This is TeX, Version 3.14159 (Web2C 7.3.1) (format=latex 2003.4.12) 21 APR 2003 20:00 **dsw_13.tex (dsw_13.tex LaTeX2e <2001/06/01> Babel and hyphenation patterns for american, french, german, ngerman, i talian, nohyphenation, loaded. (/usr/share/texmf/tex/latex/amscls/amsart.cls Document Class: amsart 2000/10/26 v2.08 \linespacing=\dimen102 \normalparindent=\dimen103 \normaltopskip=\skip41 (/usr/share/texmf/tex/latex/amsmath/amsmath.sty Package: amsmath 2000/07/18 v2.13 AMS math features \@mathmargin=\skip42 For additional information on amsmath, use the `?' option. (/usr/share/texmf/tex/latex/amsmath/amstext.sty Package: amstext 2000/06/29 v2.01 (/usr/share/texmf/tex/latex/amsmath/amsgen.sty File: amsgen.sty 1999/11/30 v2.0 \@emptytoks=\toks14 \ex@=\dimen104 )) (/usr/share/texmf/tex/latex/amsmath/amsbsy.sty Package: amsbsy 1999/11/29 v1.2d \pmbraise@=\dimen105 ) (/usr/share/texmf/tex/latex/amsmath/amsopn.sty Package: amsopn 1999/12/14 v2.01 operator names ) \inf@bad=\count79 LaTeX Info: Redefining \frac on input line 211. \uproot@=\count80 \leftroot@=\count81 LaTeX Info: Redefining \overline on input line 307. \classnum@=\count82 \DOTSCASE@=\count83 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=\dimen106 LaTeX Font Info: Redeclaring font encoding OML on input line 567. LaTeX Font Info: Redeclaring font encoding OMS on input line 568. \macc@depth=\count84 \c@MaxMatrixCols=\count85 \dotsspace@=\muskip10 \c@parentequation=\count86 \dspbrk@lvl=\count87 \tag@help=\toks15 \row@=\count88 \column@=\count89 \maxfields@=\count90 \andhelp@=\toks16 \eqnshift@=\dimen107 \alignsep@=\dimen108 \tagshift@=\dimen109 \tagwidth@=\dimen110 \totwidth@=\dimen111 \lineht@=\dimen112 \@envbody=\toks17 \multlinegap=\skip43 \multlinetaggap=\skip44 \mathdisplay@stack=\toks18 LaTeX Info: Redefining \[ on input line 2666. LaTeX Info: Redefining \] on input line 2667. ) LaTeX Font Info: Try loading font information for U+msa on input line 395. (/usr/share/texmf/tex/latex/amsfonts/umsa.fd File: umsa.fd 2002/01/19 v2.2g AMS font definitions ) (/usr/share/texmf/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. ) \copyins=\insert233 \abstractbox=\box28 \listisep=\skip45 \c@part=\count91 \c@section=\count92 \c@subsection=\count93 \c@subsubsection=\count94 \c@paragraph=\count95 \c@subparagraph=\count96 \c@figure=\count97 \c@table=\count98 \abovecaptionskip=\skip46 \belowcaptionskip=\skip47 \captionindent=\dimen113 \thm@style=\toks19 \thm@bodyfont=\toks20 \thm@headfont=\toks21 \thm@notefont=\toks22 \thm@headpunct=\toks23 \thm@preskip=\skip48 \thm@postskip=\skip49 \thm@headsep=\skip50 \dth@everypar=\toks24 ) (/usr/share/texmf/tex/latex/amsfonts/amssymb.sty Package: amssymb 2002/01/22 v2.2d ) (/usr/share/texmf/tex/latex/amsmath/amscd.sty Package: amscd 1999/11/29 v2.0 \athelp@=\toks25 \minaw@=\dimen114 \bigaw@=\dimen115 \minCDarrowwidth=\dimen116 ) \c@prop=\count99 (dsw_13.aux) \openout1 = `dsw_13.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 96. LaTeX Font Info: ... okay on input line 96. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 96. LaTeX Font Info: ... okay on input line 96. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 96. LaTeX Font Info: ... okay on input line 96. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 96. LaTeX Font Info: ... okay on input line 96. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 96. LaTeX Font Info: ... okay on input line 96. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 96. LaTeX Font Info: ... okay on input line 96. LaTeX Font Info: Checking defaults for OT2/cmr/m/n on input line 96. LaTeX Font Info: Try loading font information for OT2+cmr on input line 96. (/usr/share/texmf/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 96. LaTeX Font Info: Try loading font information for U+msa on input line 96. (/usr/share/texmf/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 96. (/usr/share/texmf/tex/latex/amsfonts/umsb.fd File: umsb.fd 2002/01/19 v2.2g AMS font definitions ) Underfull \hbox (badness 1953) in paragraph at lines 135--135 [][]\OT1/cmr/m/it/8 Key words and phrases.