This is TeX, Version 3.14159 (Web2C 7.3.1) (format=latex 2000.7.19) 20 AUG 2001 17:35 **thesisfinalversionsinglespaced.tex (thesisfinalversionsinglespaced.tex LaTeX2e <1999/12/01> patch level 1 Babel and hyphenation patterns for american, french, german, ngerman, i talian, nohyphenation, loaded. (/usr/share/texmf/tex/latex/base/report.cls Document Class: report 1999/09/10 v1.4a Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size11.clo File: size11.clo 1999/09/10 v1.4a Standard LaTeX file (size option) ) \c@part=\count79 \c@chapter=\count80 \c@section=\count81 \c@subsection=\count82 \c@subsubsection=\count83 \c@paragraph=\count84 \c@subparagraph=\count85 \c@figure=\count86 \c@table=\count87 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 ) (/usr/share/texmf/tex/latex/amsmath/amsmath.sty Package: amsmath 2000/01/15 v2.05 AMS math features \@mathmargin=\skip43 For additional information on amsmath, use the `?' option. (/usr/share/texmf/tex/latex/amsmath/amstext.sty Package: amstext 1999/11/15 v2.0 (/usr/share/texmf/tex/latex/amsmath/amsgen.sty File: amsgen.sty 1999/11/30 v2.0 \@emptytoks=\toks14 \ex@=\dimen103 )) (/usr/share/texmf/tex/latex/amsmath/amsbsy.sty Package: amsbsy 1999/11/29 v1.2d \pmbraise@=\dimen104 ) (/usr/share/texmf/tex/latex/amsmath/amsopn.sty Package: amsopn 1999/12/14 v2.01 operator names ) \inf@bad=\count88 LaTeX Info: Redefining \frac on input line 198. \uproot@=\count89 \leftroot@=\count90 LaTeX Info: Redefining \overline on input line 302. \classnum@=\count91 \DOTSCASE@=\count92 LaTeX Info: Redefining \ldots on input line 376. LaTeX Info: Redefining \dots on input line 379. LaTeX Info: Redefining \cdots on input line 464. \Mathstrutbox@=\box26 \strutbox@=\box27 \big@size=\dimen105 LaTeX Font Info: Redeclaring font encoding OML on input line 563. 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LaTeX Font Info: ... okay on input line 19. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 19. LaTeX Font Info: ... okay on input line 19. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 19. LaTeX Font Info: ... okay on input line 19. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 19. LaTeX Font Info: ... okay on input line 19. [1 ] LaTeX Font Info: Try loading font information for U+msa on input line 47. (/usr/share/texmf/tex/latex/amsfonts/umsa.fd File: umsa.fd 1995/01/05 v2.2e AMS font definitions ) LaTeX Font Info: Try loading font information for U+msb on input line 47. (/usr/share/texmf/tex/latex/amsfonts/umsb.fd File: umsb.fd 1995/01/05 v2.2e AMS font definitions ) LaTeX Font Info: Try loading font information for OMS+cmr on input line 47. (/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 <10.95> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 47. [2] [3] (thesisfinalversionsinglespaced.toc [4 ] LaTeX Warning: Reference `firstthmtwotorsion' on page 95 undefined on input lin e 66. ) \tf@toc=\write3 \openout3 = `thesisfinalversionsinglespaced.toc'. [5] [6] \openout2 = `thesischapter1finalversion.aux'. (thesischapter1finalversion.tex Chapter 1. [1 ] [2] [3] [4] [5] LaTeX Font Info: Font shape `OMS/cmr/m/it' in size <10.95> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 314. [6] [7] Underfull \hbox (badness 1087) in paragraph at lines 399--401 []\OT1/cmr/m/it/10.95 There are in-finitely many el-lip-tic curves $\OML/cmm/m/ it/10.95 F$ \OT1/cmr/m/it/10.95 over $\OML/cmm/m/it/10.95 K$ \OT1/cmr/m/it/10.9 5 with dis-tinct [] ) [8] \openout2 = `thesischapter2finalversion.aux'. (thesischapter2finalversion.tex Chapter 2. [9 ] Underfull \hbox (badness 2600) in paragraph at lines 63--71 []\OT1/cmr/m/n/10.95 The fol-low-ing is the list of prop-er-ties which we will aim for [] Overfull \hbox (1.78104pt too wide) in paragraph at lines 74--79 []\OT1/cmr/m/it/10.95 Extension of two-torsion sec-tions: \OT1/cmr/m/n/10.95 Th e two-torsion points of the generic [] [10] Underfull \hbox (badness 10000) in paragraph at lines 118--128 \OT1/cmr/m/it/10.95 num-ber field $\OML/cmm/m/it/10.95 K$\OT1/cmr/m/it/10.95 , we get (up to con-ju-ga-tion) a ho-mo-mor-phism [] Underfull \hbox (badness 1466) in paragraph at lines 147--151 \OT1/cmr/m/n/10.95 el-lip-tic curves with full level-two-structure and a nowher e van-ish-ing [] Underfull \hbox (badness 4765) in paragraph at lines 154--161 \OT1/cmr/m/n/10.95 where on $\OML/cmm/m/it/10.95 S$\OT1/cmr/m/n/10.95 , with fu ll level-two-structure (i. e. an iso-mor-phism [] [11] LaTeX Font Info: Try loading font information for U+euf on input line 216. (/usr/share/texmf/tex/latex/amsfonts/ueuf.fd File: ueuf.fd 1995/01/05 v2.2e AMS font definitions ) [12] [13] [14] [15] Underfull \hbox (badness 10000) in paragraph at lines 428--430 \OT1/cmr/m/n/10.95 Our ref-er-ences for blow-ing up schemes along sub-schemes a re [] [16] Underfull \hbox (badness 1406) in paragraph at lines 451--455 []$[][]\OT1/cmr/m/n/10.95 X$ \OT1/cmr/m/it/10.95 is cov-ered by open affines $\ OML/cmm/m/it/10.95 U[]; U[]; [] ; U[]$\OT1/cmr/m/it/10.95 , where for [] Underfull \hbox (badness 1400) in paragraph at lines 456--462 \OT1/cmr/m/it/10.95 to ob-tain $\OML/cmm/m/it/10.95 f[]$\OT1/cmr/m/it/10.95 , a poly-no-mial in $\OML/cmm/m/it/10.95 x[]; x[]; [] ; x[]$\OT1/cmr/m/it/10.95 . Then fac-tor [] [17] Underfull \hbox (badness 3088) in paragraph at lines 496--498 []\OT1/cmr/m/n/10.95 In or-der to de-duce the propo-si-tion for ir-re-ducible $ \OML/cmm/m/it/10.95 f$\OT1/cmr/m/n/10.95 , we use [] Underfull \hbox (badness 1983) in paragraph at lines 501--509 []\OT1/cmr/m/it/10.95 Let $\OML/cmm/m/it/10.95 X$ \OT1/cmr/m/it/10.95 be any sc heme, $\OML/cmm/m/it/10.95 Y \OMS/cmsy/m/n/10.95 ^^Z \OML/cmm/m/it/10.95 X$ \OT 1/cmr/m/it/10.95 a sub-scheme and [] [18] Underfull \hbox (badness 3158) in paragraph at lines 559--567 \OT1/cmr/m/n/10.95 The ``base case'' for us is the mul-ti-plic-ity of the affin e line in [] [19] [20] Underfull \hbox (badness 1231) in paragraph at lines 666--671 \OT1/cmr/m/n/10.95 inside $\U/msb/m/n/10.95 P[] \OMS/cmsy/m/n/10.95 ^^B \U/msb/ m/n/10.95 P[]$\OT1/cmr/m/n/10.95 , where the first fac-tor has ho-mo-ge-neous c o-or-di-nates [] [21] [22] [23] [24] [25] Underfull \hbox (badness 2173) in paragraph at lines 984--989 \OT1/cmr/m/n/10.95 or the two dou-ble lines $\OML/cmm/m/it/10.95 V \OT1/cmr/m/n /10.95 : \OML/cmm/m/it/10.95 