This is TeX, Version 3.14159 (Web2C 7.3.1) LaTeX2e <2001/06/01> Babel and hyphenation patterns for american, french, german, ngerman, i talian, nohyphenation, loaded. (Thesis.tex (/usr/share/texmf/tex/latex/base/report.cls Document Class: report 2001/04/21 v1.4e Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. 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Ignore all them. If uncorrectable errors occur, notify staff. [1] [2] (Thesis.toc Overfull \hbox (2.6569pt too wide) in paragraph at lines 3--3 [] []\OT1/cmr/bx/n/12 Mordell-Weil The-o-rem, Shafarevich-Tate Group and Selme r Groups (/usr/share/texmf/tex/latex/amsfonts/umsa.fd) (/usr/share/texmf/tex/latex/amsfonts/umsb.fd) (/usr/share/texmf/tex/latex/base/ulasy.fd) (/usr/share/texmf/tex/latex/cyrillic/ot2wncyr.fd) [3]) [4] :?1111 abstract (Abs.tex [1]) Acknowledgements (Ack.tex) [1] Dalthesis preface pages completed. (intro.tex Overfull \hbox (2.33617pt too wide) in paragraph at lines 44--48 \OT1/cmr/m/n/12 nec-tion be-tween the Birch and Swinnerton-Dyer con-jec-ture, a nd the Tate-Shafarevich Overfull \hbox (9.65135pt too wide) in paragraph at lines 49--61 []\OT1/cmr/m/n/12 The main goal of this the-sis is to in-tro-duce and study a s ub-group of the Shafarevich- [1]) [2] (chapter1.tex Chapter 1. [3] [4] [5] [6] (/usr/share/texmf/tex/latex/base/omscmr.fd) Overfull \hbox (5.1684pt too wide) in paragraph at lines 294--294 []\OT1/cmr/bx/n/17.28 Computing the Weak Mordell-Weil Group and Overfull \hbox (2.93066pt too wide) in paragraph at lines 304--307 []\OT1/cmr/m/n/12 The main idea for the com-pu-ta-tion of the weak Mordell-Weil group $\OML/cmm/m/it/12 E\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K\OT1/cmr/m/n/12 )\ OML/cmm/m/it/12 =mE\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K\OT1/cmr/m/n/12 )$ [7] Underfull \hbox (badness 10000) in paragraph at lines 316--325 [8] [9] [10] [11] Overfull \hbox (4.22365pt too wide) detected at line 558 [] Overfull \hbox (23.45428pt too wide) in paragraph at lines 566--571 []\OT1/cmr/m/n/12 Our ul-ti-mate goal is com-put-ing the im-age of $\OML/cmm/m/ it/12 E[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 = ^^^\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 E\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K\OT1/c mr/m/n/12 ))$ in $\OML/cmm/m/it/12 H[]\OT1/cmr/m/n/12 ([]([]\OML/cmm/m/it/12 =K \OT1/cmr/m/n/12 )\OML/cmm/m/it/12 ; E\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 ^^^\OT1/ cmr/m/n/12 ])$, [12] Overfull \hbox (5.05453pt too wide) in paragraph at lines 629--632 \OT1/cmr/m/n/12 We also con-sider the Shafarevich-Tate group of $\OML/cmm/m/it/ 12 E=K$ \OT1/cmr/m/n/12 to be the sub-group of $\OML/cmm/m/it/12 WC\OT1/cmr/m/n /12 (\OML/cmm/m/it/12 E=K\OT1/cmr/m/n/12 )$ [13] [14]) [15] (chapter2.tex Chapter 2. [16] [17] [18] [19] Overfull \hbox (18.60481pt too wide) in paragraph at lines 271--273 []\OT1/cmr/m/n/12 The the-o-rem im-plies that for all $\OML/cmm/m/it/12 P[]; P[ ]; [] ; P[]; Q[]; Q[]; [] ; Q[]$\OT1/cmr/m/n/12 , there ex-ist $\OML/cmm/m/it/1 2 R[]; [] ; R[]$\OT1/cmr/m/n/12 , [20] [21]) [22] [23] (chapter3.tex Chapter 3. (/usr/share/texmf/tex/latex/amsfonts/ueuf.fd) [24] [25] [26] [27] [28] [29] ) [30] (chapter4.tex Chapter 4. [31] Overfull \hbox (4.97887pt too wide) in paragraph at lines 55--56 \OT1/cmr/bx/n/12 Corollary 4.1.2. \OT1/cmr/m/it/12 For any em-bed-ding $\OML/cm m/m/it/12 i \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 A \OMS/cmsy/m/n/12 ! \OML/cmm/m/ it/12 J$\OT1/cmr/m/it/12 , the vis-i-ble sub-group Vis$[]\OT1/cmr/m/n/12 (\OML/ cmm/m/it/12 H[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K; A\OT1/cmr/m/n/12 ))$ [32] [33] [34] [35] [36] [37] [38] [39] Overfull \hbox (9.71442pt too wide) in paragraph at lines 519--519 []\OT1/cmr/bx/n/17.28 Producing Vis-i-ble El-e-ments of the Shafarevich- Underfull \hbox (badness 10000) in paragraph at lines 548--554 [40] [41] Overfull \hbox (6.54413pt too wide) in paragraph at lines 649--656 []\OT1/cmr/m/n/12 If $\OML/cmm/m/it/12 ^^W$ \OT1/cmr/m/n/12 is real archi-me-di an, we have $\OML/cmm/m/it/12 H[]\OT1/cmr/m/n/12 ([]([][])\OML/cmm/m/it/12 =K[] ; A\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K[]\OT1/cmr/m/n/12 )) = \OML/cmm/m/it/12 H []\OT1/cmr/m/n/12 ([](\U/msb/m/n/12 C\OML/cmm/m/it/12 =\U/msb/m/n/12 R\OT1/cmr/ m/n/12 )\OML/cmm/m/it/12 ; A\OT1/cmr/m/n/12 (\U/msb/m/n/12 R\OT1/cmr/m/n/12 ))$ . Underfull \hbox (badness 10000) in paragraph at lines 649--656 [42] Underfull \hbox (badness 10000) in paragraph at lines 726--735 [43]) [44] (chapter5.tex Chapter 5. Overfull \hbox (29.64641pt too wide) in paragraph at lines 20--20 []\OT1/cmr/bx/n/17.28 Algorithms for Com-put-ing with Mod-u-lar Abelian [45] [46] [47] [48] [49] [50] [51] Overfull \hbox (14.39882pt too wide) in paragraph at lines 371--380 []\OT1/cmr/m/n/12 Conversely, sup-pose that $\OML/cmm/m/it/12 ^^K[]$ \OT1/cmr/m /n/12 and $\OML/cmm/m/it/12 ^^K[]$ \OT1/cmr/m/n/12 sat-isfy the con-di-tion. Fi rst, choose $\OML/cmm/m/it/12 s[]; r[]; s[]; r[] \OMS/cmsy/m/n/12 2 [52] [53] Underfull \hbox (badness 10000) in paragraph at lines 494--496 [54] [55] Overfull \hbox (9.7628pt too wide) in paragraph at lines 567--570 \OT1/cmr/m/it/12 where by $\OT1/cmr/m/n/12 ^^H(\OML/cmm/m/it/12 H[]\OT1/cmr/m/n /12 (\OML/cmm/m/it/12 X[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 N\OT1/cmr/m/n/12 ); \U/msb/m/n/12 Z\OT1/cmr/m/n/12 ))[]$ \OT1/cmr/m/it/12 we mean the pos-i-tive ei genspace of $[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 S[]\OT1/cmr/m/n/12 (^^@[](\OML /cmm/m/it/12 N\OT1/cmr/m/n/12 ))[\OML/cmm/m/it/12 I[]\OT1/cmr/m/n/12 ]\OML/cmm/ m/it/12 ; \U/msb/m/n/12 C\OT1/cmr/m/n/12 )$ Underfull \hbox (badness 10000) in paragraph at lines 579--581 Underfull \hbox (badness 10000) in paragraph at lines 582--590 [56] Underfull \hbox (badness 10000) in paragraph at lines 603--605 Underfull \hbox (badness 10000) in paragraph at lines 606--612 Underfull \hbox (badness 10000) in paragraph at lines 630--636 Underfull \hbox (badness 10000) in paragraph at lines 637--640 Underfull \hbox (badness 10000) in paragraph at lines 641--654 [57] Overfull \hbox (7.953pt too wide) in paragraph at lines 667--670 \OT1/cmr/m/n/12 Thus, $\OMS/cmsy/m/n/12 j[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 A[ ]=\U/msb/m/n/12 Q\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 j \OT1/cmr/m/n/12 = 2[] \OMS /cmsy/m/n/12 ^^A \OT1/cmr/m/n/12 5[]$. Us-ing the in-jec-tion from Step 6, we g et $5[]\OMS/cmsy/m/n/12 jj[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 A[]=\U/msb/m/n/12 Q\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 j$\OT1/cmr/m/n/12 , Underfull \hbox (badness 10000) in paragraph at lines 674--677 Underfull \hbox (badness 10000) in paragraph at lines 678--683 Underfull \hbox (badness 10000) in paragraph at lines 684--688 Underfull \hbox (badness 10000) in paragraph at lines 689--695 [58] Underfull \hbox (badness 10000) in paragraph at lines 696--706 Underfull \hbox (badness 10000) in paragraph at lines 715--717 Underfull \hbox (badness 10000) in paragraph at lines 724--728 [59] Underfull \hbox (badness 10000) in paragraph at lines 775--780 [60]) [61] (concl.tex Underfull \hbox (badness 10000) in paragraph at lines 7--9 Underfull \hbox (badness 10000) in paragraph at lines 16--18 Underfull \hbox (badness 10000) in paragraph at lines 30--37 [62]) [63] (Thesis.bbl [64] Underfull \hbox (badness 1308) in paragraph at lines 89--91 [][]\OT1/cmr/m/n/12 , \OT1/cmr/m/it/12 Ja-co-bian va-ri-eties\OT1/cmr/m/n/12 , Arith-metic ge-om-e-try (Storrs, Conn., 1984), [65]) [66] (Thesis.aux (Abs.aux) (Ack.aux) (intro.aux) (chapter1.aux) (chapter2.aux) (chapter3.aux) (chapter4.aux) (chapter5.aux) (concl.aux)) ) (see the transcript file for additional information) Output written on Thesis.dvi (72 pages, 264276 bytes). Transcript written on Thesis.log.