This is pdfTeXk, Version 3.1415926-1.40.9 (Web2C 7.5.7) (format=pdflatex 2008.12.28) 30 JUL 2009 14:57 entering extended mode %&-line parsing enabled. **kolyvagin-on_the_structure_of_sha.tex (./kolyvagin-on_the_structure_of_sha.tex LaTeX2e <2005/12/01> Babel and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, german-x-2008-06-18, ngerman-x-2008-06-18, ancientgreek, ibycus, ar abic, basque, bulgarian, catalan, pinyin, coptic, croatian, czech, danish, dutc h, esperanto, estonian, farsi, finnish, french, galician, german, ngerman, mono greek, greek, hungarian, icelandic, indonesian, interlingua, irish, italian, la tin, lithuanian, mongolian, mongolian2a, bokmal, nynorsk, polish, portuguese, r omanian, russian, sanskrit, serbian, slovak, slovenian, spanish, swedish, turki sh, ukenglish, ukrainian, uppersorbian, welsh, loaded. (/usr/local/texlive/2008/texmf-dist/tex/latex/base/article.cls Document Class: article 2005/09/16 v1.4f Standard LaTeX document class (/usr/local/texlive/2008/texmf-dist/tex/latex/base/size12.clo File: size12.clo 2005/09/16 v1.4f Standard LaTeX file (size option) ) \c@part=\count79 \c@section=\count80 \c@subsection=\count81 \c@subsubsection=\count82 \c@paragraph=\count83 \c@subparagraph=\count84 \c@figure=\count85 \c@table=\count86 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 ) \@input{macros.aux} \openout2 = `macros.aux'. (./macros.tex (/usr/local/texlive/2008/texmf-dist/tex/latex/amsmath/amsmath.sty Package: amsmath 2000/07/18 v2.13 AMS math features \@mathmargin=\skip43 For additional information on amsmath, use the `?' option. (/usr/local/texlive/2008/texmf-dist/tex/latex/amsmath/amstext.sty Package: amstext 2000/06/29 v2.01 (/usr/local/texlive/2008/texmf-dist/tex/latex/amsmath/amsgen.sty File: amsgen.sty 1999/11/30 v2.0 \@emptytoks=\toks14 \ex@=\dimen103 )) (/usr/local/texlive/2008/texmf-dist/tex/latex/amsmath/amsbsy.sty Package: amsbsy 1999/11/29 v1.2d \pmbraise@=\dimen104 ) (/usr/local/texlive/2008/texmf-dist/tex/latex/amsmath/amsopn.sty Package: amsopn 1999/12/14 v2.01 operator names ) \inf@bad=\count87 LaTeX Info: Redefining \frac on input line 211. \uproot@=\count88 \leftroot@=\count89 LaTeX Info: Redefining \overline on input line 307. \classnum@=\count90 \DOTSCASE@=\count91 LaTeX Info: Redefining \ldots on input line 379. LaTeX Info: Redefining \dots on input line 382. LaTeX Info: Redefining \cdots on input line 467. \Mathstrutbox@=\box26 \strutbox@=\box27 \big@size=\dimen105 LaTeX Font Info: Redeclaring font encoding OML on input line 567. LaTeX Font Info: Redeclaring font encoding OMS on input line 568. \macc@depth=\count92 \c@MaxMatrixCols=\count93 \dotsspace@=\muskip10 \c@parentequation=\count94 \dspbrk@lvl=\count95 \tag@help=\toks15 \row@=\count96 \column@=\count97 \maxfields@=\count98 \andhelp@=\toks16 \eqnshift@=\dimen106 \alignsep@=\dimen107 \tagshift@=\dimen108 \tagwidth@=\dimen109 \totwidth@=\dimen110 \lineht@=\dimen111 \@envbody=\toks17 \multlinegap=\skip44 \multlinetaggap=\skip45 \mathdisplay@stack=\toks18 LaTeX Info: Redefining \[ on input line 2666. LaTeX Info: Redefining \] on input line 2667. ) (/usr/local/texlive/2008/texmf-dist/tex/latex/amsfonts/amsfonts.sty