\OT1/cmr/m/n/8 modular form, $\OML/cm m/m/it/8 L$\OT1/cmr/m/n/8 -function, vis-i-bil-ity, Bloch-Kato con-jec-ture, [] LaTeX Font Info: Try loading font information for U+euf on input line 175. (/usr/share/texmf/tex/latex/amsfonts/ueuf.fd File: ueuf.fd 2002/01/19 v2.2g AMS font definitions ) [1] LaTeX Font Info: Try loading font information for OT2+wncyr on input line 19 6. (/usr/share/texmf/tex/latex/cyrillic/ot2wncyr.fd File: ot2wncyr.fd 1999/12/17 v3.2a Computer Modern Cyrillic font definitions ) [2] [3] [4] Overfull \hbox (2.26244pt too wide) in paragraph at lines 482--501 \OT1/cmr/m/n/10 since this en-sures that $\OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 ( \OML/cmm/m/it/10 I[]; A[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 j\OT1/cmr/m/n/10 )) = \OML/cmm/m/it/10 V[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 j\OT1/cmr/m/n/10 )[]\OM L/cmm/m/it/10 =T[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 j\OT1/cmr/m/n/10 )[]$, henc e that $\OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\U/msb/m/n/10 Q[]\OML/cmm/m/it/10 ; A[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 j\OT1/cmr/m/n/10 )) = [] [5] [6] [7] LaTeX Font Info: Try loading font information for OMS+cmr on input line 650. (/usr/share/texmf/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 <10> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 650. [8] Overfull \hbox (1.88966pt too wide) in paragraph at lines 716--723 \OT1/cmr/m/n/10 jects into $\OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\U/msb/m/n/10 Q\OML/cmm/m/it/10 ; A[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 k=\OT1/cmr/m/n/10 2))$ , and we get a nonzero, $\U/euf/m/n/10 q$\OT1/cmr/m/n/10 -torsion class $\OML/c mm/m/it/10 ^^M \OMS/cmsy/m/n/10 2 \OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\U/msb/m /n/10 Q\OML/cmm/m/it/10 ; A[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 k=\OT1/cmr/m/n/1 0 2))$. [] [9] Overfull \hbox (40.2837pt too wide) in paragraph at lines 865--872 []\OT1/cmr/m/n/10 It fol-lows that the class $[][](\OML/cmm/m/it/10 c\OT1/cmr/m /n/10 ) \OMS/cmsy/m/n/10 2 \OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\U/msb/m/n/10 Q []\OML/cmm/m/it/10 ; A\OT1/cmr/m/n/10 [\U/euf/m/n/10 q\OT1/cmr/m/n/10 ](\OML/cm m/m/it/10 k=\OT1/cmr/m/n/10 2))$ is in the im-age of $\OML/cmm/m/it/10 h[]\OT1/ cmr/m/n/10 (\OML/cmm/m/it/10 D\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 k=\OT1/cmr/m/n/ 10 2)\OML/cmm/m/it/10 =\U/euf/m/n/10 q\OML/cmm/m/it/10 D\OT1/cmr/m/n/10 (\OML/c mm/m/it/10 k=\OT1/cmr/m/n/10 2))$ [] [10] [11] [12] [13] Overfull \hbox (45.41705pt too wide) detected at line 1203 [] [] [14] [15] [16] [17] [18] Underfull \vbox (badness 10000) has occurred while \output is active [] [19] Overfull \hbox (4.50038pt too wide) in paragraph at lines 1541--1541 []\OT1/cmr/m/sc/8 University of Sheffield, De-part-ment of Pure Math-e-mat-ics, Hicks Build-ing, Hounsfield [] [20] (dsw_13.aux) LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right. ) (\end occurred inside a group at level 1) Here is how much of TeX's memory you used: 1793 strings out of 20880 18649 string characters out of 197306 88889 words of memory out of 350001 4589 multiletter control sequences out of 10000+15000 14738 words of font info for 55 fonts, out of 400000 for 1000 175 hyphenation exceptions out of 10000 34i,17n,32p,272b,460s stack positions out of 3000i,100n,1500p,50000b,4000s Output written on dsw_13.dvi (20 pages, 115764 bytes).