This is TeX, Version 3.14159 (Web2C 7.3.1) (format=latex 2000.8.10) 13 NOV 2007 16:00 **EllipticCurves (EllipticCurves.tex LaTeX2e <2000/06/01> Babel and hyphenation patterns for american, british, nohyphenation, lo aded. (/common/teTeX/texmf/tex/latex/LaTeX2000/latex2e/base/report.cls Document Class: report 2000/05/19 v1.4b Standard LaTeX document class (/common/teTeX/texmf/tex/latex/LaTeX2000/latex2e/base/size12.clo File: size12.clo 2000/05/19 v1.4b 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 ) (/common/teTeX/texmf/tex/latex/required/amslatex/math/amsmath.sty Package: amsmath 2000/01/15 v2.05 AMS math features \@mathmargin=\skip43 For additional information on amsmath, use the `?' option. 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(/common/teTeX/texmf/tex/latex/LaTeX2000/graphics/dvips.def File: dvips.def 1999/02/16 v3.0i Driver-dependant file (DPC,SPQR) )) \Gin@req@height=\dimen113 \Gin@req@width=\dimen114 ) \mp@file=\write3 \c@mpfig=\count99 ) (/common/TeX/texmf/tex/latex/askinclude.sty Enter file(s) to include (separated by commas and no spaces), or enter * for all files, or enter - for none, or press Return if this run is the same as the last. \inputfiles= ) (/common/teTeX/texmf/tex/latex/LaTeX2000/tools/theorem.sty Package: theorem 1995/11/23 v2.2c Theorem extension package (FMi) \theorem@style=\toks21 \theorem@bodyfont=\toks22 \theorempreskipamount=\skip46 \theorempostskipamount=\skip47 (/common/teTeX/texmf/tex/latex/LaTeX2000/tools/thp.sty File: thp.sty 1995/11/23 v2.2c Theorem extension package (FMi) )) (EllipticCurves.sty \c@theorem=\count100 \c@proposition=\count101 \c@definition=\count102 \c@corollary=\count103 \c@Corollary=\count104 \c@lemma=\count105 \c@Lemma=\count106 \c@sublemma=\count107 ) (EllipticCurves.aux (NewIntro.aux) (ch01.aux) (ch02.aux) (ch03.aux) (ch04.aux) (ch05.aux) (ch06.aux) (ch07.aux) (ch08.aux) (ch09.aux) (ComputingRank.aux) (ch10.aux) (ComputingRank-II.aux) (ch11.aux) (appA.aux) (appB.aux) (appC.aux) (appH.aux)) \openout1 = `EllipticCurves.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 32. LaTeX Font Info: ... okay on input line 32. \openout3 = `EllipticCurves.mp'. \openout2 = `NewIntro.aux'. (NewIntro.tex Chapter 1. LaTeX Font Info: Try loading font information for U+msa on input line 7. (/common/teTeX/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 7. (/common/teTeX/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 U+rsfs on input line 7. (/common/teTeX/texmf/tex/latex/jknappen/ursfs.fd File: ursfs.fd 1998/03/24 rsfs font definition file (jk) ) [1 ] LaTeX Warning: Reference `p-adic-case' on page 428--99\hskip 1em\relax 1--2 und efined on input line 79. [2] [3] [4] [5] [6] [7] [8] [9]) [10] \openout2 = `ch01.aux'. (ch01.tex Chapter 1. [1 ] [2] [3] [4] [5] Overfull \hbox (19.24626pt too wide) in paragraph at lines 324--327 \OT1/cmr/m/n/12 Hence $2 \OMS/cmsy/m/n/12 j \OML/cmm/m/it/12 Z$\OT1/cmr/m/n/12 ; and so $2 \OMS/cmsy/m/n/12 j \OML/cmm/m/it/12 X; Y; Z$\OT1/cmr/m/n/12 , con-t ra-dict-ing our as-sump-tion that $[](\OML/cmm/m/it/12 