loading font information for U+lasy on input line 139. (:TeX:Inputs:LaTeX:base:ulasy.fd File: ulasy.fd 1998/08/17 v2.2eLaTeX symbol font definitions ) LaTeX Font Warning: Font shape `U/lasy/m/n' in size <0> not available (Font) size <5> substituted on input line 139. ! Missing number, treated as zero. \def l.139 \maketitle A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.139 \maketitle Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Missing number, treated as zero. \def l.141 ... groups of connected components.} A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.141 ... groups of connected components.} Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.173 a morphism of free $\ZZ $-modules: The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.175 \alpha\colon \ZZ ^\Ccaltilde\lto\ZZ^\Ccaltilde, \qquad The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.175 \alpha\colon \ZZ^\Ccaltilde\lto\ZZ ^\Ccaltilde, \qquad The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.182 \ZZ ^\Ccaltilde\;\;\stackrel{\alpha}{\lto}\;\;\ZZ^\Ccaltilde\;\; The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.182 ...;\;\stackrel{\alpha}{\lto}\;\;\ZZ ^\Ccaltilde\;\; The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.184 \ZZ , The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.208 s_*,t_*\colon \ZZ ^\Scaltilde\lto\ZZ^\Ccaltilde,\qquad The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.208 s_*,t_*\colon \ZZ^\Scaltilde\lto\ZZ ^\Ccaltilde,\qquad The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.209 s^*,t^*\colon \ZZ ^\Ccaltilde\lto\ZZ^\Scaltilde , The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.209 s^*,t^*\colon \ZZ^\Ccaltilde\lto\ZZ ^\Scaltilde , The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.213 on $\ZZ ^\Scaltilde$ and $\ZZ^\Ccaltilde$, of~$s_*$. With this The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.213 on $\ZZ^\Scaltilde$ and $\ZZ ^\Ccaltilde$, of~$s_*$. With this The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.217 d_*:=t_*-s_*\colon \ZZ ^\Scaltilde\lto\ZZ^\Ccaltilde, \qquad The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.217 ...*-s_*\colon \ZZ^\Scaltilde\lto\ZZ ^\Ccaltilde, \qquad The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.218 d^*:=t^*-s^*\colon \ZZ ^\Ccaltilde\lto\ZZ^\Scaltilde. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.218 ...*-s^*\colon \ZZ^\Ccaltilde\lto\ZZ ^\Scaltilde. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.221 $\rH_1(\Gtilde,\ZZ )$ and The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.222 $\coker(d^*)=\rH^1(\Gtilde,\ZZ )$ (the isomorphisms depend on the The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.225 $\ker(d^*)=\rH^0(\Gtilde,\ZZ )$ is the diagonal in $\ZZ^\Ccaltil... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.225 ...lde,\ZZ)$ is the diagonal in $\ZZ ^\Ccaltilde$, The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.226 and $\coker(d_*)=\rH_0(\Gtilde,\ZZ )$ is $\ZZ$, via the map The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.226 ...(d_*)=\rH_0(\Gtilde,\ZZ)$ is $\ZZ $, via the map The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.227 $+\colon\ZZ ^\Ccaltilde\to\ZZ$. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.227 $+\colon\ZZ^\Ccaltilde\to\ZZ $. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.234 character group. We denote the $\ZZ $-linear dual of $M$\/ by The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Font Info: Try loading font information for OMS+cmr on input line 237. (:TeX:Inputs:LaTeX:base:omscmr.fd File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions ) LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <0> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 237. LaTeX Font Warning: Font shape `OMS/cmsy/m/n' in size <0> not available (Font) size <5> substituted on input line 237. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.238 that $M$ and $\rH_1(\Gtilde,\ZZ )$ are canonically isomorphic. Let The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.258 $\rH^1(\Gtilde,\ZZ )$ which does not depend on the orientation The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.268 & & & & \ZZ & & \stackrel{0}{\lto} & & M & & & & \\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. \def l.269 & & & \mbox{\scriptsize\id} \nearrow & & A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.269 & & & \mbox{\scriptsize\id} \nearrow & & Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Missing number, treated as zero. \def l.270 \searrow\mbox{\scriptsize\rm diag} & A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.270 \searrow\mbox{\scriptsize\rm diag} & Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.272 0 & \lto & \ZZ & & \stackrel{\rm diag}{\lto} & & \ZZ^\Ccaltilde & The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.272 ...\stackrel{\rm diag}{\lto} & & \ZZ ^\Ccaltilde & The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.273 \stackrel{d^*}{\lto} & \ZZ ^\Scaltilde & \lto & M^\vee & \to & 0\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.276 & & & & & & & & \ZZ ^\Ccaltilde[+] & & \downarrow & & \\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. \def l.283 \end{eqn} A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.283 \end{eqn} Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) [1 ] ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.349 $\rH_1(G,\ZZ )$ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.350 and $\rH_1(\Gtilde,\ZZ )$ are canonically isomorphic, and that this The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.354 ...hism between $M$ and $\rH_1(G,\ZZ )$ defined just as The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.360 h_\Ccal\colon\ZZ ^\Ccal\lto\ZZ^\Ccaltilde, \quad C\mapsto\Ctilde, The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.360 h_\Ccal\colon\ZZ^\Ccal\lto\ZZ ^\Ccaltilde, \quad C\mapsto\Ctilde, The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.362 h_\Scal\colon\ZZ ^\Scal\lto\ZZ^\Scaltilde, \quad The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.362 h_\Scal\colon\ZZ^\Scal\lto\ZZ ^\Scaltilde, \quad