This is TeX, Version 3.14159 (Web2C 7.4.5) (format=latex 2003.9.11) 3 DEC 2003 16:25 **\nonstopmode\input katz.tex (./katz.tex (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2001/04/21 v1.4e Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size11.clo File: size11.clo 2001/04/21 v1.4e Standard LaTeX file (size option) ) \c@part=\count79 \c@section=\count80 \c@subsection=\count81 \c@subsubsection=\count82 \c@paragraph=\count83 \c@subparagraph=\count84 \c@figure=\count85 \c@table=\count86 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 ) (/usr/share/texmf/tex/latex/pstricks/pstricks.sty Package: pstricks 1997/03/25 package wrapper for PSTricks pstricks.tex (/usr/share/texmf/tex/generic/pstricks/pstricks.tex `PSTricks' v97 patch 14 <1999/12/23> (tvz) \pst@dima=\dimen103 \pst@dimb=\dimen104 \pst@dimc=\dimen105 \pst@dimd=\dimen106 \pst@dimg=\dimen107 \pst@dimh=\dimen108 \pst@hbox=\box26 \pst@boxg=\box27 \pst@cnta=\count87 \pst@cntb=\count88 \pst@cntc=\count89 \pst@cntd=\count90 \pst@cntg=\count91 \pst@cnth=\count92 \pst@toks=\toks14 (/usr/share/texmf/tex/generic/pstricks/pstricks.con) \psunit=\dimen109 \psxunit=\dimen110 \psyunit=\dimen111 \pslinewidth=\dimen112 \pst@customdefs=\toks15 \pslinearc=\dimen113 \everypsbox=\toks16 \psframesep=\dimen114 \pslabelsep=\dimen115 \theoverlaybox=\box28 )) (/usr/share/texmf/tex/latex/amsmath/amsmath.sty Package: amsmath 2000/07/18 v2.13 AMS math features \@mathmargin=\skip43 For additional information on amsmath, use the `?' option. (/usr/share/texmf/tex/latex/amsmath/amstext.sty Package: amstext 2000/06/29 v2.01 (/usr/share/texmf/tex/latex/amsmath/amsgen.sty File: amsgen.sty 1999/11/30 v2.0 \@emptytoks=\toks17 \ex@=\dimen116 )) (/usr/share/texmf/tex/latex/amsmath/amsbsy.sty Package: amsbsy 1999/11/29 v1.2d \pmbraise@=\dimen117 ) (/usr/share/texmf/tex/latex/amsmath/amsopn.sty Package: amsopn 1999/12/14 v2.01 operator names ) \inf@bad=\count93 LaTeX Info: Redefining \frac on input line 211. \uproot@=\count94 \leftroot@=\count95 LaTeX Info: Redefining \overline on input line 307. \classnum@=\count96 \DOTSCASE@=\count97 LaTeX Info: Redefining \ldots on input line 379. LaTeX Info: Redefining \dots on input line 382. LaTeX Info: Redefining \cdots on input line 467. \Mathstrutbox@=\box29 \strutbox@=\box30 \big@size=\dimen118 LaTeX Font Info: Redeclaring font encoding OML on input line 567. LaTeX Font Info: Redeclaring font encoding OMS on input line 568. \macc@depth=\count98 \c@MaxMatrixCols=\count99 \dotsspace@=\muskip10 \c@parentequation=\count100 \dspbrk@lvl=\count101 \tag@help=\toks18 \row@=\count102 \column@=\count103 \maxfields@=\count104 \andhelp@=\toks19 \eqnshift@=\dimen119 \alignsep@=\dimen120 \tagshift@=\dimen121 \tagwidth@=\dimen122 \totwidth@=\dimen123 \lineht@=\dimen124 \@envbody=\toks20 \multlinegap=\skip44 \multlinetaggap=\skip45 \mathdisplay@stack=\toks21 LaTeX Info: Redefining \[ on input line 2666. LaTeX Info: Redefining \] on input line 2667. ) (./katz.aux) \openout1 = `katz.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 76. LaTeX Font Info: ... okay on input line 76. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 76. LaTeX Font Info: ... okay on input line 76. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 76. LaTeX Font Info: ... okay on input line 76. 