Package: amsfonts 2001/10/25 v2.2f \symAMSa=\mathgroup4 \symAMSb=\mathgroup5 LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' (Font) U/euf/m/n --> U/euf/b/n on input line 132. ) (/usr/local/texlive/2008/texmf-dist/tex/latex/amsfonts/amssymb.sty Package: amssymb 2002/01/22 v2.2d ) (/usr/local/texlive/2008/texmf-dist/tex/latex/amscls/amsthm.sty Package: amsthm 2004/08/06 v2.20 \thm@style=\toks19 \thm@bodyfont=\toks20 \thm@headfont=\toks21 \thm@notefont=\toks22 \thm@headpunct=\toks23 \thm@preskip=\skip46 \thm@postskip=\skip47 \thm@headsep=\skip48 \dth@everypar=\toks24 ) (/usr/local/texlive/2008/texmf-dist/tex/latex/tools/xspace.sty Package: xspace 2006/05/08 v1.12 Space after command names (DPC,MH) ) \c@theorem=\count99 \c@listnum=\count100 ) (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xy.sty (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xy.tex Bootstrap'ing: catcodes, docmode, (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xyrecat.tex) (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xyidioms.tex) Xy-pic version 3.7 <1999/02/16> Copyright (c) 1991-1998 by Kristoffer H. Rose Xy-pic is free software: see the User's Guide for details. Loading kernel: messages; fonts; allocations: state, \X@c=\dimen112 \Y@c=\dimen113 \U@c=\dimen114 \D@c=\dimen115 \L@c=\dimen116 \R@c=\dimen117 \Edge@c=\toks25 \X@p=\dimen118 \Y@p=\dimen119 \U@p=\dimen120 \D@p=\dimen121 \L@p=\dimen122 \R@p=\dimen123 \Edge@p=\toks26 \X@origin=\dimen124 \Y@origin=\dimen125 \X@xbase=\dimen126 \Y@xbase=\dimen127 \X@ybase=\dimen128 \Y@ybase=\dimen129 \X@min=\dimen130 \Y@min=\dimen131 \X@max=\dimen132 \Y@max=\dimen133 \lastobjectbox@=\box28 \zerodotbox@=\box29 \almostz@=\dimen134 direction, \d@X=\dimen135 \d@Y=\dimen136 \K@=\count101 \KK@=\count102 \Direction=\count103 \K@dXdY=\dimen137 \K@dYdX=\dimen138 \xyread@=\read1 \xywrite@=\write3 \csp@=\count104 \quotPTK@=\dimen139 utility macros; pictures: \xy, positions, \swaptoks@@=\toks27 \connectobjectbox@@=\box30 objects, \styletoks@=\toks28 decorations; kernel objects: directionals, circles, text; options; algorithms: directions, edges, connections; Xy-pic loaded) Package: xy 1999/02/16 Xy-pic version 3.7 (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xyall.tex Xy-pic option: All features v.3.3 (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xycurve.tex Xy-pic option: Curve and Spline extension v.3.7 curve, \crv@cnt@=\count105 \crvpts@=\toks29 \splinebox@=\box31 \splineval@=\dimen140 \splinedepth@=\dimen141 \splinetol@=\dimen142 \splinelength@=\dimen143 circles, \L@=\dimen144 loaded) (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xyframe.tex Xy-pic option: Frame and Bracket extension v.3.7 loaded) (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xycmtip.tex Xy-pic option: Computer Modern tip extension v.3.3 (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xytips.tex Xy-pic option: More Tips extension v.3.3 loaded) loaded) (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xyline.tex Xy-pic option: Line styles extension v.3.6 \xylinethick@=\dimen145 loaded) (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xyrotate.tex