The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.371 0 & \lto & M & \lto & \ZZ ^\Scal & \stackrel{d_*}{\lto} & \ZZ^\Ccal The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.371 ...Scal & \stackrel{d_*}{\lto} & \ZZ ^\Ccal The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.373 \stackrel{+}{\lto} & \ZZ & \lto & 0 \\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.377 0 & \lto & M & \lto & \ZZ ^\Scaltilde & \stackrel{d_*}{\lto} & The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.378 \ZZ ^\Ccaltilde & The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.379 \stackrel{+}{\lto} & \ZZ & \lto & 0 \\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.387 0 & \lto & \ZZ & \stackrel{\diag}{\lto} & \ZZ^\Ccaltilde & The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.387 ...ZZ & \stackrel{\diag}{\lto} & \ZZ ^\Ccaltilde & The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.389 & \ZZ ^\Scaltilde & \lto & M^\vee & \lto & 0 \\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.392 0 & \lto & \ZZ & \stackrel{\diag}{\lto} & \ZZ^\Ccal & The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.392 ...ZZ & \stackrel{\diag}{\lto} & \ZZ ^\Ccal & The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.394 & \ZZ ^\Scal & \lto & M^\vee & \lto & 0 \\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.400 h_\Scal^\vee\colon \ZZ ^\Scaltilde\lto\ZZ^\Scal, \quad y\mapsto The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.400 ...^\vee\colon \ZZ^\Scaltilde\lto\ZZ ^\Scal, \quad y\mapsto The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.402 h_\Ccal^\vee\colon \ZZ ^\Ccaltilde\lto\ZZ^\Ccal, \quad C\mapsto The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.402 ...^\vee\colon \ZZ^\Ccaltilde\lto\ZZ ^\Ccal, \quad C\mapsto The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. \def l.404 \end{eqn} A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.404 \end{eqn} Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) [2] ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.409 product on $\ZZ ^\Scaltilde$ induces the following inner product on The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.410 $\ZZ ^\Scal$: The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.412 \ld\cdot,\cdot\rd\colon \ZZ ^\Scal\times\ZZ^\Scal \lto \ZZ, \quad The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.412 ...\cdot\rd\colon \ZZ^\Scal\times\ZZ ^\Scal \lto \ZZ, \quad The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.412 ...\ZZ^\Scal\times\ZZ^\Scal \lto \ZZ , \quad The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.415 ...e view $M$ as a submodule of $\ZZ ^\Scal$ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. \def l.428 ...tion of divisors of degree zero.} \label{section2} A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.428 ...tion of divisors of degree zero.} \label{section2} Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.443 $(C\mapsto (C\cdot\Dtilde))$ of $\ZZ ^\Ccaltilde$. Our aim is now The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.444 ...on for $\phi(D)$ in terms of $\ZZ ^\Ccal$ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.445 and~$\ZZ ^\Scal$. For simplicity, we suppose that $D$ has support The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.456 ...lement $\sum_P n_P\,c(P)$ of~$\ZZ ^\Ccaltilde$. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.501 $\ZZ ^\Ccal[+]$ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.502 of~$\ZZ ^\Ccaltilde[+]$ and the second from~$\ZZ^\Scal$. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.502 ...ilde[+]$ and the second from~$\ZZ ^\Scal$. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. \def l.538 \section{Modular curves.} \label{section3} A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.538 \section{Modular curves.} \label{section3} Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Missing number, treated as zero. \def l.543 {}F rom now on, let $p$ be a prime number. We suppose that $K$ A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.543 {}F rom now on, let $p$ be a prime number. We suppose that $K$ Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) [3] ! Missing number, treated as zero. \def l.640 \section{Shimura curves.} \label{section4} A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.640 \section{Shimura curves.} \label{section4} Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.645 ...ection~\ref{section3}. Let $B_\QQ $ be an The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.646 ...nite quaternion algebra over $\QQ $, of discriminant prime The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.648 Let $B$ be a maximal order in~$B_\QQ $. First we consider the The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.671 ... a non-trivial idempotent in $\ZZ _p\otimes B$ (recall The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \Bbb@ #1->{\msbm #1} l.674 $\ZZ _p\otimes B$ acts on the $p$-divisible group $E[p^\infty]$ of The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing number, treated as zero. \def l.744 A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.744 Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) [4] ! Missing number, treated as zero. \def l.766 \begin{thebibliography}{99} A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.766 \begin{thebibliography}{99} Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) ! Missing number, treated as zero. \def l.812 \end{document} A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). \def l.812 \end{document} Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) [5] (appendix.aux) LaTeX Font Warning: Size substitutions with differences (Font) up to 5.0pt have occured. ) Here is how much of TeX's memory you used: 814 strings out of 5909 7500 string characters out of 28632 60258 words of memory out of 200001 3666 multiletter control sequences out of 10000 16814 words of font info for 67 fonts, out of 80000 for 256 14 hyphenation exceptions out of 2551 24i,13n,19p,177b,267s stack positions out of 600i,150n,200p,3000b,2000s Output written on appendix.dvi (5 pages, 33384 bytes).