Defining L-functions in GP
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1 A tutorial B. Allombert IMB CNRS/Université de Bordeaux 17/01/2018 This project has received funding from the European Union s Horizon 2020 research and innovation programme under grant agreement N
2 Riemann ζ function? Zeta = lfuncreate(1) %1 = [[Vecsmall([1]),1],0,[0],1,1,1,1]? lfun(zeta,2) %2 = ? lfun(zeta,0,1) %3 = ? lfun(zeta,1) %4 = *x^-1+O(x^0)? lfun(zeta,1+x+o(x^10)) %5 = *x^ *x ? lfunzeros(zeta,20) %6 = [ ]? lfunlambda(zeta,2) %7 =
3 Dirichlet L functions? G=znstar(4,1); G.clgp %7 = [2, [2], [3]]? Dir=lfuncreate([G,[1]]); Dir[2..5] %8 = [0,[1],1,4]? lfunan(dir,30) %9 = [1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1? lfun(dir,2) %10 = ? Catalan %11 = ? znconreyexp(g,[1]) %12 = 3? lfun(mod(3,4),2) %13 =
4 Dedekind ζ functions? Dedek = lfuncreate(x^2+1); Dedek[2..5] %14 = [0,[0,1],1,4]? lfun(dedek,2) %15 = ? zeta(2)*catalan %16 = ? L=lfunmul(Zeta,Mod(3,4));? lfun(l,2) %18 = ? L2=lfundiv(Dedek,1);? lfun(l2,2) %20 =
5 elliptic curves For the elliptic curve y 2 + y = x 3 x 2 10x 20:? E = ellinit([0,-1,1,-10,-20]);? lfun(e,1) %21 = ? ellbsd(e) %22 = ? lfuncreate(e)[2..5] %23 = [0,[0,1],2,11]
6 Elliptic curves over number fields We define the elliptic curve y 2 + xy + φx = x 3 + (φ + 1)x 2 + x over the field Q( 5) where φ = ? nf = nfinit(a^2-5);? phi = (1+a)/2;? E = ellinit([1,phi+1,phi,phi,0],nf);? E.j %27 = Mod(-53104/31*a-1649/31,a^2-5)? E.disc %28 = Mod(-8*a+17,a^2-5)? N = ellglobalred(e)[1] %29 = [31,13;0,1]? tor = elltors(e) \\ Z/8Z %30 = [8,[8],[[-1,Mod(-1/2*a+1/2,a^2-5)]]]
7 Elliptic curves over number fields We check the BSD conjecture for E.? om = E.omega %16 = [[ , *I], % [ , *I]]? per = om[1][1]*om[2][1];? tam = elltamagawa(e) %18 = 2? bsd = (per*tam) / (tor[1]^2*sqrt(abs(nf.disc))) %19 = ? ellbsd(e) %20 = ? L1 = lfun(e,1) %21 =
8 lfuntwist lfuntwist allows to twist an L function by a Dirichlet character. The conductors need to be coprime.? E = ellinit([0,-1,1,-10,-20]);? L=lfuntwist(E,Mod(2,5));? lfunan(e,10) %3 = [1,-2,-1,2,1,2,-2,0,-2,-2]? lfunan(mod(2,5),10) %4 = [1,I,-I,-1,0,1,I,-I,-1,0]? lfunan(l,10) %5 = [1,-2*I,I,-2,0,2,-2*I,0,2,0]
9 lfuntwist We redefine the curve over Q(ζ 5 ).? nf=nfinit(polcyclo(5, a));? E2=ellinit(E[1..5],nf);? localbitprec(64); lfun(e2,2) %8 = ? L2=lfuntwist(E,Mod(4,5));? lfun(e,2)*lfun(l2,2)*norm(lfun(l,2)) %10 =
10 Genus-2 curve For the genus-2 curve y 2 + (x 3 + 1)y = x 2 + x:? L=lfungenus2([x^2+x,x^3+1]);? L[2..5] %12 = [0,[0,0,1,1],2,249]? lfun(l,1) %13 = ? lfunan(l,5) %14 = [1,-2,-2,1,0]
11 Hecke L functions? bnf = bnfinit(a^2+23);? bnr = bnrinit(bnf, 1);? bnr.clgp %3 = [3,[3]]? Hecke = lfuncreate([bnr,[1]]);? Hecke[2..5]? z=lfun(hecke,0,1) %4 = E-60? algdep(exp(z),3) %5 = x^3-x-1
12 Galois group We start with a Galois extension of the rationals, here Q( 3 2, ζ 3 ) = Q( 6 108), with Galois group isomorphic to S 3.? N = nfinit(x^6+108);? G = galoisinit(n); G is the Galois group of N.