Xy-pic option: Rotate and Scale extension v.3.3 loaded) (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xycolor.tex Xy-pic option: Colour extension v.3.3 loaded) (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xymatrix.tex Xy-pic option: Matrix feature v.3.4 \Row=\count106 \Col=\count107 \queue@=\toks30 \queue@@=\toks31 \qcount@=\count108 \qcount@@=\count109 \matrixsize@=\count110 loaded) (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xyarrow.tex Xy-pic option: Arrow and Path feature v.3.5 path, \ar, loaded) (/usr/local/texlive/2008/texmf-dist/tex/generic/xypic/xygraph.tex Xy-pic option: Graph feature v.3.7 loaded) loaded)) (./kolyvagin-on_the_structure_of_sha.aux) \openout1 = `kolyvagin-on_the_structure_of_sha.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 25. LaTeX Font Info: ... okay on input line 25. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 25. LaTeX Font Info: ... okay on input line 25. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 25. LaTeX Font Info: ... okay on input line 25. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 25. LaTeX Font Info: ... okay on input line 25. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 25. LaTeX Font Info: ... okay on input line 25. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 25. LaTeX Font Info: ... okay on input line 25. LaTeX Font Info: Checking defaults for OT2/cmr/m/n on input line 25. LaTeX Font Info: Try loading font information for OT2+cmr on input line 25. (/usr/local/texlive/2008/texmf-dist/tex/latex/cyrillic/ot2cmr.fd File: ot2cmr.fd 2001/08/11 v3.2c Computer Modern Cyrillic font definitions ) LaTeX Font Info: ... okay on input line 25. LaTeX Font Info: Try loading font information for U+msa on input line 26. (/usr/local/texlive/2008/texmf-dist/tex/latex/amsfonts/umsa.fd File: umsa.fd 2002/01/19 v2.2g AMS font definitions ) LaTeX Font Info: Try loading font information for U+msb on input line 26. (/usr/local/texlive/2008/texmf-dist/tex/latex/amsfonts/umsb.fd File: umsb.fd 2002/01/19 v2.2g AMS font definitions ) (./kolyvagin-on_the_structure_of_sha.toc LaTeX Warning: Citation `kolyvagin:structureofsha' on page 1 undefined on input line 2. LaTeX Font Info: Try loading font information for OT2+wncyr on input line 9. (/usr/local/texlive/2008/texmf-dist/tex/latex/cyrillic/ot2wncyr.fd File: ot2wncyr.fd 1999/12/17 v3.2a Computer Modern Cyrillic font definitions )) \tf@toc=\write4 \openout4 = `kolyvagin-on_the_structure_of_sha.toc'. [1 {/usr/local/texlive/2008/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] Overfull \hbox (1.65984pt too wide) in paragraph at lines 45--47 []\OT1/cmr/m/n/12 Another im-por-tant arith-metic in-vari-ant of $\OML/cmm/m/it /12 E$ \OT1/cmr/m/n/12 is the Shafarevich-Tate group [] [2] LaTeX Warning: Reference `thM:A' on page 3 undefined on input line 115. [3] LaTeX Font Info: Try loading font information for OMS+cmr on input line 186. (/usr/local/texlive/2008/texmf-dist/tex/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 <12> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 186. LaTeX Warning: Citation `kolyvagin:structureofsha' on