X; Y; Z\OT1/cmr/m/n/12 ) = [] [6] [7] Overfull \hbox (34.89749pt too wide) in paragraph at lines 446--449 \OT1/cmr/m/n/12 It fol-lows that we can al-ways find a conic pass-ing through 5 points $\OML/cmm/m/it/12 Q[]; Q[]; Q[]; Q[]; Q[]$\OT1/cmr/m/n/12 . [] [8]) [9] \openout2 = `ch02.aux'. (ch02.tex Chapter 2. [1 ] [2] [3] [4] Overfull \hbox (5.42297pt too wide) in paragraph at lines 274--276 \OT1/cmr/m/n/12 where the par-tial dif-fer-en-tial co-ef-fi-cients are com-pute d at the point $[\OML/cmm/m/it/12 X[]; Y[]; Z[]\OT1/cmr/m/n/12 ]$. [] [5] [6] [7] [8] [9] [10] [11] [12] Overfull \hbox (70.35779pt too wide) in paragraph at lines 755--755 [] [] [13] [14] Overfull \hbox (7.95956pt too wide) detected at line 856 \OML/cmm/m/it/12 F[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 X; Y; Z\OT1/cmr/m/n/12 ) = \OML/cmm/m/it/12 M\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 X; Y; Z\OT1/cmr/m/n/12 )\ OML/cmm/m/it/12 G[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 X; Y; Z\OT1/cmr/m/n/12 )\O ML/cmm/m/it/12 ; F[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 X; Y; Z\OT1/cmr/m/n/12 ) = \OML/cmm/m/it/12 M\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 X; Y; Z\OT1/cmr/m/n/12 ) \OML/cmm/m/it/12 G[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 X; Y; Z\OT1/cmr/m/n/12 )\ OML/cmm/m/it/12 : [] [15] [16] Overfull \hbox (94.96423pt too wide) detected at line 997 []$[][]\OML/cmm/m/it/12 u[]v[] [] \OT1/cmr/m/n/12 + [] $ [] Overfull \hbox (89.6096pt too wide) detected at line 1016 \OML/cmm/m/it/12 M\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 X[]; Y[]; Z[]\OT1/cmr/m/n/1 2 ) \OMS/cmsy/m/n/12 ^^Q [][][] \OT1/cmr/m/n/12 + [][][] + [][][] + 2[]\OML/cmm /m/it/12 X[]Y[] \OT1/cmr/m/n/12 + 2[]\OML/cmm/m/it/12 X[]Z[] \OT1/cmr/m/n/12 + 2[]\OML/cmm/m/it/12 Y[]Z[] \OT1/cmr/m/n/12 = 0\OML/cmm/m/it/12 : [] [17] Overfull \hbox (3.90529pt too wide) in paragraph at lines 1049--1052 \OT1/cmr/m/it/12 Proof of Lemma \U/msa/m/n/10 B \OT1/cmr/m/n/12 To avoid con-fu -sion, let us for a mo-ment set $\OML/cmm/m/it/12 P \OT1/cmr/m/n/12 = [\OML/cmm /m/it/12 X[]; Y[]; Z[]\OT1/cmr/m/n/12 ]$. [] Overfull \hbox (16.9633pt too wide) detected at line 1062 []$[] [] \OML/cmm/m/it/12 X \OT1/cmr/m/n/12 + [] \OML/cmm/m/it/12 Y$ [] [18] [19] Overfull \hbox (2.82162pt too wide) in paragraph at lines 1206--1209 \OT1/cmr/m/n/12 its co-or-di-nates $(\OML/cmm/m/it/12 x; y\OT1/cmr/m/n/12 )$, D escartes con-verted ge-o-met-ric into al-ge-braic prob- [] Overfull \hbox (2.9784pt too wide) in paragraph at lines 1211--1216 []\OT1/cmr/m/n/12 Although not ex-plic-itly ge-o-met-ri-cal, Rie- [] Overfull \hbox (4.78441pt too wide) in paragraph at lines 1230--1232 []\OT1/cmr/m/n/12 studied gen-eral \OT1/cmr/m/it/12 al-ge-braic va- [] [20]) [21] \openout2 = `ch03.aux'. (ch03.tex Chapter 3. [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] Overfull \hbox (9.26886pt too wide) detected at line 629 \OML/cmm/m/it/12 P \OT1/cmr/m/n/12 = (0\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 0)\OM L/cmm/m/it/12 ; Q \OT1/cmr/m/n/12 = (1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 0)\OM L/cmm/m/it/12 ; R \OT1/cmr/m/n/12 = (\OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML /cmm/m/it/12 ; \OT1/cmr/m/n/12 0)\OML/cmm/m/it/12 ; S \OT1/cmr/m/n/12 = (0\OML /cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 ; T \OT1/cmr/m/n/12 = (1\OML /cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 ; U \OT1/cmr/m/n/12 = (\OMS/ cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\OML/cmm/m /it/12 : [] [12] [13] [14]) [15] \openout2 = `ch04.aux'. (ch04.tex Chapter 4. [1 ] Overfull \hbox (74.44589pt too wide) detected at line 90 \OML/cmm/m/it/12 `[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 X[]X[]X[]; `[] \OT1/cm r/m/n/12 = \OML/cmm/m/it/12 X[]X[]X[]; `[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 X[]X[]X[]; `[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 X[]X[]X[]; `[] \OT1/cmr/m/n /12 = \OML/cmm/m/it/12 X[]X[]X[]; `[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 X[]X[ ]X[]: [] [2] Overfull \hbox (55.67265pt too wide) detected at line 126 \OT1/cmr/m/n/12 ^^@ : \OML/cmm/m/it/12 a[]X[] \OT1/cmr/m/n/12 + \OML/cmm/m/it/1 2 a[]X[]Y \OT1/cmr/m/n/12 + \OML/cmm/m/it/12 a[]X[]Za[]XY[] \OT1/cmr/m/n/12 + \ OML/cmm/m/it/12 a[]XYZ \OT1/cmr/m/n/12 + \OML/cmm/m/it/12 a[]XZ[] \OT1/cmr/m/n/ 12 + \OML/cmm/m/it/12 a[]Y[] \OT1/cmr/m/n/12 + \OML/cmm/m/it/12 a[]Y[]Z \OT1/cm r/m/n/12 + \OML/cmm/m/it/12 a[]YZ[] \OT1/cmr/m/n/12 + \OML/cmm/m/it/12 a[]Z[] \ OT1/cmr/m/n/12 = 0\OML/cmm/m/it/12 : [] ) [3] \openout2 = `ch05.aux'. (ch05.tex Chapter 5. [1 ] [2] [3] [4] [5] [6] [7] [8] [9] Overfull \hbox (4.14864pt too wide) in paragraph at lines 557--559 []\OT1/cmr/m/it/12 Suppose $\OMS/cmsy/m/n/12 k\OML/cmm/m/it/12 X[]\OMS/cmsy/m/n /12 k[]\OML/cmm/m/it/12 ; \OMS/cmsy/m/n/12 k\OML/cmm/m/it/12 X[]\OMS/cmsy/m/n/1 2 k[] \OML/cmm/m/it/12 < \OT1/cmr/m/n/12 1$\OT1/cmr/m/it/12 . Then we can ex-pr ess $\OML/cmm/m/it/12 S\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 X[]; X[]\OT1/cmr/m/n/1 2 )$ [] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] Overfull \hbox (7.04172pt too wide) in paragraph at lines 1299--1306 []\OT1/cmr/m/n/12 The Struc-ture The-o-rem for finitely-generated abelian group s, ie $\OML/cmm/m/it/12 Z$\OT1/cmr/m/n/12 -modules, [] ) [23] \openout2 = `ch06.aux'. (ch06.tex Chapter 6. [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] Overfull \hbox (15.7339pt too wide) in paragraph at lines 698--700 []\OT1/cmr/m/it/12 Suppose $\OML/cmm/m/it/12 f\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 ; g\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/ cmr/m/n/12 ) \OMS/cmsy/m/n/12 2 \U/msb/m/n/12 Z\OT1/cmr/m/n/12 [\OML/cmm/m/it/1 2 x\OT1/cmr/m/n/12 ]$\OT1/cmr/m/it/12 . Then there ex-ist poly-no-mi-als $\OML/ cmm/m/it/12 u\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )\OML/cmm/m/it/ 12 ; v\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 ) \OMS/cmsy/m/n/12 2 [] [12] [13] [14] [15] [16] [17] Overfull \hbox (57.86493pt too wide) in paragraph at lines 1111--1114 \OT1/cmr/m/n/12 This has 6 ob-vi-ous in-te-gral points, namely $(0\OML/cmm/m/it /12 ; \OT1/cmr/m/n/12 0)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (0\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (1\OML/cmm/m/it/12 ; \OT1 /cmr/m/n/12 0)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (1\OML/cmm/m/it/12 ; \OT1/cmr/ m/n/12 1)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/1 2 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 0)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (\OM S/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1)$. [] [18] [19] [20] [21]) [22] \openout2 = `ch07.aux'. (ch07.tex Chapter 7. [1 ] [2] [3] LaTeX Warning: Reference `Reduce' on page 428--99\hskip 1em\relax 7--4 undefine d on input line 246. [4] [5] [6] [7] [8] [9] [10] [11]) [12] \openout2 = `ch08.aux'. (ch08.tex Chapter 8. [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] Overfull \hbox (10.47914pt too wide) in paragraph at lines 1036--1038 \OT1/cmr/m/n/12 The func-tional equa-tion can be in-ter-preted as say-ing that the point $(\OML/cmm/m/it/12 '\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 z\OT1/cmr/m/n/1 2 )\OML/cmm/m/it/12 ; '[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 z\OT1/cmr/m/n/12 ))$ [] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44]) [45] \openout2 = `ch09.aux'. (ch09.tex Chapter 9. [1 ] [2] Overfull \hbox (55.84038pt too wide) in alignment at lines 176--180 [][][] [] [] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] Package amsfonts Warning: Obsolete command \frak; \mathfrak should be used inst ead on input line 766. LaTeX Font Info: Try loading font information for U+euf on input line 766. (/common/teTeX/texmf/tex/latex/amsfonts/ueuf.fd File: ueuf.fd 1995/01/05 v2.2e AMS font definitions ) [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]) [26] \openout2 = `ComputingRank.aux'. (ComputingRank.tex [27 ] [28] [29] Overfull \hbox (49.1741pt too wide) detected at line 215 \OML/cmm/m/it/12 p[]\OT1/cmr/m/n/12 (0) = (0 \OMS/cmsy/m/n/12 ^^@ \OT1/cmr/m/n/ 12 1)(0 + 1) = \OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; p[]\OT1 /cmr/m/n/12 (1) = (1 \OMS/cmsy/m/n/12 ^^@ \OT1/cmr/m/n/12 0)(1 + 1) = 2\OML/cmm /m/it/12 ; p[]\OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1) = (\OMS/ cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1 \OMS/cmsy/m/n/12 ^^@ \OT1/cmr/m/n/12 0)(\OMS/c msy/m/n/12 ^^@\OT1/cmr/m/n/12 1 \OMS/cmsy/m/n/12 ^^@ \OT1/cmr/m/n/12 1) = 2\OML /cmm/m/it/12 : [] Overfull \hbox (137.86993pt too wide) detected at line 257 \OT1/cmr/m/n/12 (\OML/cmm/m/it/12 d; e; f\OT1/cmr/m/n/12 ) = (1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 ; \O T1/cmr/m/n/12 (1\OML/cmm/m/it/12 ; \OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cm m/m/it/12 ; \OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 ; \OT1/cmr/ m/n/12 (\OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OMS/c msy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\OML/cmm/m/ it/12 ; \OT1/cmr/m/n/12 (1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 