13 Linear representation? [T,o] = galoischartable(g);? T~ %4 = [1 1 1] % [1 1-1] % [2-1 0] T is the character table of G = S 3, which is defined over Z. The first character is related to the trivial representation, the second to the signature, and the third to a faithful irreducible representation of dimension 2. The ordering of the conjugacy classes is given by galoisconjclasses(g).? galoisconjclasses(g) %4 = [[Vecsmall([1,2,3,4,5,6])],[Vecsmall([3,1,2,6,4
14 Artin L-function We compute the Artin function associated to the 3 rd character.? L = lfunartin(n,g,t[,3],o);? lfuncheckfeq(l) %6 = -127? L[2..5] %7 = [0, [0, 1], 1, 108]? z = lfun(l,0,1) %8 = ? algdep(exp(z),3) %9 = x^3-3*x^2-3*x-1 which suggests that this function is equal to a Hecke L-function.? bnr = bnrinit(bnfinit(a^2+a+1),6);? lfunan([bnr,[1]],100)==lfunan(l,100) %11 = 1
15 A more interesting example Let E be a model of the curve X 0 (11) E : y 2 + y = x 3 x 2 10x 20, we build the field Q(E[3]) generated by the coordinates of the points of 3-torsions.? E=ellinit([0,-1,1,-10,-20]); \\ or ellinit("11a1") if elldata is available? P=elldivpol(E,3) %13 = 3*x^4-4*x^3-60*x^2-237*x-21? Q=polresultant(P,y^2-elldivpol(E,2)); %14 = 27*y^8+108*y^7-4813*y^ *y^ *y^4+3? R=nfsplitting(Q) %15 = y^48-36*y^46+558*y^ *y^ *y^ This defines a Galois extension of Q with Galois group GL 2 (F 3 ).
16 Non monomial representation? N=nfinit(R); G=galoisinit(N);? [T,o]=galoischartable(G); T~ %17 = [1,1,1,1,1,1,1,1; % 1,1,-1,1,1,1,-1,-1; % 2,0,-y^3-y,1,-1,-2,0, y^3+y; % 2,0, y^3+y,1,-1,-2,0,-y^3-y; % 2,2,0,-1,-1,2,0,0; % 3,-1,-1,0,0,3,1,-1; % 3,-1,1,0,0,3,-1,1; % 4,0,0,-1,1,-4,0,0]? o %18 = 8 o = 8 means that the variable y denotes a 8-th root of unity.
17 Non monomial representation? minpoly(mod(y^3+y, polcyclo(o,y))) %19 = x^2+2 So the coefficients are in Q( 2). We use the third irreducible representation.? L = lfunartin(n,g,t[,3],o);? L[2..5] %21 = [0,[0,1],1,3267]? lfuncheckfeq(l) %22 = -127
18 Determinant? dt = galoischardet(g,t[,3],o) %23 = [1,-1,-1,1,1,1,1,-1]~? dl = lfunartin(n,g,dt,o);? dl[2..5] %25 = [0,[1],1,3]; So L is associate to a modular form of weight 1, level 3267 and Nebentypus ( ) 3..? mf=mfinit([3267,1,-3],1);? M=mfeigenbasis(mf);? C=mfcoefs(M[3],100);? mfembed(m[3],c)[2][2..-1]==lfunan(l,100) %29 = 1
19 Link to E We reduce the coefficients of L modulo of norm 3.? S = lfunan(l,1000); SE = lfunan(e,1000);? Smod3 = round(real(s))+round(imag(s)/sqrt(2));? [(Smod3[i]-SE[i])%3 i<-[1..#smod3],gcd(i,33)==1] %29 = [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,... The coefficients of L are congruent to the coefficients of the L-function associated to E modulo
20 Quotient of Hecke L-function We will write L as L 1 /L 2, where L 1 and L 2 are two Hecke L-functions.? bnf6=bnfinit(a^6-3*a^5+6*a^4+4*a^3+6*a^2-3*a+1);? bnr6=bnrinit(bnf6,1);? L1=lfuncreate([bnr6,[1]]);? L1[2..5] %33 = [1, [0, 0, 0, 1, 1, 1], 1, ]? bnf4=bnfinit(a^4-a^3+3*a^2+a-1);? pr4 = idealprimedec(bnf4,3)[1];? bnr4=bnrinit(bnf4,[pr4,[0,1]]);? L2=lfuncreate([bnr4,[1]]);? L2[2..5] %38 = [0, [0, 0, 1, 1], 1, 9801]
21 Quotient of Hecke L-function? LL = lfundiv(l1,l2); LL[2..5] %39 = [0,[0,1],1,3267]? round(lfunan(l,1000)-lfunan(ll,1000),&e) %40 = [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,...? e %41 = -125 So L is equal to a quotient of two Hecke L-functions.
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