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161--162 []$[]$ [] Underfull \hbox (badness 10000) in paragraph at lines 161--162 [] [23] Overfull \hbox (27.58952pt too wide) in paragraph at lines 163--163 [][] [] Overfull \hbox (17.0pt too wide) in paragraph at lines 163--164 []$[]$ [] Underfull \hbox (badness 10000) in paragraph at lines 163--164 [] [24] Overfull \hbox (27.58952pt too wide) in paragraph at lines 165--165 [][] [] Overfull \hbox (17.0pt too wide) in paragraph at lines 165--166 []$[]$ [] Underfull \hbox (badness 10000) in paragraph at lines 165--166 [] [25] Overfull \hbox (27.58952pt too wide) in paragraph at lines 167--167 [][] [] Overfull \hbox (17.0pt too wide) in paragraph at lines 167--168 []$[]$ [] Underfull \hbox (badness 10000) in paragraph at lines 167--168 [] [26] [27] (./katz.m [28] [29] [30] [31] [32] ! Missing $ inserted. $ l.173 ...n. Let p=5. I'd love to compute (mod p^ 4 = 725, I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.175 I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.176 1/p U_ p(j-744) I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.177 I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.178 ...l divided by a power of j. Ditto for 1/p^ 2 U_p^2(j-744). I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.181 I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. Overfull \hbox (131.29549pt too wide) in paragraph at lines 178--181 []\OT1/cmr/m/n/10.95 as a poly-no-mial di-vided by a power of j. Ditto for 1/p$ []\OML/cmm/m/it/10.95 U[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 j \OMS/cmsy/m/ n/10.95 ^^@ \OT1/cmr/m/n/10.95 744)\OML/cmm/m/it/10.95 :Dittofor\OT1/cmr/m/n/10 .95 1\OML/cmm/m/it/10.95 =p[]U[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 j \OMS/ cmsy/m/n/10.95 ^^@ \OT1/cmr/m/n/10.95 744)\OML/cmm/m/it/10.95 :\OT1/cmr/m/n/10. 95 (\OML/cmm/m/it/10.95 Untiltheysettledownmod\OT1/cmr/m/n/10.95 725\OML/cmm/m/ it/10.95 :\OT1/cmr/m/n/10.95 )\OML/cmm/m/it/10.95 Infact; thelimit$ [] ! Missing $ inserted. $ l.182 Lim_ n 1/p^n U_p^n(j-744) I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.183 I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.184 ...e eigenfunction that Gouvea and I call f_ 1 in I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.187 I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. Overfull \hbox (287.54347pt too wide) in paragraph at lines 184--187 []\OT1/cmr/m/n/10.95 exists 5-adically (it is the eigen-func-tion that Gou-vea and I call f$[]\OML/cmm/m/it/10.95 inourpaper\OT1/cmr/m/n/10.95 "\OML/cmm/m/it/ 10.95 Searchingforeigenfunctions\OT1/cmr/m/n/10.95 ")\OML/cmm/m/it/10.95 andhas anexpansionasapolynomialinjplusapowerseriesinj[]:$ [] ! Missing $ inserted. $ l.193 K := IntegerRing(p^ pow); I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.196 I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. [33] ! Misplaced alignment tab character &. l.200 return & +[Coefficient(f,n*p)*q^n : n in [1..d]] + O(q^(d+1)); I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Missing $ inserted. $ l.200 return &+[Coefficient(f,n*p)*q^ n : n in [1..d]] + O(q^(d+1)); I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.202 I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. Overfull \hbox (53.77173pt too wide) in paragraph at lines 197--202 []\OT1/cmr/m/n/10.95 intrinsic Up(f::RngSerLaurElt) -> RngSer-Lau-rElt d := Fl oor((AbsolutePrecision(f)-1)/p); re-turn +[Coefficient(f,n*p)*q$[] : [] ! Missing $ inserted. $ l.205 ...turn Max([i : i in [0..pow] | Integers(p^ i)!x eq 0]); I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.207 I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.214 h := j^ r*g; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Missing $ inserted. $ l.237 I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. Overfull \hbox (31.04933pt too wide) in paragraph at lines 208--237 \OT1/cmr/m/n/10.95 1; while true do print r; h := j$[] \OMS/cmsy/m/n/10.95 ^^C \OML/cmm/m/it/10.95 g\OT1/cmr/m/n/10.95 ; \OML/cmm/m/it/10.95 v \OT1/cmr/m/n/10 .95 := []; \OML/cmm/m/it/10.95 S < X >\OT1/cmr/m/n/10.95 := \OML/cmm/m/it/10.95 PolynomialRing\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 F \OT1/cmr/m/n/10.95 := 0; \OML/cmm/m/it/10.95 print\OT1/cmr /m/n/10.95 "\OML/cmm/m/it/10.95 h \OT1/cmr/m/n/10.95 = "\OML/cmm/m/it/10.95 ; h \OT1/cmr/m/n/10.95 ; \OML/cmm/m/it/10.95 