B[] \OT1/cmr/m/n/10.95 = 0\OML/cmm/m/it/10.95 ; \O T1/cmr/m/n/10.95 ^^@[] = \OML/cmm/m/it/10.95 x[] \OMS/cmsy/m/n/10.95 ^^@ \OT1/c mr/m/n/10.95 1$ [] [26] Underfull \hbox (badness 1803) in paragraph at lines 1041--1047 \OT1/cmr/bx/n/10.95 Two-torsion sec-tions. \OT1/cmr/m/n/10.95 We want to lift t he two-torsion sec-tions [] [27] Underfull \hbox (badness 2057) in paragraph at lines 1116--1122 \OT1/cmr/m/it/10.95 down map. Then $\OML/cmm/m/it/10.95 ^^^[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 Y\OT1/cmr/m/n/10.95 )$ \OT1/cmr/m/it/10.95 is a Cartier s ub-scheme of $[][]\OML/cmm/m/it/10.95 X$\OT1/cmr/m/it/10.95 , and [] [28] [29] [30] Overfull \hbox (4.04092pt too wide) in paragraph at lines 1252--1257 \OT1/cmr/m/n/10.95 on this chart, given by $(\OML/cmm/m/it/10.95 x[]; \OT1/cmr/ m/n/10.95 ^^A[]\OML/cmm/m/it/10.95 ; ^^Z[]; A[]\OT1/cmr/m/n/10.95 ) = (0\OML/cm m/m/it/10.95 ; \OT1/cmr/m/n/10.95 0\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 0\O ML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 0)$ and $(0\OML/cmm/m/it/10.95 ; \OT1/cm r/m/n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 1\OML/cmm/m/it/10.95 ; \O T1/cmr/m/n/10.95 0)$. [] [31] [32] [33] [34] [35] [36] [37] Overfull \hbox (0.40088pt too wide) in paragraph at lines 1721--1725 \OT1/cmr/m/n/10.95 inside $[](\U/msb/m/n/10.95 Z\OT1/cmr/m/n/10.95 [1\OML/cmm/m /it/10.95 =\OT1/cmr/m/n/10.95 2][\OML/cmm/m/it/10.95 x[]; \OT1/cmr/m/n/10.95 ^^ @[]\OML/cmm/m/it/10.95 ; s[]; B[]\OT1/cmr/m/n/10.95 ])$. A straight-for-ward co m-pu-ta-tion gives [] [38] [39] [40] [41] Underfull \hbox (badness 2951) in paragraph at lines 1979--1985 \OT1/cmr/m/n/10.95 From the shape of the fibers we see that it is im-pos-si-ble that [] [42] Underfull \hbox (badness 1231) in paragraph at lines 2047--2058 \OT1/cmr/m/n/10.95 the mod-uli in-ter-pre-ta-tion of their points, for $\OML/cm m/m/it/10.95 ^^U \OMS/cmsy/m/n/10.95 2 \OML/cmm/m/it/10.95 SL[]\OT1/cmr/m/n/10. 95 (\U/msb/m/n/10.95 Z\OML/cmm/m/it/10.95 =n\U/msb/m/n/10.95 Z\OT1/cmr/m/n/10.9 5 )$, let [] [43] [44] Underfull \hbox (badness 2460) in paragraph at lines 2107--2119 \OT1/cmr/m/it/10.95 Using the $\OML/cmm/m/it/10.95 S[]$\OT1/cmr/m/it/10.95 -act ion on $\OMS/cmsy/m/n/10.95 F[]$\OT1/cmr/m/it/10.95 , $\OML/cmm/m/it/10.95 ^^Z[ ]$ \OT1/cmr/m/it/10.95 gives rise to a co-ho-mol-ogy class [] Underfull \hbox (badness 2253) in paragraph at lines 2107--2119 \OML/cmm/m/it/10.95 c[] \OMS/cmsy/m/n/10.95 2 \OML/cmm/m/it/10.95 H[]\OT1/cmr/m /n/10.95 (\OML/cmm/m/it/10.95 G[]; [][]\OT1/cmr/m/n/10.95 (\OMS/cmsy/m/n/10.95 F[]\OT1/cmr/m/n/10.95 ))$\OT1/cmr/m/it/10.95 ; like-wise, we ob-tain a co-ho-mo l-ogy class [] Underfull \hbox (badness 6893) in paragraph at lines 2152--2169 \OT1/cmr/m/n/10.95 a nowhere van-ish-ing reg-u-lar rel-a-tive dif-fer-en-tial o n $\OML/cmm/m/it/10.95 F$ \OT1/cmr/m/n/10.95 over $\OML/cmm/m/it/10.95 S$\OT1/c mr/m/n/10.95 , [] [45] [46] [47] [48] [49] [50]) [51] \openout2 = `thesischapter3finalversionsinglespaced.aux'. (thesischapter3finalversionsinglespaced.tex Chapter 3. [52 ] Overfull \hbox (0.91173pt too wide) detected at line 90 [] [] [53] [54] [55] Underfull \hbox (badness 2460) in paragraph at lines 218--222 []\OT1/cmr/m/it/10.95 Let $\OMS/cmsy/m/n/10.95 B$ \OT1/cmr/m/it/10.95 be the pu ll-back of the pro-jec-tive line bun-dle [] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] Underfull \hbox (badness 2277) in paragraph at lines 755--758 []\OT1/cmr/m/n/10.95 Similarly, the Zariski open set $\OML/cmm/m/it/10.95 U[]$ \OT1/cmr/m/n/10.95 of $\OML/cmm/m/it/10.95 D$ \OT1/cmr/m/n/10.95 given by the s ame [] [66] [67] [68] [69] [70] [71] [72] Underfull \hbox (badness 1365) in paragraph at lines 1142--1147 \OT1/cmr/m/n/10.95 inside $\U/msb/m/n/10.95 A[]\OT1/cmr/m/n/10.95 [\OML/cmm/m/i t/10.95 d[]; d[]; ^^Z[]; s[]\OT1/cmr/m/n/10.95 ]$. But note that on the co-or-d i-nate [] Underfull \hbox (badness 1242) in paragraph at lines 1142--1147 \OT1/cmr/m/n/10.95 chart $\U/euf/m/n/10.95 C[]$ \OT1/cmr/m/n/10.95 (in-tro-duce d in Sec-tion 2.5[]), $\OMS/cmsy/m/n/10.95 E\OT1/cmr/m/n/10.95 (2\OML/cmm/m/it/ 10.95 ; !\OT1/cmr/m/n/10.95 )$ is given by the [] [73] Underfull \hbox (badness 5741) in paragraph at lines 1172--1178 \OT1/cmr/m/n/10.95 inside $\U/msb/m/n/10.95 A[]\OT1/cmr/m/n/10.95 [\OML/cmm/m/i t/10.95 A[]; A[]; ^^Z[]; s[]\OT1/cmr/m/n/10.95 ]$. Hence, the map given by [] [74] Underfull \hbox (badness 1112) in paragraph at lines 1207--1214 \OT1/cmr/m/n/10.95 that the map given by $\OML/cmm/m/it/10.95 x[] \OT1/cmr/m/n/ 10.95 = \OML/cmm/m/it/10.95 d[]; \OT1/cmr/m/n/10.95 ^^A[] = \OML/cmm/m/it/10.95 ^^N[]; [] [75] [76] Underfull \hbox (badness 2922) in paragraph at lines 1341--1350 \OT1/cmr/m/it/10.95 si-tion, and $\OMS/cmsy/m/n/10.95 B$ \OT1/cmr/m/it/10.95 is the pull-back to $\OML/cmm/m/it/10.95 V$ \OT1/cmr/m/it/10.95 of the pro-jec-ti ve line bun-dle [] [77] Underfull \hbox (badness 1609) in paragraph at lines 1381--1395 \OT1/cmr/m/n/10.95 the lift $\OML/cmm/m/it/10.95 ^^Q \OMS/cmsy/m/n/10.95 2 \OML /cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G[]; E\OT1/cmr/m/n/1 0.95 [2])$ of $\OML/cmm/m/it/10.95 ^^[$ \OT1/cmr/m/n/10.95 gives rise to a co-h o-mol-ogy class [] Overfull \hbox (1.47093pt too wide) in paragraph at lines 1419--1426 \OT1/cmr/m/n/10.95 A the-o-rem proved in-de-pen-dently by A. Sko-roboga-tov (se e [[]]), N. Shepherd- [] [78] Underfull \hbox (badness 1199) in paragraph at lines 1481--1489 \OML/cmm/m/it/10.95 ^^[ \OMS/cmsy/m/n/10.95 2 []\OT1/cmr/m/n/10.95 (\OML/cmm/m/ it/10.95 E=K\OT1/cmr/m/n/10.95 )[2]$ of the \OT1/cmr/m/it/10.95 Shafarevich-Tat e group \OT1/cmr/m/n/10.95 of any el-lip-tic [] [79] [80]) [81] \openout2 = `thesischapter4finalversion.aux'. (thesischapter4finalversion.tex Chapter 4. [82 ] [83] Overfull \hbox (2.449pt too wide) in paragraph at lines 110--118 \OT1/cmr/m/n/10.95 Chap-ter $1$ for lifts of el-e-ments of $\OML/cmm/m/it/10.95 WC\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 E=K\OT1/cmr/m/n/10.95 )[\OML/cmm/m/i t/10.95 m\OT1/cmr/m/n/10.95 ]$ to el-e-ments in $\OML/cmm/m/it/10.95 H[]\OT1/cm r/m/n/10.95 (\OML/cmm/m/it/10.95 G[]; E\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 m\OT1/cmr/m/n/10.95 ])$. [] [84] [85] Underfull \hbox (badness 1478) in paragraph at lines 230--236 []\OT1/cmr/m/it/10.95 