This is TeX, Version 3.14159 (Web2C 7.4.5) (format=latex 2004.6.18) 30 JUN 2004 15:43 **new10 (./new10.tex LaTeX2e <2001/06/01> Babel and hyphenation patterns for american, nohyphenation, loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2001/04/21 v1.4e Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size11.clo File: size11.clo 2001/04/21 v1.4e 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/share/texmf/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. 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LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 9. LaTeX Font Info: ... okay on input line 9. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 9. LaTeX Font Info: ... okay on input line 9. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 9. LaTeX Font Info: ... okay on input line 9. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 9. LaTeX Font Info: ... okay on input line 9. LaTeX Font Info: Checking defaults for OT2/cmr/m/n on input line 9. LaTeX Font Info: Try loading font information for OT2+cmr on input line 9. (/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 9. LaTeX Font Info: Try loading font information for OT2+wncyr on input line 13 . (/usr/share/texmf/tex/latex/cyrillic/ot2wncyr.fd File: ot2wncyr.fd 1999/12/17 v3.2a Computer Modern Cyrillic font definitions ) LaTeX Font Info: Try loading font information for U+msa on input line 15. (/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 15. (/usr/share/texmf/tex/latex/amsfonts/umsb.fd File: umsb.fd 2002/01/19 v2.2g AMS font definitions ) Overfull \hbox (7.7993pt too wide) in paragraph at lines 15--30 \OT1/cmr/m/n/10.95 For an el-lip-tic curve $\OML/cmm/m/it/10.95 E$ \OT1/cmr/m/n /10.95 de-fined over a num-ber field $\OML/cmm/m/it/10.95 K$\OT1/cmr/m/n/10.95 , let $[](\OML/cmm/m/it/10.95 E=K\OT1/cmr/m/n/10.95 ) = [](\OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 K; E\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m /n/10.95 ! [] LaTeX Warning: Citation `MR31:3420' on page 1 undefined on input line 36. LaTeX Warning: Citation `rubin:main-conjectures' on page 1 undefined on input l ine 39. LaTeX Warning: Citation `coates-wiles:bsd' on page 1 undefined on input line 42 . LaTeX Warning: Citation `rubin:cmsha' on page 1 undefined on input line 42. LaTeX Warning: Citation `gross-zagier' on page 1 undefined on input line 42. [1 ] LaTeX Warning: Citation `kolyvagin:euler_systems' on page 2 undefined on input line 51. LaTeX Warning: Citation `gross:kolyvagin' on page 2 undefined on input line 51. LaTeX Warning: Citation `serre:propgal' on page 2 undefined on input line 51. LaTeX Warning: Citation `MR92e:11050' on page 2 undefined on input line 51. LaTeX Warning: Citation `MR92a:11058' on page 2 undefined on input line 51. [2] LaTeX Warning: Citation `kolyvagin:weil' on page 3 undefined on input line 57. Overfull \hbox (15.49132pt too wide) in paragraph at lines 57--59 \OML/cmm/m/it/10.95 E[] \OT1/cmr/m/n/10.95 : \OMS/cmsy/m/n/10.95 ^^@\OML/cmm/m/ it/10.95 Dy[] \OT1/cmr/m/n/10.95 = 4\OML/cmm/m/it/10.95 x[] \OMS/cmsy/m/n/10.95 ^^@ \OT1/cmr/m/n/10.95 4\OML/cmm/m/it/10.95 x \OT1/cmr/m/n/10.95 + 1$ with $\O ML/cmm/m/it/10.95 D \OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 7\OML/cmm/m/it/10 .95 ; \OT1/cmr/m/n/10.95 11\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 47\OML/cmm/ m/it/10.95 ; \OT1/cmr/m/n/10.95 71\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 83\O ML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 84\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10. 95 127\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 159\OML/cmm/m/it/10.95 ; \OT1/cm r/m/n/10.95 164\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 219\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 231\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 263\OML/cmm/m/ it/10.95 ; \OT1/cmr/m/n/10.95 271\OML/cmm/m/it/10.95 ; [] [] Overfull \hbox (3.03333pt too wide) in paragraph at lines 63--63 []\OT1/cmr/bx/n/14.4 Weakening the Hy-pothe-ses for Triv-i-al-ity of $[]\OT1/cm r/m/n/14.4 (\OML/cmm/m/it/14.4 E=K\OT1/cmr/m/n/14.4 )[\OML/cmm/m/it/14.4 p\OT1/ cmr/m/n/14.4 ]$ [] [3] LaTeX Warning: Citation `ribet-stein:serre' on page 4 undefined on input line 6 8. [4] LaTeX Warning: Citation `serre:propgal' on page 5 undefined on input line 111. [5] [6] LaTeX Warning: Citation `gross:kolyvagin' on page 7 undefined on input line 179 . [7] Overfull \hbox (12.00082pt too wide) in paragraph at lines 193--193 []\OT1/cmr/m/n/10.95 Denote com-plex con-ju-ga-tion by $\OML/cmm/m/it/10.95 ^^\ $\OT1/cmr/m/n/10.95 . We will de-com-pose $\OML/cmm/m/it/10.95 E\OT1/cmr/m/n/10 .95 (\OML/cmm/m/it/10.95 K\OT1/cmr/m/n/10.95 )$ into its eigenspaces [] Overfull \hbox (5.53236pt too wide) in paragraph at lines 198--198 \OT1/cmr/m/n/10.95 In light of Don-nelly's ob-ser-va-tion, we can ap-ply Koly-v a-gin's re-sult that $[](\OML/cmm/m/it/10.95 E=K\OT1/cmr/m/n/10.95 )[\OML/cmm/m /it/10.95 p\OT1/cmr/m/n/10.95 ] = [] [8] No file new10.bbl. [9] (./new10.aux) LaTeX Warning: There were undefined references. ) Here is how much of TeX's memory you used: 1631 strings out of 95870 16343 string characters out of 1196419 65186 words of memory out of 1000001 4564 multiletter control sequences out of 10000+50000 13828 words of font info for 52 fonts, out of 500000 for 1000 14 hyphenation exceptions out of 1000 31i,15n,24p,1493b,268s stack positions out of 1500i,500n,5000p,200000b,5000s Output written on new10.dvi (9 pages, 37192 bytes).