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(pdftex.def) Requested size: 285.12267pt x 83.52266pt. [39 <./graphics/uhp.pdf>] [40] Overfull \hbox (2.76831pt too wide) in paragraph at lines 2441--2446 []\OT1/cmr/m/it/10 The non-cuspidal points on $\OML/cmm/m/it/10 X\OT1/cmr/m/n/1 0 (\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 cor-re-spond to iso-mo r-phism classes of triples [] [41] Overfull \hbox (2.65828pt too wide) in paragraph at lines 2480--2483 []\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.2.7\OT1/cmr/m/it/10 . []\OT1/cmr/m/ n/10 The ana-logue of Lemma []5.2.6[] is also true for $[][](\OT1/cmr/bx/n/10 Z \OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ! [][]\OT1/cmr/m/n/10 (\OT1/cmr/bx/n/10 Z\OM L/cmm/m/it/10 =N\OT1/cmr/bx/n/10 Z\OT1/cmr/m/n/10 )$ [] [42] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 2567. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 2567. [43] [44] [45] [46 ] 6. [47] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 2797. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\Gamma' on input line 2797. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 2797. [48] [49] Overfull \hbox (7.00468pt too wide) in paragraph at lines 2920--2927 []\OT1/cmr/m/n/10 How is the space $\OML/cmm/m/it/10 S[]\OT1/cmr/m/n/10 (\OT1/c mr/bx/n/10 C\OT1/cmr/m/n/10 )$ of cusp forms re-lated to the co-ho-mol-ogy grou p $\OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X[]; \OT1/cmr/bx/n/10 R\OT1/cmr/m/n/10 )$? [] [50] [51] [52 ] 7. [53] [54] Overfull \hbox (0.57304pt too wide) in paragraph at lines 3123--3129 []\OT1/cmr/m/n/10 By Propo-si-tion []7.1.4[], ev-ery el-e-ment $\OML/cmm/m/it/1 0 h$ \OT1/cmr/m/n/10 of $[][](\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 (\OML/cmm/m/it/ 10 G\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; \OT1/cmr/bx/n/10 Z\OT1/cmr/m/n/10 )$ is of the form $\OMS/cmsy/m/n/10 f\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 ; g\OT1/cmr/m /n/10 (0)\OMS/cmsy/m/n/10 g$ [] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 3167. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 3167. [55] File: graphics/manin_symbol_proof.pdf Graphic file (type pdf) Package pdftex.def Info: graphics/manin_symbol_proof.pdf used on input line 320 7. (pdftex.def) Requested size: 118.22844pt x 236.7177pt. [56 <./graphics/manin_symbol_proof.pdf>] Overfull \hbox (21.7909pt too wide) detected at line 3306 \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 ) \OMS/c msy/m/n/10 ^^@ \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 rs\OT1/cmr /m/n/10 ) = \OML/cmm/m/it/10 ^^Y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 g\OT1/cmr/m/n /10 (\OMS/cmsy/m/n/10 1\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; g\OT1/cmr/m/n/10 (\O ML/cmm/m/it/10 i\OT1/cmr/m/n/10 )) + \OML/cmm/m/it/10 ^^Y\OT1/cmr/m/n/10 (\OML/ cmm/m/it/10 gs\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 i\OT1/cmr/m/n/10 )\OML/cmm/m/it /10 ; gs\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 1\OT1/cmr/m/n/10 )) = \OML/cmm/m/it/1 0 ^^Y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 1\OT1 /cmr/m/n/10 )\OML/cmm/m/it/10 ; g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 i\OT1/cmr/m/ n/10 )) + \OML/cmm/m/it/10 ^^Y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 g\OT1/cmr/m/n/1 0 (1 + \OML/cmm/m/it/10 i\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; g\OT1/cmr/m/n/10 ( \OMS/cmsy/m/n/10 1\OT1/cmr/m/n/10 ))\OML/cmm/m/it/10 : [] [57] Overfull \hbox (5.96455pt too wide) detected at line 3361 \OML/cmm/m/it/10 '\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 ) = \OML/c mm/m/it/10 e[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 rs\OT1/cmr/m/n/10 ) \OMS/cmsy/m /n/10 ^^@ \OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r\OT1/cmr/m/n/1 0 ) = \OML/cmm/m/it/10 ^^Y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 rs\OT1/cmr/m/n/10 ( \OMS/cmsy/m/n/10 1\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; rs\OT1/cmr/m/n/10 (\OML/c mm/m/it/10 i\OT1/cmr/m/n/10 )) \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 ^^Y\OT1/cm r/m/n/10 (\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 1\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; r\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 i\OT1/cmr/m/n/10 )) = \O ML/cmm/m/it/10 ^^Y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 (0)\OML/cm m/m/it/10 ; r\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 i\OT1/cmr/m/n/10 )) \OMS/cmsy/m/ n/10 ^^@ \OML/cmm/m/it/10 ^^Y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 r\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 1\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; r\OT1/cmr/m/n/10 (\OML/ cmm/m/it/10 i\OT1/cmr/m/n/10 )) [] [58] Overfull \hbox (6.7021pt too wide) in paragraph at lines 3390--3396 \OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 []; ^^L[]\OMS/cmsy/m/n/10 g \OT1/cmr/m/n/10 = \OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 ; ^^L\OMS/cmsy/m/n/10 g 2 [][]\OT1/cmr/m /n/10 (\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 ) \OML/cmm/m/it/10 ; \OT1/cmr/bx/n/10 R\OT1/cmr/m/n/10 )$ does not im-ply that $\ OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 []\OT1/cm r/m/n/10 )\OML/cmm/m/it/10 ; g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^L[]\OT1/cmr/m /n/10 )\OMS/cmsy/m/n/10 g \OT1/cmr/m/n/10 = \OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 \OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; g\OT1/c mr/m/n/10 (\OML/cmm/m/it/10 ^^L\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 g$\OT1/cmr/m/n /10 . [] [59] Overfull \hbox (19.76059pt too wide) in paragraph at lines 3431--3433 []\OT1/cmr/bx/n/10 Definition 7.3.2 \OT1/cmr/m/n/10 (Hecke op-er-a-tors)\OT1/cm r/bx/n/10 . []\OT1/cmr/m/n/10 We de-fine the \OT1/cmr/m/it/10 Hecke op-er-a-tor $\OML/cmm/m/it/10 T[]$ \OT1/cmr/m/n/10 on $[][](\OML/cmm/m/it/10 X[]\OT1/cmr/m /n/10 (\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; \OT1/cmr/bx/n/10 R \OT1/cmr/m/n/10 )$ [] Overfull \hbox (0.59491pt too wide) in paragraph at lines 3452--3452 [] \OT1/cmtt/m/n/10 M!<1,[Cusps()|0,Infinity()]> + M!<1,[Cusps()|1/2,Infini ty()]>);[] [] [60] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 3520. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 3520. [61] LaTeX Font Info: Try loading font information for OT2+wncyr on input line 35 53. (/usr/local/texlive/2011/texmf-dist/tex/latex/cyrillic/ot2wncyr.fd File: ot2wncyr.fd 1999/12/17 v3.2a Computer Modern Cyrillic font definitions ) [62] [63] [64 ] 8. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 3633. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\Gamma' on input line 3633. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 3633. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 3633. [65] Underfull \hbox (badness 10000) in paragraph at lines 3750--3751 [] [66] [67] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 3844. 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Overfull \hbox (0.67111pt too wide) in paragraph at lines 3991--3997 []\OT1/cmr/m/n/10 If $\OML/cmm/m/it/10 "$ \OT1/cmr/m/n/10 is a Dirich-let char- ac-ter mod-ulo $\OML/cmm/m/it/10 MN$ \OT1/cmr/m/n/10 such that $\OML/cmm/m/it/1 0 "\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1) = (\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1)[]$, let $\OML/cmm/m/it/10 S[]\OT1/cmr/m/n/10 (\OML/cmm/m/ it/10 M; N; "\OT1/cmr/m/n/10 )$ [] [71] [72] Overfull \hbox (10.85173pt too wide) in paragraph at lines 4128--4131 []\OT1/cmr/m/n/10 Since $\OML/cmm/m/it/10 T[]$ \OT1/cmr/m/n/10 com-mutes with $ \OML/cmm/m/it/10 B[]$ \OT1/cmr/m/n/10 and $\OML/cmm/m/it/10 C[]$\OT1/cmr/m/n/10 , the Hecke op-er-a-tors $\OML/cmm/m/it/10 T[]$ \OT1/cmr/m/n/10 pre-serve $\OM L/cmm/m/it/10 S[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 M; N; "\OT1/cmr/m/n/10 )[]$, [] [73] [74] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4251. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 4251. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4251. [75] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4372. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 4372. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4372. [76] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4402. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 4402. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4402. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4412. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4412. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4412. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 4412. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4412. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4414. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4414. [77] [78] [79] [80] Overfull \hbox (11.26907pt too wide) in paragraph at lines 4694--4702 \OT1/cmr/m/n/10 tion []9.3.2[], the space $\OML/cmm/m/it/10 S[]\OT1/cmr/m/n/10 (^^@[](\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 ))$ is free of rank $1$ over $\OT1/cmr /bx/n/10 T[]$\OT1/cmr/m/n/10 , so $[] \OT1/cmr/bx/n/10 T[] \OT1/cmr/m/n/10 = [] \OML/cmm/m/it/10 S[]\OT1/cmr/m/n/10 (^^@[](\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 ) )$. [] [81] [82 ] 10. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4727. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 4727. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\Gamma' on input line 4727. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4727. Underfull \vbox (badness 10000) has occurred while \output is active [] [83] [84] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4827. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4827. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4859. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 4859. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4859. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 4859. 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Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 5096. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 5096. LaTeX Font Info: Try loading font information for OMS+cmss on input line 514 0. LaTeX Font Info: No file OMScmss.fd. on input line 5140. LaTeX Font Warning: Font shape `OMS/cmss/m/n' undefined (Font) using `OMS/cmsy/m/n' instead (Font) for symbol `textsection' on input line 5140. [91] Overfull \hbox (8.05022pt too wide) in paragraph at lines 5156--5164 []\OT1/cmr/m/n/10 Let $\OML/cmm/m/it/10 E$ \OT1/cmr/m/n/10 be an el-lip-tic cur ve over $\OML/cmm/m/it/10 K$ \OT1/cmr/m/n/10 with $\OML/cmm/m/it/10 j$\OT1/cmr/ m/n/10 -invariant $\OML/cmm/m/it/10 j$\OT1/cmr/m/n/10 . We will show that $[](\ OMS/cmsy/m/n/10 F[]\OML/cmm/m/it/10 =\OMS/cmsy/m/n/10 F[]\OT1/cmr/m/n/10 )$ [] [92] [93] [94] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 5350. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\mathop' on input line 5350. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\mathcode' on input line 5350. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\mathcode' on input line 5350. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\mathcode' on input line 5350. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\mathchardef' on input line 5350. 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[100] [101] [102] [103] File: graphics/deligne-rapaport.pdf Graphic file (type pdf) Package pdftex.def Info: graphics/deligne-rapaport.pdf used on input line 5904. (pdftex.def) Requested size: 320.7596pt x 49.19412pt. [104 <./graphics/deligne-rapaport.pdf>] [ 105] [106] [107] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 6207. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 6207. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 6207. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 6207. [108] [109] [110 ] 13. [111] Underfull \vbox (badness 10000) has occurred while \output is active [] [112] Underfull \hbox (badness 10000) in paragraph at lines 6351--6352 [] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] Overfull \hbox (21.59473pt too wide) in paragraph at lines 7197--7197 [] \OT1/cmtt/m/n/10 (0.59153223605591049412844857432 - 1.6874592734680125399 3135357636*i[] [] Overfull \hbox (100.34404pt too wide) in paragraph at lines 7197--7197 [] \OT1/cmtt/m/n/10 0.762806324458047168681080323846571478727 - 0.603687 64497868211035115379488*i),[] [] Overfull \hbox (26.84468pt too wide) in paragraph at lines 7197--7197 [] \OT1/cmtt/m/n/10 (-0.59153223605591049412844857432 - 1.687459273468012539 93135357636*i[] [] Overfull \hbox (105.594pt too wide) in paragraph at lines 7197--7197 [] \OT1/cmtt/m/n/10 -0.762806324458047168681080323846571478727 - 0.60368 764497868211035115379488*i),[] [] Overfull \hbox (89.84413pt too wide) in paragraph at lines 7197--7197 [] \OT1/cmtt/m/n/10 (-1.354338560513957662809528899804 - 1.08377162848933042 95801997808568748714097*i[] [] Overfull \hbox (58.3444pt too wide) in paragraph at lines 7197--7197 [] \OT1/cmtt/m/n/10 -0.59153223605591049412844857401 + 0.480083983510648 319229045987467*i),[] [] Overfull \hbox (16.34477pt too wide) in paragraph at lines 7197--7197 [] \OT1/cmtt/m/n/10 (-1.52561264891609433736216065099 0.34254817680427334910 5263499648)[] [] [125] [126] [127] [128] Overfull \hbox (6.81845pt too wide) in paragraph at lines 7455--7463 \OT1/cmtt/m/n/10 http://modular.fas.harvard.edu/scans/antwerp/\OT1/cmr/m/n/10 , and look at Sil-ver-man, chap- [] [129] [130] [131] [132] 14. [133 ] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 7810. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 7810. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 7810. LaTeX Warning: Reference `' on page 134 undefined on input line 7836. [134] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 7867. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 7867. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 7867. [135] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 7984. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 7984. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 7984. [136] [137] Overfull \hbox (3.53947pt too wide) in paragraph at lines 8082--8087 \OT1/cmr/m/n/10 Using the Eichler-Shimura con-gru-ence re-la-tion (see [][][] ) we will show that $[](\OML/cmm/m/it/10 ^^Z[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 '\OT1/cmr/m/n/10 )) = [] [138] [139] [140] LaTeX Warning: Reference `sec:int_con' on page 141 undefined on input line 8287 . [141] [142] 15. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 8366. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 8366. Overfull \hbox (5.23723pt too wide) in paragraph at lines 8401--8406 []\OT1/cmr/m/n/10 As dis-cussed in [[], \OMS/cmsy/m/n/10 x\OT1/cmr/m/n/10 2], $ \OML/cmm/m/it/10 K$ \OT1/cmr/m/n/10 acts faith-fully on the tan-gent space $[][ ](\OML/cmm/m/it/10 A=\OT1/cmr/bx/n/10 Q\OT1/cmr/m/n/10 )$ [] [143 ] [144] [145] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 8569. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\@@overline' on input line 8569. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 8569. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 8623. 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Underfull \vbox (badness 1701) has occurred while \output is active [] [149] Overfull \hbox (0.56735pt too wide) in paragraph at lines 8741--8744 []\OT1/cmr/m/n/10 Since $\OML/cmm/m/it/10 N[]\OT1/cmr/m/n/10 (2\OML/cmm/m/it/10 ^^Y\OT1/cmr/m/n/10 )[]^^@(\OML/cmm/m/it/10 s\OT1/cmr/m/n/10 )$ is ev-ery-where nonzero, The-o-rem []16.1.4[] im-plies that $\OML/cmm/m/it/10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f; s\OT1/cmr/m/n/10 )$ [] Overfull \hbox (3.19969pt too wide) in paragraph at lines 8762--8763 []\OT1/cmr/m/n/10 We fol-low [[], \OMS/cmsy/m/n/10 x\OT1/cmr/m/n/10 VIII.5] clo sely. [] [150] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 8817. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 8817. [151] Overfull \hbox (1.52274pt too wide) in paragraph at lines 8848--8851 []\OT1/cmr/bx/n/10 Theorem 16.1.6. []\OT1/cmr/m/it/10 Let $\OML/cmm/m/it/10 f \ OT1/cmr/m/n/10 = [] \OML/cmm/m/it/10 a[]q[]$ \OT1/cmr/m/it/10 be a new-form in $\OML/cmm/m/it/10 S[]\OT1/cmr/m/n/10 (^^@[](\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 ) \OML/cmm/m/it/10 ; "\OT1/cmr/m/n/10 )$\OT1/cmr/m/it/10 , and let $\OML/cmm/m/it /10 L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f; s\OT1/cmr/m/n/10 ) = [] [152] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 8905. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 8905. LaTeX Warning: Reference `fig:visLabs' on page 153 undefined on input line 8909 . [153] [154 ] 17. [155] [156] [157] [158] LaTeX Warning: Citation `agashe-stein:manin' on page 159 undefined on input lin e 10448. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 10455. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\Omega' on input line 10455. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 10455. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 10455. [159] Overfull \hbox (2.48738pt too wide) in paragraph at lines 10514--10519 \OT1/cmr/m/n/10 Recall the Manin-Drinfeld the-o-rem, which we proved long ago, as-serts that $\OMS/cmsy/m/n/10 f\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 ; \OMS/cmsy/ m/n/10 1g 2 [] [160] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 10557. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 10557. 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Package hyperref Warning: Composite letter `\textacute+3' (hyperref) not defined in PD1 encoding, (hyperref) removing `\textacute' on input line 10898. Package hyperref Warning: Composite letter `\textdotaccent+"' (hyperref) not defined in PD1 encoding, (hyperref) removing `\textdotaccent' on input line 10898. [165] [166] [167] [168] Overfull \hbox (31.39096pt too wide) in paragraph at lines 11251--11311 []$ [] Overfull \hbox (30.76596pt too wide) in paragraph at lines 11319--11378 []$ [] [169] [170] [171] [172] [173] [174] [175] [176] [177] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 11732. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 11732. Overfull \hbox (47.8445pt too wide) in paragraph at lines 11768--11768 []\OT1/cmtt/m/n/10 Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 ov er Rational Field[] [] Overfull \hbox (89.84413pt too wide) in paragraph at lines 11768--11768 []\OT1/cmtt/m/n/10 > P := [* *]; for D in A do time p,q := HeegnerPoint(E,D); A ppend(~P,); end for;[] [] Overfull \hbox (89.84413pt too wide) in paragraph at lines 11768--11768 []\OT1/cmtt/m/n/10 [* <(0.500000000000000000023668707008012648357 + 1.322875655 532295292727422603610*i :[] [] Overfull \hbox (100.34404pt too wide) in paragraph at lines 11768--11768 [] \OT1/cmtt/m/n/10 -2.0000000000000000043473199962953622702 + 5.29150262212 9181186268103643387*i : 1),[] [] Overfull \hbox (26.84468pt too wide) in paragraph at lines 11768--11768 []\OT1/cmtt/m/n/10 <(-2.99999999999999999717372417577 + 1.414213562373095051128 215430668*i :[] [] Overfull \hbox (53.09445pt too wide) in paragraph at lines 11768--11768 [] \OT1/cmtt/m/n/10 2.99999999999999999372415320819 + 4.24264068711928515852 9184817212*i : 1),[] [] Overfull \hbox (21.59473pt too wide) in paragraph at lines 11768--11768 []\OT1/cmtt/m/n/10 <(-6.99999999999999999753082167926 + 8.717797887081347098291 81547345*i :[] [] Overfull \hbox (84.59418pt too wide) in paragraph at lines 11768--11768 [] \OT1/cmtt/m/n/10 37.000000000000000044867139452136100922 + 8.7177978870813 4710182445372185*i : 1),[] [] Overfull \hbox (74.09427pt too wide) in paragraph at lines 11768--11768 []\OT1/cmtt/m/n/10 <(0.500000000000000000023668707008012648357 + 1.322875655532 295292727422603610*i :[] [] Overfull \hbox (95.09409pt too wide) in paragraph at lines 11768--11768 [] \OT1/cmtt/m/n/10 -2.0000000000000000043473199962953622702 + 5.291502622129 181186268103643387*i : 1),[] [] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 11770. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 11770. Underfull \vbox (badness 1769) has occurred while \output is active [] [178] [179] [180 ] 18. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 11784. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\ell' on input line 11784. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 11784. [181] [182] Overfull \hbox (8.70174pt too wide) in paragraph at lines 11965--11976 \OT1/cmr/m/n/10 idem-po-tents $\OML/cmm/m/it/10 e[]$\OT1/cmr/m/n/10 , in this d e-com-po-si-tion, de-com-pose $[][]$ as a prod-uct $[][] [][](\OML/cmm/m/it/10 A[]\OT1/cmr/m/n/10 )$. [] [183] LaTeX Warning: Command \ss invalid in math mode on input line 12047. LaTeX Warning: Command \ss invalid in math mode on input line 12047. LaTeX Warning: Command \ss invalid in math mode on input line 12052. [184] LaTeX Warning: Command \ss invalid in math mode on input line 12071. LaTeX Warning: Command \ss invalid in math mode on input line 12071. LaTeX Warning: Command \ss invalid in math mode on input line 12078. LaTeX Warning: Command \ss invalid in math mode on input line 12078. LaTeX Warning: Command \ss invalid in math mode on input line 12107. [185] Overfull \hbox (26.69293pt too wide) in paragraph at lines 12157--12172 []\OT1/cmr/m/n/10 (That $\OML/cmm/m/it/10 J\OT1/cmr/m/n/10 [\U/euf/m/n/10 m\OT1 /cmr/m/n/10 ] \OMS/cmsy/m/n/10 6\OT1/cmr/m/n/10 = 0$.) The point is that the co n-travari-ant Tate mod-ule $[](\OML/cmm/m/it/10 J[]; \OT1/cmr/bx/n/10 Q[]\OML/c mm/m/it/10 =\OT1/cmr/bx/n/10 Z[]\OT1/cmr/m/n/10 )$ [] Overfull \hbox (2.53041pt too wide) in paragraph at lines 12157--12172 [][] \OT1/cmr/bx/n/10 T[]$\OT1/cmr/m/n/10 , $[][] \OML/cmm/m/it/10 J [] [][] [] [] J$ \OT1/cmr/m/n/10 so we can de-fine $[][] \OML/cmm/m/it/10 J \OT1/cmr/m/n/1 0 := [][]([][] \OML/cmm/m/it/10 J; \OT1/cmr/bx/n/10 Z[]\OT1/cmr/m/n/10 )$. [] [186] LaTeX Warning: Command \ss invalid in math mode on input line 12214. [187] Overfull \hbox (22.22363pt too wide) in paragraph at lines 12263--12275 \OT1/cmr/m/n/10 over $\OT1/cmr/bx/n/10 T\OML/cmm/m/it/10 =\U/euf/m/n/10 m$\OT1/ cmr/m/n/10 . The sec-ond is to prove that $\OT1/cmr/bx/n/10 T[]$ \OT1/cmr/m/n/1 0 is Goren-stein, i.e., that $\OT1/cmr/bx/n/10 T[] [] [][]\OT1/cmr/m/n/10 (\OT1 /cmr/bx/n/10 T[]\OML/cmm/m/it/10 ; \OT1/cmr/bx/n/10 Z[]\OT1/cmr/m/n/10 )$. [] Overfull \hbox (24.1505pt too wide) in paragraph at lines 12291--12297 \OT1/cmr/m/n/10 Thus the dual of $[][] \OML/cmm/m/it/10 =\U/euf/m/n/10 m [][]$ \OT1/cmr/m/n/10 is two di-men-sional over $\OT1/cmr/bx/n/10 T\OML/cmm/m/it/10 = \U/euf/m/n/10 m$ \OT1/cmr/m/n/10 and hence $[][] \OML/cmm/m/it/10 =\U/euf/m/n/1 0 m [][]$ [] [188] [189] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 12417. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 12417. [190] [191] [192] Overfull \hbox (60.85648pt too wide) in paragraph at lines 12603--12613 \OML/cmm/m/it/10 D\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 W\OT1/cmr/m/n/10 )\OML/cmm/ m/it/10 = [] D\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 W\OT1/cmr/m/n/10 )$. By the pre -vi-ous lemma $[] \OML/cmm/m/it/10 D\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 W\OT1/cmr /m/n/10 )\OML/cmm/m/it/10 = [] D\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 W\OT1/cmr/m/n /10 ) = 1$. Thus $\OML/cmm/m/it/10 D\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 W\OT1/cmr /m/n/10 )\OML/cmm/m/it/10 = [] D\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 W\OT1/cmr/m/n /10 ) \OMS/cmsy/m/n/10 ! [] [193] [194] [195] [196 ] 19. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 12663. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\rho' on input line 12663. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 12663. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\lambda' on input line 12663. 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Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 15358. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 15358. [248] [249] Overfull \hbox (5.33716pt too wide) in paragraph at lines 15495--15497 []\OT1/cmr/m/n/10 Gorenstein as a $\OT1/cmr/bx/n/10 Z[]$\OT1/cmr/m/n/10 -module , i.e., there ex-ists an iso-mor-phism $[][](\OT1/cmr/bx/n/10 T[]\OML/cmm/m/it/ 10 ; \OT1/cmr/bx/n/10 Z[]\OT1/cmr/m/n/10 ) [] [] [250] [251] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 15585. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 15585. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `\Sigma' on input line 15585. 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Underfull \vbox (badness 10000) has occurred while \output is active [] [259] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 15892. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 15892. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 15892. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 15900. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 15900. File: graphics/389/cong.pdf Graphic file (type pdf) Package pdftex.def Info: graphics/389/cong.pdf used on input line 15949. (pdftex.def) Requested size: 270.00807pt x 178.66705pt. [260] [261 <./graphics/389/cong.pdf>] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 16084. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 16084. Overfull \hbox (13.80864pt too wide) in paragraph at lines 16104--16106 []\OT1/cmr/bx/n/10 Proposition 26.2.1. []\OT1/cmr/m/it/10 The dis-crim-i-nant o f the Hecke al-ge-bra as-so-ci-ated to $\OML/cmm/m/it/10 S[]\OT1/cmr/m/n/10 (^^ @[](389))$ [] LaTeX Warning: Citation `agashe-stein:schoof-appendix' on page 262 undefined on input line 16112. [262] Overfull \hbox (15.94171pt too wide) in alignment at lines 16163--16173 [][][] [] [] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 16218. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 16218. 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Overfull \hbox (6.27884pt too wide) in paragraph at lines 16328--16332 \OT1/cmr/m/n/10 For any pos-i-tive in-te-ger $\OML/cmm/m/it/10 N$\OT1/cmr/m/n/1 0 , let $\OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 N\OT1/cmr/m/n/10 ) = \OML/cmm/m/it/10 H[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ( \OML/cmm/m/it/10 N\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; \OT1/cmr/bx/n/10 Z\OT1/cm r/m/n/10 )[]$ be the $+1$ eigen-submodule [] (./main.bbl [265] [266 ] [267] [268] Underfull \hbox (badness 10000) in paragraph at lines 163--165 []\OT1/cmr/m/n/10 J. El-len-berg, $\OT1/cmr/bx/n/10 q$\OT1/cmr/m/it/10 -curves and Ga-lois Rep-re-sen-ta-tions\OT1/cmr/m/n/10 , [] Underfull \hbox (badness 10000) in paragraph at lines 163--165 \OT1/cmtt/m/n/10 http://www.math.princeton.edu/ ellenber/papers.html#MCAV [] Underfull \hbox (badness 2173) in paragraph at lines 167--169 []\OT1/cmr/m/n/10 J. El-len-berg and C. Skin-ner, \OT1/cmr/m/it/10 On the Mod-u -lar-ity of $\OT1/cmr/bx/n/10 Q$\OT1/cmr/m/it/10 -curves\OT1/cmr/m/n/10 , [] Underfull \hbox (badness 10000) in paragraph at lines 167--169 \OT1/cmtt/m/n/10 http://www.math.princeton.edu/ ellenber/papers.html#QCURVE [] [269] Underfull \hbox (badness 2469) in paragraph at lines 225--227 []\OT1/cmr/m/n/10 H. Hida, \OT1/cmr/m/it/10 Mod-u-lar-ity Prob-lems of $\OT1/cm r/bx/n/10 Q$\OT1/cmr/m/it/10 -Motives and Base-Change\OT1/cmr/m/n/10 , [] [270] Underfull \hbox (badness 1127) in paragraph at lines 323--326 \OT1/cmr/m/n/10 con-jec-ture for odd 2-dimensional rep-re-sen-ta-tions, Springe r, 1994, [] [271] Underfull \hbox (badness 10000) in paragraph at lines 364--367 []\OT1/cmr/m/n/10 E. Pyle, \OT1/cmr/m/it/10 Abelian va-ri-eties over $\OT1/cmr/ bx/n/10 Q$ \OT1/cmr/m/it/10 with large en-do-mor- [] Underfull \hbox (badness 10000) in paragraph at lines 364--367 \OT1/cmr/m/it/10 phism al-ge-bras and their sim-ple com-po-nents over $[]$\OT1/ cmr/m/n/10 , [] [272] [273]) Package atveryend Info: Empty hook `BeforeClearDocument' on input line 16349. [274] Package atveryend Info: Empty hook `AfterLastShipout' on input line 16349. (./main.aux) Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 16349. Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 16349 . Package rerunfilecheck Info: File `main.out' has not changed. (rerunfilecheck) Checksum: DC1889DDAF0D36DDA02886B68A302113;20708. LaTeX Font Warning: Some font shapes were not available, defaults substituted. LaTeX Warning: There were undefined references. LaTeX Warning: There were multiply-defined labels. 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