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[7] Overfull \hbox (0.32317pt too wide) in paragraph at lines 439--444 \T1/aer/m/n/10.95 duce the gen-eral case from the iden-tity $\OML/cmm/m/it/10.9 5 H[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G; A \OMS/cmsy/m/n/10.95 ^^H \OML/ cmm/m/it/10.95 B\OT1/cmr/m/n/10.95 ) = \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.9 5 (\OML/cmm/m/it/10.95 G; A\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ^^H \OML/cm m/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G; B\OT1/cmr/m/n/10.95 )$\T1/aer/m/n/10.95 . [] Overfull \hbox (3.1075pt too wide) in paragraph at lines 470--472 []\T1/aer/m/n/10.95 We now ex-plic-itly con-sider $\OML/cmm/m/it/10.95 H[]\OT1/ cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G; A\OT1/cmr/m/n/10.95 )$ \T1/aer/m/n/10.95 as a quo-tient of co-cy-cles by cobound- [] [8] Overfull \hbox (20.82722pt too wide) in paragraph at lines 519--521 []\T1/aer/m/n/10.95 Recall that $[][]\OT1/cmr/m/n/10.95 (\OMS/cmsy/m/n/10.95 ^^ @\OML/cmm/m/it/10.95 ; A\OT1/cmr/m/n/10.95 )$ \T1/aer/m/n/10.95 is a con-travar i-ant func-tor and let $\OML/cmm/m/it/10.95 K[] \OT1/cmr/m/n/10.95 = [][](\OML/ cmm/m/it/10.95 P[]; A\OT1/cmr/m/n/10.95 )$\T1/aer/m/n/10.95 . [] Overfull \hbox (9.7759pt too wide) in paragraph at lines 526--529 []\T1/aer/m/n/10.95 Suppose that $\OML/cmm/m/it/10.95 f \OMS/cmsy/m/n/10.95 2 [ ][]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 P[]; A\OT1/cmr/m/n/10.95 )$ \T1/aer/ m/n/10.95 and de-fine the set-theoretic map [] [] Overfull \hbox (25.48735pt too wide) in paragraph at lines 536--536 [] [] Overfull \hbox (3.99452pt too wide) in paragraph at lines 540--542 []\T1/aer/m/n/10.95 The map $\OML/cmm/m/it/10.95 d [] [] P[] \OMS/cmsy/m/n/10.9 5 ! \OML/cmm/m/it/10.95 P[]$ \T1/aer/m/n/10.95 in-duces a map $[][]\OT1/cmr/m/n /10.95 (\OML/cmm/m/it/10.95 P[]; A\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ! [] []\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 P[]; A\OT1/cmr/m/n/10.95 )$ [] Overfull \hbox (2.05011pt too wide) in paragraph at lines 551--551 [] [] [9] Overfull \hbox (8.82007pt too wide) in paragraph at lines 577--581 \T1/aer/m/n/10.95 A mor-phism of pairs $\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G; A\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ! \OT1/cmr/m/n/10.95 (\OML/cmm/m/ it/10.95 G[]; A[]\OT1/cmr/m/n/10.95 )$ \T1/aer/m/n/10.95 to-gether with ho-mo-m or-phisms $\OML/cmm/m/it/10.95 f [] [] G[] \OMS/cmsy/m/n/10.95 ! [] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] Overfull \hbox (18.61697pt too wide) detected at line 1113 [] [] [20] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 1151. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 1151. [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] Overfull \hbox (23.45978pt too wide) detected at line 1895 []\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G[]; \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/10.95 )[] [] \OMS/cmsy/m/n/10.95 f[] g\OML/cmm/m/it/10.95 : [] [35] Overfull \hbox (18.26195pt too wide) in paragraph at lines 1968--1975 \T1/aer/m/n/10.95 Another stan-dard fact about el-lip-tic curves---the (weak) M ordell-Weil theorem--- [] LaTeX Font Info: Try loading font information for OT2+wncyr on input line 19 79. (/usr/share/texmf-texlive/tex/latex/cyrillic/ot2wncyr.fd File: ot2wncyr.fd 1999/12/17 v3.2a Computer Modern Cyrillic font definitions ) [36] [37] Overfull \hbox (2.84361pt too wide) in paragraph at lines 2075--2079 \T1/aer/m/n/10.95 Recall that if $\OML/cmm/m/it/10.95 G$ \T1/aer/m/n/10.95 is a fi-nite \T1/aer/m/it/10.95 cyclic \T1/aer/m/n/10.95 group and $\OML/cmm/m/it/1 0.95 A$ \T1/aer/m/n/10.95 is a $\OML/cmm/m/it/10.95 G$\T1/aer/m/n/10.95 -module , then $[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G; A\OT1/cmr/m/n/10.95 ) \OMS /cmsy/m/n/10.95 ^^Y [] Overfull \hbox (24.5075pt too wide) in paragraph at lines 2103--2107 []\T1/aer/m/n/10.95 Since, as was men-tioned above, $[]\OT1/cmr/m/n/10.95 (\OML /cmm/m/it/10.95 G; A\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ^^Y []\OT1/cmr/m/n /10.95 (\OML/cmm/m/it/10.95 G; A\OT1/cmr/m/n/10.95 )$ \T1/aer/m/n/10.95 and $[] \OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G; A\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/ 10.95 ^^Y [] [38] [39] [40] LaTeX Font Info: Try loading font information for T1+aett on input line 2263 . (/usr/share/texmf-texlive/tex/latex/ae/t1aett.fd File: t1aett.fd 1997/11/16 Font definitions for T1/aett. ) [41] Overfull \hbox (20.0981pt too wide) in paragraph at lines 2298--2299 []\T1/aer/bx/n/10.95 Theorem 20.1 \T1/aer/m/n/10.95 (Lang--Tate, 1958)\T1/aer/b x/n/10.95 . []\T1/aer/m/it/10.95 There is a nat-u-ral iso-mor-phism $[]\OT1/cmr /m/n/10.95 (\OML/cmm/m/it/10.95 A=k\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ! [] [42] [43] Overfull \hbox (51.1268pt too wide) detected at line 2401 [][]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 A\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/ 10.95 n\OT1/cmr/m/n/10.95 ]) = [][]([][](\OML/cmm/m/it/10.95 A\OT1/cmr/m/n/10.9 5 [\OML/cmm/m/it/10.95 n\OT1/cmr/m/n/10.95 ])) = [][]([\OML/cmm/m/it/10.95 n\OT 1/cmr/m/n/10.95 ]\OML/cmm/m/it/10.95 A\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 n \OT1/cmr/m/n/10.95 ]) = [][](0) = 0\OML/cmm/m/it/10.95 : [] [44] [45] Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 2560. Package hyperref Warning: Token not allowed in a PDFDocEncoded string, (hyperref) removing `math shift' on input line 2560. [46] [47] [48] [49] [50] [51] [52] [53] File: etale.png Graphic file (type png) [54] [55 <./etale.png (PNG copy)>] [56] LaTeX Font Warning: Command \scriptsize invalid in math mode on input line 3153 . [57] [58] [59] [60] [61] Overfull \hbox (17.03876pt too wide) in paragraph at lines 3461--3463 []\T1/aer/bx/n/10.95 Definition 25.3 \T1/aer/m/n/10.95 (Étale Co-ho-mol-ogy)\T1 /aer/bx/n/10.95 . []\T1/aer/m/n/10.95 The Étale co-ho-mol-ogy groups $[][]\OT1/ cmr/m/n/10.95 (\OML/cmm/m/it/10.95 X[]; F\OT1/cmr/m/n/10.95 )$ [] [62] [63] [64] [65] [66] Overfull \hbox (54.50468pt too wide) detected at line 3744 [][]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 j[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/i t/10.95 M[]\OT1/cmr/m/n/10.95 )\OML/cmm/m/it/10.95 ; I\OT1/cmr/m/n/10.95 ) = [] ((\OML/cmm/m/it/10.95 M[]; \OT1/cmr/m/n/10.95 0\OML/cmm/m/it/10.95 ; \OT1/cmr/m /n/10.95 0)\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 M; N; '\OT1/cmr/m/n/10.95 )) [] [][](\OML/cmm/m/it/10.95 M[]; M\OT1/cmr/m/n/10.95 ) [ ] [][](\OML/cmm/m/it/10.95 M[]; j[]I\OT1/cmr/m/n/10.95 )\OML/cmm/m/it/10.95 : [] [67] [68] [69] [70] [71] [72] [73] (./galois_cohomology.bbl [74]) [75] (./galois_cohomology.aux) ) Here is how much of TeX's memory you used: 8142 strings out of 94834 99287 string characters out of 1179181 216846 words of memory out of 1500000 10758 multiletter control sequences out of 10000+50000 47447 words of font info for 119 fonts, out of 1200000 for 2000 212 hyphenation exceptions out of 8191 29i,19n,42p,245b,842s stack positions out of 5000i,500n,6000p,200000b,5000s Output written on galois_cohomology.pdf (75 pages, 909310 bytes). PDF statistics: 1122 PDF objects out of 1200 (max. 8388607) 354 named destinations out of 1000 (max. 131072) 542 words of extra memory for PDF output out of 10000 (max. 10000000)