2\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 2)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (1\OML/cmm/m/it/12 ; \ OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 2\OML/cmm/m/it/12 ; \OMS/cmsy/m/n/12 ^^@\OT1 /cmr/m/n/12 2)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 ^^@\OT1/cmr /m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 2\OML/cmm/m/it/12 ; \OMS/cmsy/m/n/1 2 ^^@\OT1/cmr/m/n/12 2)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 ^^ @\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 2\OML/ cmm/m/it/12 ; \OT1/cmr/m/n/12 2)\OML/cmm/m/it/12 : [] [30] Overfull \hbox (100.5365pt too wide) detected at line 337 \OT1/cmr/m/n/12 (\OML/cmm/m/it/12 d; e; f\OT1/cmr/m/n/12 ) = (1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 ; \O T1/cmr/m/n/12 (\OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OMS/cms y/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\OML/cmm/m/it /12 ; \OT1/cmr/m/n/12 (1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 2\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 2)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 ^^@\OT 1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 2\OML/cmm/ m/it/12 ; \OT1/cmr/m/n/12 2)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (2\OML/cmm/m/it /12 ; \OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 2)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 2\OML/cmm/m/it/12 ; \OMS/ cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 2)\OML/cmm/m /it/12 ; \OT1/cmr/m/n/12 (2\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 2\OML/cmm/m/it/1 2 ; \OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 ^^@ \OT1/cmr/m/n/12 2\OML/cmm/m/it/12 ; \OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 2\OML/c mm/m/it/12 ; \OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 : [] [31] [32] [33] [34] [35] [36] [37]) [38] \openout2 = `ch10.aux'. (ch10.tex Chapter 10. [1 ] [2] [3] [4] [5] [6] Overfull \hbox (18.48004pt too wide) in paragraph at lines 392--392 [] [] [7] Overfull \hbox (3.81674pt too wide) in paragraph at lines 495--498 \OT1/cmr/m/n/12 Our for-mu-lae be-come much sim-pler if we work with el-lip-tic curves in `constant- [] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] Overfull \hbox (82.89856pt too wide) in paragraph at lines 1635--1638 \OT1/cmr/m/n/12 It is read-ily ver-i-fied that the only pos-si-ble points of fi -nite or-der are: $\OML/cmm/m/it/12 O; \OT1/cmr/m/n/12 (0\OML/cmm/m/it/12 ; \O T1/cmr/m/n/12 0)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (1\OML/cmm/m/it/12 ; \OMS/c msy/m/n/12 ^^F\OT1/cmr/m/n/12 2)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (3\OML/cmm/ m/it/12 ; \OMS/cmsy/m/n/12 ^^F\OT1/cmr/m/n/12 6)$. [] [24]) [25] \openout2 = `ComputingRank-II.aux'. (ComputingRank-II.tex Overfull \hbox (5.01434pt too wide) detected at line 25 \OT1/cmr/m/n/12 ^^B(\OML/cmm/m/it/12 x; y\OT1/cmr/m/n/12 ) = [] \OML/cmm/m/it/1 2 ; \OT1/cmr/m/n/12 ^^H(\OML/cmm/m/it/12 x[]; y[]\OT1/cmr/m/n/12 ) = [] \OML/cm m/m/it/12 ; [] [26 ] [27] [28] [29] [30] [31]) [32] \openout2 = `ch11.aux'. (ch11.tex Chapter 11. [1 ] [2] [3] Overfull \hbox (120.03583pt too wide) detected at line 315 \OML/cmm/m/it/12 X[] \OMS/cmsy/m/n/12 ^^Q \OML/cmm/m/it/12 Y[]\OT1/cmr/m/n/12 ( []\OML/cmm/m/it/12 m\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 ; X[] \OMS/cmsy/m/n/12 ^^ Q \OML/cmm/m/it/12 Y[]\OT1/cmr/m/n/12 ([]\OML/cmm/m/it/12 m\OT1/cmr/m/n/12 ) =[ ]\OMS/cmsy/m/n/12 ) \OML/cmm/m/it/12 X[] \OT1/cmr/m/n/12 + \OML/cmm/m/it/12 X[] \OMS/cmsy/m/n/12 ^^Q \OML/cmm/m/it/12 Y[] \OT1/cmr/m/n/12 + \OML/cmm/m/it/12 Y []\OT1/cmr/m/n/12 ([]\OML/cmm/m/it/12 m\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 ; X[]X [] \OMS/cmsy/m/n/12 ^^Q \OML/cmm/m/it/12 Y[]Y[]\OT1/cmr/m/n/12 ([]\OML/cmm/m/it /12 m\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 : [] [4]) [5] \openout2 = `appA.aux'. (appA.tex Appendix A. Overfull \hbox (0.60434pt too wide) in paragraph at lines 13--17 $\OML/cmm/m/it/12 A$ \OT1/cmr/m/it/12 is said to be \OT1/cmr/m/n/12 finitely-ge nerated \OT1/cmr/m/it/12 if it is gen-er-ated by a fi-nite set $\OMS/cmsy/m/n/1 2 f\OML/cmm/m/it/12 a[]; [] ; a[]\OMS/cmsy/m/n/12 g ^^Z [] [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10] Overfull \hbox (51.51854pt too wide) in paragraph at lines 735--739 \OT1/cmr/m/n/12 Choose el-e-ments $\OML/cmm/m/it/12 a[]; [] ; a[]$ \OT1/cmr/m/n /12 in $\OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 map-ping onto the el-e-ments $(1\OM L/cmm/m/it/12 ; \OT1/cmr/m/n/12 0\OML/cmm/m/it/12 ; [] ; \OT1/cmr/m/n/12 0)\OML /cmm/m/it/12 ; [] ; \OT1/cmr/m/n/12 (0\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 0\OML/ cmm/m/it/12 ; [] ; \OT1/cmr/m/n/12 1)$ [] [11] [12] [13]) [14] \openout2 = `appB.aux'. (appB.tex Appendix B. [1 ] [2]) [3] \openout2 = `appC.aux'. (appC.tex Appendix C. [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10] Overfull \hbox (2.92609pt too wide) in paragraph at lines 649--652 \OT1/cmr/m/n/12 Suppose first $(\OML/cmm/m/it/12 x; y; z\OT1/cmr/m/n/12 )$ is a so-lu-tion in $\U/msb/m/n/12 Z$\OT1/cmr/m/n/12 . As usual we as-sume that $[]( \OML/cmm/m/it/12 x; y; z\OT1/cmr/m/n/12 ) = [] [11] [12] [13] [14]) [15] \openout2 = `appH.aux'. (appH.tex Appendix H. [1 ] [2]) [3] [4] (EllipticCurves.aux (NewIntro.aux) (ch01.aux) (ch02.aux) (ch03.aux) (ch04.aux) (ch05.aux) (ch06.aux) (ch07.aux) (ch08.aux) (ch09.aux) (ComputingRank.aux) (ch10.aux) (ComputingRank-II.aux) (ch11.aux) (appA.aux) (appB.aux) (appC.aux) (appH.aux)) LaTeX Warning: There were undefined references. Overfull \hbox (236.66016pt too wide) in paragraph at lines 2--62 []\OT1/cmr/m/n/12 * [] ) Here is how much of TeX's memory you used: 1818 strings out of 61443 20047 string characters out of 948236 78129 words of memory out of 263001 4674 multiletter control sequences out of 10000+60000 21310 words of font info for 84 fonts, out of 400000 for 1000 22 hyphenation exceptions out of 1000 32i,17n,22p,227b,341s stack positions out of 300i,100n,500p,50000b,10000s Output written on EllipticCurves.dvi (271 pages, 637168 bytes).