print\OT1/cmr/m/n/10.95 "\OML/cmm/m/it /10.95 Valuation\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 h\OT1/cmr/m/n/10.95 ) = [] Overfull \hbox (28.70647pt too wide) in paragraph at lines 208--237 \OT1/cmr/m/n/10.95 "\OML/cmm/m/it/10.95 ; Valuation\OT1/cmr/m/n/10.95 (\OML/cmm /m/it/10.95 h\OT1/cmr/m/n/10.95 ); \OML/cmm/m/it/10.95 foriin\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 Valuation\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 h\OT1/cm r/m/n/10.95 )\OML/cmm/m/it/10.95 ::\OT1/cmr/m/n/10.95 0]\OML/cmm/m/it/10.95 doA ppend\OT1/cmr/m/n/10.95 ( []\OML/cmm/m/it/10.95 v; Coefficient\OT1/cmr/m/n/10.9 5 (\OML/cmm/m/it/10.95 h; i\OT1/cmr/m/n/10.95 )); \OML/cmm/m/it/10.95 F \OT1/cm r/m/n/10.95 := \OML/cmm/m/it/10.95 F \OT1/cmr/m/n/10.95 + \OML/cmm/m/it/10.95 C oefficient\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 h; i\OT1/cmr/m/n/10.95 ) \OMS /cmsy/m/n/10.95 ^^C \OML/cmm/m/it/10.95 X[] \OMS/cmsy/m/n/10.95 ^^@ \OML/cmm/m/ it/10.95 i\OT1/cmr/m/n/10.95 ); \OML/cmm/m/it/10.95 print\OT1/cmr/m/n/10.95 "\O ML/cmm/m/it/10.95 F \OT1/cmr/m/n/10.95 = [] Overfull \hbox (137.45018pt too wide) in paragraph at lines 208--237 \OT1/cmr/m/n/10.95 "\OML/cmm/m/it/10.95 ; h\OT1/cmr/m/n/10.95 ; \OML/cmm/m/it/1 0.95 z \OT1/cmr/m/n/10.95 := \OML/cmm/m/it/10.95 Min\OT1/cmr/m/n/10.95 ([\OML/c mm/m/it/10.95 Val\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 Coefficient\OT1/cmr/m/ n/10.95 (\OML/cmm/m/it/10.95 h; n\OT1/cmr/m/n/10.95 )) : \OML/cmm/m/it/10.95 ni n\OT1/cmr/m/n/10.95 [1\OML/cmm/m/it/10.95 ::prec\OT1/cmr/m/n/10.95 ]]); \OML/cm m/m/it/10.95 print\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 Val\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 Coefficient\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 h; n\O T1/cmr/m/n/10.95 )) : \OML/cmm/m/it/10.95 nin\OT1/cmr/m/n/10.95 [1\OML/cmm/m/it /10.95 ::prec\OT1/cmr/m/n/10.95 ]]; \OML/cmm/m/it/10.95 ifzgeprecisionthenretur nF; r\OT1/cmr/m/n/10.95 ; \OML/cmm/m/it/10.95 endif\OT1/cmr/m/n/10.95 ; \OML/cm m/m/it/10.95 r \OT1/cmr/m/n/10.95 := [] ! Missing $ inserted. $ l.244 r := p^ n; I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. l.246 f := & +[(Integers()!(Coefficient(f,m)) div r)*q^m : m in I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ) ! Missing $ inserted. $ l.172 I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. Overfull \hbox (142.84215pt too wide) in paragraph at lines 238--172 \OML/cmm/m/it/10.95 AbsolutePrecision\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 f\ OT1/cmr/m/n/10.95 ); \OML/cmm/m/it/10.95 f \OT1/cmr/m/n/10.95 := +[(\OML/cmm/m/ it/10.95 Integers\OT1/cmr/m/n/10.95 ()!(\OML/cmm/m/it/10.95 Coefficient\OT1/cmr /m/n/10.95 (\OML/cmm/m/it/10.95 f; m\OT1/cmr/m/n/10.95 ))\OML/cmm/m/it/10.95 di vr\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ^^C \OML/cmm/m/it/10.95 q[] \OT1/cmr /m/n/10.95 : \OML/cmm/m/it/10.95 min\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 Val uation\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 f\OT1/cmr/m/n/10.95 )\OML/cmm/m/i t/10.95 ::prec \OMS/cmsy/m/n/10.95 ^^@ \OT1/cmr/m/n/10.95 1]] + \OML/cmm/m/it/1 0.95 O\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 q[]rec\OT1/cmr/m/n/10.95 ); \OML/ cmm/m/it/10.95 printf\OT1/cmr/m/n/10.95 "1\OML/cmm/m/it/10.95 =p[]rint\OT1/cmr/ m/n/10.95 "146560\OML/cmm/m/it/10.95 =j \OT1/cmr/m/n/10.95 = [] [34] (./katz.aux) ) Here is how much of TeX's memory you used: 1661 strings out of 95796 18002 string characters out of 1191756 246118 words of memory out of 1000001 4638 multiletter control sequences out of 10000+50000 12182 words of font info for 44 fonts, out of 500000 for 1000 26 hyphenation exceptions out of 1000 27i,6n,20p,11423b,305s stack positions out of 3000i,1500n,5000p,200000b,15000s Output written on katz.dvi (34 pages, 825368 bytes).