page 4 undefined on input line 198. [4] [5] LaTeX Font Info: Try loading font information for U+euf on input line 328. (/usr/local/texlive/2008/texmf-dist/tex/latex/amsfonts/ueuf.fd File: ueuf.fd 2002/01/19 v2.2g AMS font definitions ) Overfull \hbox (10.0031pt too wide) in paragraph at lines 328--328 [] [] [6] Overfull \vbox (111.35754pt too high) has occurred while \output is active [] [7] [8] [9] [10] [11] Overfull \hbox (3.38004pt too wide) detected at line 570 \OML/cmm/m/it/12 H[] \OT1/cmr/m/n/12 := \OML/cmm/m/it/12 H[]\OT1/cmr/m/n/12 (\O MS/cmsy/m/n/12 K\OML/cmm/m/it/12 ; E\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 M\OT1/cmr /m/n/12 ]) = [](\OML/cmm/m/it/12 G[]=\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 G[]\OT1/ cmr/m/n/12 )[]\OML/cmm/m/it/12 ; E\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 M\OT1/cmr/m /n/12 ]) [] [](\OMS/cmsy/m/n/12 K[]\OML/cmm/m/it/12 =\OT1/cmr/m/n/12 (\OMS/cmsy /m/n/12 K[]\OT1/cmr/m/n/12 )[]\OML/cmm/m/it/12 ; E\OT1/cmr/m/n/12 [\OML/cmm/m/i t/12 M\OT1/cmr/m/n/12 ])\OML/cmm/m/it/12 ; [] [12] LaTeX Warning: Citation `kolyvagin:subclass' on page 13 undefined on input line 655. [13] Overfull \hbox (52.90642pt too wide) in paragraph at lines 711--722 []\OT1/cmr/m/n/12 Let $\OML/cmm/m/it/12 S[]$ \OT1/cmr/m/n/12 be the sub-group o f $\OML/cmm/m/it/12 \OMS/cmsy/m/n/12 2 []\OML/cmm/m/it/12 H[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K; E\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 M\OT1/cmr/m/n/12 ])$ su ch that $\OML/cmm/m/it/12 \OT1/cmr/m/n/12 (\OML/cmm/m/it/12 v\OT1/cmr/m/n/12 ) \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 E\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K\OT1/c mr/m/n/12 (\OML/cmm/m/it/12 v\OT1/cmr/m/n/12 ))\OML/cmm/m/it/12 =ME\OT1/cmr/m/n /12 (\OML/cmm/m/it/12 K\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 v\OT1/cmr/m/n/12 ))$ [] [14] [15] Overfull \hbox (34.33353pt too wide) detected at line 842 \OML/cmm/m/it/12 E\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 R\OT1/cmr/m/n/12 )[\OML/cmm /m/it/12 M[]\OT1/cmr/m/n/12 ]\OML/cmm/m/it/12 =ME\OT1/cmr/m/n/12 (\OML/cmm/m/it /12 R\OT1/cmr/m/n/12 )[\OML/cmm/m/it/12 M[]\OT1/cmr/m/n/12 ] \OML/cmm/m/it/12 , []\OMS/cmsy/m/n/12 ! \OML/cmm/m/it/12 H[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 R; E \OT1/cmr/m/n/12 [\OML/cmm/m/it/12 M\OT1/cmr/m/n/12 ]) [] \OML/cmm/m/it/12 H[]\O T1/cmr/m/n/12 (\OML/cmm/m/it/12 R; E\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 M[]\OT1/c mr/m/n/12 ]) [] \OML/cmm/m/it/12 H[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 R; E\OT1/ cmr/m/n/12 [\OML/cmm/m/it/12 M[]\OT1/cmr/m/n/12 ])\OML/cmm/m/it/12 : [] [16] [17] [18] [19] LaTeX Warning: Citation `kolyvagin:weil' on page 20 undefined on input line 107 2. [20] [21] [22] [23] LaTeX Warning: Reference `' on page 24 undefined on input line 1357. LaTeX Warning: Reference `' on page 24 undefined on input line 1359. [24] LaTeX Warning: Citation `kolyvagin:structureofsha' on page 25 undefined on inpu t line 1397. [25] LaTeX Warning: Citation `kolyvagin:mordellweil' on page 26 undefined on input l ine 1503. LaTeX Warning: Citation `kolyvagin:euler_systems' on page 26 undefined on input line 1504. [26] [27] LaTeX Warning: Reference `eq:' on page 28 undefined on input line 1623. [28] LaTeX Warning: Reference `' on page 29 undefined on input line 1683. [29] LaTeX Warning: Reference `' on page 30 undefined on input line 1714. LaTeX Warning: Reference `' on page 30 undefined on input line 1714. LaTeX Warning: Reference `' on page 30 undefined on input line 1721. LaTeX Warning: Reference `' on page 30 undefined on input line 1722. LaTeX Warning: Reference `' on page 30 undefined on input line 1726. LaTeX Warning: Reference `' on page 30 undefined on input line 1727. LaTeX Warning: Reference `' on page 30 undefined on input line 1728. LaTeX Warning: Reference `' on page 30 undefined on input line 1728. LaTeX Warning: Reference `' on page 30 undefined on input line 1730. LaTeX Warning: Reference `' on page 30 undefined on input line 1745. [30] Overfull \hbox (13.06567pt too wide) in paragraph at lines 1760--1762 []\OT1/cmr/m/n/12 Note that the or-thog-o-nal-ity be-tween el-e-ments of $\OML/ cmm/m/it/12 S[]$ \OT1/cmr/m/n/12 and $\OT1/cmr/bx/n/12 x\OT1/cmr/m/n/12 ([]\OML /cmm/m/it/12 ME\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K\OT1/cmr/m/n/12 ))$, [] LaTeX Warning: Reference `' on page 31 undefined on input line 1765. LaTeX Warning: Reference `eq:sumciaij' on page 31 undefined on input line 1766. LaTeX Warning: Reference `' on page 31 undefined on input line 1778. LaTeX Warning: Reference `' on page 31 undefined on input line 1783. Overfull \hbox (10.8449pt too wide) detected at line 1791 [] [] [31] [32] Overfull \hbox (1.80199pt too wide) in paragraph at lines 1924--1932 []\OT1/cmr/m/n/12 Recall (cf. The-o-rem ?) that the $\OML/cmm/m/it/12 `$\OT1/cm r/m/n/12 -component of the Birch and Swinnerton- [] [33] Overfull \hbox (36.34886pt too wide) in paragraph at lines 1941--1943 []\OT1/cmr/m/n/12 for all $\OML/cmm/m/it/12 ` \OMS/cmsy/m/n/12 j \OML/cmm/m/it/ 12 M$ \OT1/cmr/m/n/12 and $\OML/cmm/m/it/12 p \OMS/cmsy/m/n/12 2 \OT1/cmr/m/n/1 2 ^^C[]$ we can cal-cu-late the value $\OML/cmm/m/it/12 z\OT1/cmr/m/n/12 (\OML/ cmm/m/it/12 p\OT1/cmr/m/n/12 ) \OMS/cmsy/m/n/12 2 \OML/cmm/m/it/12 E\OT1/cmr/m/ n/12 (\OML/cmm/m/it/12 K\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 p\OT1/cmr/m/n/12 ))\O ML/cmm/m/it/12 =ME\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K\OT1/cmr/m/n/12 (\OML/cmm/ m/it/12 p\OT1/cmr/m/n/12 ))$. [] [34] (./kolyvagin-on_the_structure_of_sha.aux) LaTeX Warning: There were undefined references. ) Here is how much of TeX's memory you used: 4064 strings out of 493876 46435 string characters out of 1150567 138667 words of memory out of 3000000 7281 multiletter control sequences out of 10000+50000 22503 words of font info for 88 fonts, out of 3000000 for 5000 714 hyphenation exceptions out of 8191 31i,16n,25p,338b,440s stack positions out of 5000i,500n,10000p,200000b,50000s Output written on kolyvagin-on_the_structure_of_sha.pdf (34 pages, 279248 bytes ). PDF statistics: 199 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 131072) 1 words of extra memory for PDF output out of 10000 (max. 10000000)