Let $\OML/cmm/m/it/10.95 E$ \OT1/cmr/m/it/10.95 be an el- lip-tic curve over a num-ber field $\OML/cmm/m/it/10.95 K$ [] [86] Underfull \hbox (badness 1087) in paragraph at lines 278--280 []\OT1/cmr/m/it/10.95 There are in-finitely many el-lip-tic curves $\OML/cmm/m/ it/10.95 F$ \OT1/cmr/m/it/10.95 over $\OML/cmm/m/it/10.95 K$ \OT1/cmr/m/it/10.9 5 with dis-tinct [] [87] [88] [89] [90] [91] Underfull \hbox (badness 1038) in paragraph at lines 503--512 \OT1/cmr/m/n/10.95 where $\OML/cmm/m/it/10.95 t[]$ \OT1/cmr/m/n/10.95 is trans- la-tion by the point $(\OML/cmm/m/it/10.95 ^^W; ^^W\OT1/cmr/m/n/10.95 )$ on $\O ML/cmm/m/it/10.95 E[] \OMS/cmsy/m/n/10.95 ^^B \OML/cmm/m/it/10.95 E[]$\OT1/cmr/ m/n/10.95 , and let [] [92] Underfull \hbox (badness 10000) in paragraph at lines 571--583 \OT1/cmr/bx/n/10.95 PROOF. \OT1/cmr/m/n/10.95 Us-ing equa-tion (4.4[]) de-scri b-ing the ra-tio-nal map [] [93] [94] Underfull \hbox (badness 1424) in paragraph at lines 690--696 \OML/cmm/m/it/10.95 ^^Q \OMS/cmsy/m/n/10.95 2 \OML/cmm/m/it/10.95 H[]\OT1/cmr/m /n/10.95 (\OML/cmm/m/it/10.95 G[]; \OMS/cmsy/m/n/10.95 h\OML/cmm/m/it/10.95 P\O MS/cmsy/m/n/10.95 i\OT1/cmr/m/n/10.95 ) [] \OML/cmm/m/it/10.95 K[]=\OT1/cmr/m/n /10.95 (\OML/cmm/m/it/10.95 K[]\OT1/cmr/m/n/10.95 )[]$ cor-re-spond-ing to the class of the [] [95] Underfull \hbox (badness 10000) in paragraph at lines 735--740 \OT1/cmr/m/n/10.95 sult-ing from the rows of the fol-low-ing com-mu-ta-tive di- a-gram [] [96] [97] [98] [99]) [100] Underfull \hbox (badness 1571) in paragraph at lines 176--179 []\OT1/cmr/m/n/10.95 Cremona, J. E., Mazur, B.: \OT1/cmr/m/it/10.95 Vi-su-al-iz -ing el-e-ments in the [] Underfull \hbox (badness 1448) in paragraph at lines 185--189 []\OT1/cmr/m/n/10.95 Grothendieck, A., Dieudonn^^Se, J.: \OT1/cmr/m/it/10.95 El ^^Sements de G^^Seom^^Setrie [] [101 ] Underfull \hbox (badness 1286) in paragraph at lines 217--221 \OT1/cmr/m/it/10.95 bi-ra-tional ge-om-e-try of de-gen-er-a-tions\OT1/cmr/m/n/1 0.95 , ed. R. Fried-man and [] Underfull \hbox (badness 1297) in paragraph at lines 217--221 \OT1/cmr/m/n/10.95 D. R. Mor-ri-son, Progress in Math-e-mat-ics \OT1/cmr/bx/n/1 0.95 29\OT1/cmr/m/n/10.95 , Birkh^^?auser, [] Underfull \hbox (badness 1769) in paragraph at lines 242--244 []\OT1/cmr/m/n/10.95 Shioda, T.: \OT1/cmr/m/it/10.95 On el-lip-tic mod-u-lar su r-faces\OT1/cmr/m/n/10.95 , J. Math. Soc. [] [102] [103] (thesisfinalversionsinglespaced.aux (thesischapter1finalversion.aux ) (thesischapter2finalversion.aux) (thesischapter3finalversionsinglespaced.aux) (thesischapter4finalversion.aux)) LaTeX Warning: There were undefined references. LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right. ) Here is how much of TeX's memory you used: 1394 strings out of 20898 15238 string characters out of 196686 124963 words of memory out of 350001 4330 multiletter control sequences out of 10000+15000 23233 words of font info for 89 fonts, out of 400000 for 1000 14 hyphenation exceptions out of 1000 27i,14n,27p,332b,426s stack positions out of 1500i,100n,500p,50000b,4000s Output written on thesisfinalversionsinglespaced.dvi (109 pages, 470484 bytes).