This is TeX, Version 3.14159 (Web2C 7.3.1) (format=latex 2000.7.19) 29 JAN 2001 20:36 **lseries-code.tex (lseries-code.tex LaTeX2e <1999/12/01> patch level 1 Babel and hyphenation patterns for american, french, german, ngerman, i talian, nohyphenation, loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 1999/09/10 v1.4a Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size10.clo File: size10.clo 1999/09/10 v1.4a 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 ) (lseries-code.aux) \openout1 = `lseries-code.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 2. LaTeX Font Info: ... okay on input line 2. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 2. LaTeX Font Info: ... okay on input line 2. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 2. LaTeX Font Info: ... okay on input line 2. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 2. LaTeX Font Info: ... okay on input line 2. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 2. LaTeX Font Info: ... okay on input line 2. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 2. LaTeX Font Info: ... okay on input line 2. Overfull \hbox (32.9967pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 /*********************************************************** ************[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 The D represents twisting by a quadratic character of conduc tor D.[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 Then Lseries(V,p,a_p,n,D) returns in the ordinary case a pol ynomial[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 of degree p^(n-1)-1 approximating the L-series. In the supe rsingular[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 case, it returns two polynomials say G and H so that G + H * alpha[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 approximates the L-series where alpha is a root of x^2 - a_p x + p.[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 D represents twisting by a quadratic character of condu ctor D[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 the p-adic valuation of the leading non-zero term of the L-series,[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 -this is returned as [ [m_1,s_1] , [m_2,s_2 ] , ... ][] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 where m_1 is the number of roots of slope s _1...;[] [] [1 ] Overfull \hbox (27.74675pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 (note if mu is positive then a "true" only means th at the data[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 p^(n-1)-1; that is it could have a another root of smaller slope)[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 same data for the g and h functions i construct in my thesis . i should[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 in how they are defined. basically, they are the "real" and "imaginary"[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 parts of the L-series with their trivial zeroes removed. Th ey are[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 ************************************************************ *********/[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 //T_2 acts as a_2, T_3 acts as a_3, T_5 acts as a_5 and T_7 acts as a_7[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 //Output: space of modular symbols of level N and specified eigenvalues[] [] [2] Overfull \hbox (27.74675pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 temp := ModularSymbolEven(V,(D*a-q*b)/(D*q))[ 1] * scale;[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 print "PROBLEMS! There are still denomina tors in ",[] [] [3] [4] [5] [6] Overfull \hbox (11.99689pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 scale(alpha(n+1,ALPHA),modsym((a mod q1) * p ,MOD SYM))));[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 //L-series of the elliptic curve corresponding to V twisted by a quadratic[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 //character of discriminant D. The degree of the polynomial is p^n-1.[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 //If a_p is prime to p (ordinary) then the answer is simply a polynomial.[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 //Otherwise (in the supersingular case) the answer is in the form [G,H][] [] [7] Overfull \hbox (1.49698pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 scale(measure(teich(a,TEICH)*gamma(j,GAMM A) mod[] [] [8] Overfull \hbox (17.24684pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 if (Valuation(Coefficient(f,j-1),p) ge bound) and (j n e 1) then[] [] [9] [10] Overfull \hbox (6.74693pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 if (r eq 0) and (rank eq 1) and (KroneckerSymbol(D,N) eq 1) then[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 if (r eq 1) and (rank eq 1) and (KroneckerSymbol(D,N) eq -1) then[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 //checks if lowering the valuation of any unknown coeffic ient to its[] [] [11] Overfull \hbox (1.49698pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 C_min := minimize_ord(C,p,n); //lowers all unknown coeffi cients[] [] [12] [13] Overfull \hbox (32.9967pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 rank := C[2]; //changes valuation of a_1 to 10000 if gue sses rank 2.[] [] [14] Overfull \hbox (48.74657pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 good := check_NP(NP,C,list_bad,rank,p,a_p,n,0); //0 repr esents nothing.[] [] [15] Overfull \hbox (6.74693pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 CG := compute_rank(CG,N,rank,D); //changes a_1 if guesses rank 2[] [] [16] Overfull \hbox (27.74675pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 goodg := check_NP(NPG,CG,list_badg,rank,p,a_p,n,0); //0 r epresents g[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 goodh := check_NP(NPH,CH,list_badh,rank,p,a_p,n,1); //1 r epresents h[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 if (leadingg eq -1) and (in_list(list_badg,rank+1) eq false) then[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 if (leadingh eq -1) and (in_list(list_badh,rank+1) eq false) then[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 878--878 [] \OT1/cmtt/m/n/10 return();[] [] [17] Overfull \hbox (11.99689pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 // This procedure returns many invariants of the curve corre sponding[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 878--878 []\OT1/cmtt/m/n/10 //