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(/usr/share/texmf/tex/latex/amsfonts/umsb.fd File: umsb.fd 2002/01/19 v2.2g AMS font definitions ) LaTeX Font Info: Try loading font information for OT2+wncyr on input line 21 . (/usr/share/texmf/tex/latex/cyrillic/ot2wncyr.fd File: ot2wncyr.fd 1999/12/17 v3.2a Computer Modern Cyrillic font definitions ) Overfull \hbox (26.72964pt too wide) detected at line 26 [] [] LaTeX Warning: Citation `MR31:3420' on page 1 undefined on input line 47. [1 ] LaTeX Warning: Citation `gross-zagier' on page 2 undefined on input line 57. Overfull \hbox (12.00082pt too wide) in paragraph at lines 68--70 [][]\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 [] [2] LaTeX Warning: Citation `mccallum' on page 3 undefined on input line 89. [3] LaTeX Warning: Citation `cremona-algorithms' on page 4 undefined on input line 119. LaTeX Warning: Citation `cremona-mwrank' on page 4 undefined on input line 121. LaTeX Warning: Citation `cremona-algorithms' on page 4 undefined on input line 123. LaTeX Warning: Citation `cremona-algorithms' on page 4 undefined on input line 125. LaTeX Warning: Citation `kolyvagin:euler_systems' on page 4 undefined on input line 150. LaTeX Warning: Citation `gross:kolyvagin' on page 4 undefined on input line 152 . [4] LaTeX Warning: Citation `serre:propgal' on page 5 undefined on input line 154. Overfull \hbox (48.44183pt too wide) detected at line 167 \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 K[]=K; E\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 p\O T1/cmr/m/n/10.95 ]) \OMS/cmsy/m/n/10.95 ! \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/1 0.95 (\OML/cmm/m/it/10.95 K; E\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 p\OT1/cmr /m/n/10.95 ]) \OMS/cmsy/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 [\OML/cmm/m/it/10.95 p\OT1/cmr/m/n /10.95 ])[] \OMS/cmsy/m/n/10.95 ! \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\O ML/cmm/m/it/10.95 K[]=K; E\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 p\OT1/cmr/m/n/10.95 ])\OML/cmm/m/it/10.95 : [] Overfull \hbox (4.4099pt too wide) detected at line 175 \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 L=K; E\OT1/cmr/m /n/10.95 (\OML/cmm/m/it/10.95 L\OT1/cmr/m/n/10.95 )[\OML/cmm/m/it/10.95 p\OT1/c mr/m/n/10.95 ]) \OMS/cmsy/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 [\OML/cmm/m/it/10.95 p\OT1/cmr/m/n /10.95 ]) \OMS/cmsy/m/n/10.95 ! \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML /cmm/m/it/10.95 L; E\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 p\OT1/cmr/m/n/10.95 ])[] \OMS/cmsy/m/n/10.95 ! \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML/cmm /m/it/10.95 L=K; E\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 L\OT1/cmr/m/n/10.95 ) [\OML/cmm/m/it/10.95 p\OT1/cmr/m/n/10.95 ])\OML/cmm/m/it/10.95 : [] [5] LaTeX Warning: Citation `gross:kolyvagin' on page 6 undefined on input line 187 . LaTeX Warning: Citation `grigor' on page 6 undefined on input line 189. LaTeX Warning: Citation `cha:kolyvagin' on page 6 undefined on input line 204. [6] LaTeX Warning: Citation `kolyvagin:euler_systems' on page 7 undefined on input line 226. LaTeX Warning: Citation `mazur:rational' on page 7 undefined on input line 234. [7] LaTeX Warning: Citation `serre:propgal' on page 8 undefined on input line 256. Overfull \hbox (0.13979pt too wide) in paragraph at lines 261--262 [][]\OT1/cmr/m/n/10.95 Let $\OML/cmm/m/it/10.95 Z$ \OT1/cmr/m/n/10.95 be the su b-group of scalars. Clearly $\OML/cmm/m/it/10.95 E\OT1/cmr/m/n/10.95 [\OML/cmm/ m/it/10.95 p\OT1/cmr/m/n/10.95 ][] = 0$. Con-sider the Hochshild- [] [8] Overfull \hbox (23.49307pt too wide) in paragraph at lines 299--300 \OT1/cmr/bx/n/10.95 Proposition 4.5. []\OT1/cmr/m/it/10.95 Sup-pose $\OML/cmm/m /it/10.95 E$ \OT1/cmr/m/it/10.95 has no $\OT1/cmr/bx/n/10.95 Q$\OT1/cmr/m/it/10 .95 -rational $\OML/cmm/m/it/10.95 p$\OT1/cmr/m/it/10.95 -isogeny. Then $\OML/c mm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OT1/cmr/bx/n/10.95 Q\OT1/cmr/m/n/10.95 (\ OML/cmm/m/it/10.95 E\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 p\OT1/cmr/m/n/10.95 ])\OML/cmm/m/it/10.95 =\OT1/cmr/bx/n/10.95 Q\OML/cmm/m/it/10.95 ; E\OT1/cmr/m/ n/10.95 [\OML/cmm/m/it/10.95 p\OT1/cmr/m/n/10.95 ]) = [] [9] [10] Overfull \hbox (91.67513pt too wide) in paragraph at lines 360--361 [][]\OT1/cmr/m/n/10.95 The re-sult will now fol-low from the Hochschild-Serre s pec-tral se-quence $\OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10 .95 P[]; H[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 P; E\OT1/cmr/m/n/10.95 (\OM L/cmm/m/it/10.95 L\OT1/cmr/m/n/10.95 )[\OML/cmm/m/it/10.95 p\OT1/cmr/m/n/10.95 ][])) =[]\OMS/cmsy/m/n/10.95 ) [] [11] LaTeX Warning: Citation `gross:kolyvagin' on page 12 undefined on input line 38 0. LaTeX Warning: Citation `MR92e:11050' on page 12 undefined on input line 393. LaTeX Warning: Citation `MR92a:11058' on page 12 undefined on input line 393. 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(/usr/share/texmf/tex/latex/amsfonts/ueuf.fd File: ueuf.fd 2002/01/19 v2.2g AMS font definitions ) LaTeX Font Warning: Font shape `OT1/cmss/m/it' in size <10.95> not available (Font) Font shape `OT1/cmss/m/sl' tried instead on input line 447. [14] LaTeX Warning: Citation `cremona:algs' on page 15 undefined on input line 524. [15] LaTeX Warning: Citation `MR2004g:11045' on page 16 undefined on input line 530. LaTeX Warning: Citation `cremona:algorithms' on page 16 undefined on input line 533. :<+ all3.1415.bbl (./all3.1415.bbl LaTeX Warning: Empty `thebibliography' environment on input line 10. ) :<- [16] [17] (./all3.1415.aux) LaTeX Warning: There were undefined references. ) Here is how much of TeX's memory you used: 5421 strings out of 95755 64903 string characters out of 1194953 149081 words of memory out of 1000001 8331 multiletter control sequences out of 10000+50000 18893 words of font info for 73 fonts, out of 500000 for 1000 14 hyphenation exceptions out of 1000 31i,15n,30p,1230b,398s stack positions out of 1500i,500n,5000p,200000b,5000s Output written on all3.1415.dvi (17 pages, 68136 bytes).