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// lambda.h Declarations of functions which compute Silverman's
//////////////////////////////////////////////////////////////////////////
//
// Copyright 1990-2012 John Cremona
//
// This file is part of the eclib package.
//
// eclib is free software; you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by the
// Free Software Foundation; either version 2 of the License, or (at your
// option) any later version.
//
// eclib is distributed in the hope that it will be useful, but WITHOUT
// ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
// FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
// for more details.
//
// You should have received a copy of the GNU General Public License
// along with eclib; if not, write to the Free Software Foundation,
// Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
//
//////////////////////////////////////////////////////////////////////////
// finite set Lambda_bad for a curve
// N.B. (1) Uses my height normalization, double S's.
// (3) Uses the local height normalization WITHOUT the log|Delta|
// (2) Intended for use in computing Heegner points (not yet implemented)
#include <eclib/points.h>
#include <eclib/lambda.h>
#define MAX_NUM_LAMBDA 1000
vector<bigfloat> lambda_bad_1(const bigint& p, long kcode, long npd, long& nlambda)
{
bigfloat logp = log(I2bigfloat(p)), n=to_bigfloat(npd);
if((kcode%10)==0) // Type I_m
{
long i, m = kcode/10;
nlambda=1+(m/2);
vector<bigfloat> ans; ans.reserve(nlambda);
for(i=0; i<nlambda; i++) ans.push_back( ( (i*i)/n - i ) * logp);
return ans;
}
if((kcode%10)==1) // Type I*m
{
bigfloat m = to_bigfloat(kcode-1)/10;
nlambda = 3;
vector<bigfloat> ans; ans.reserve(nlambda);
ans.push_back(to_bigfloat(0));
ans.push_back(-logp);
ans.push_back(-(1 +m/4)*logp);
return ans;
}
if((kcode==2)||(kcode==7)||(p>3))
{
nlambda = 1;
vector<bigfloat> ans(1,to_bigfloat(0));
return ans;
}
nlambda = 2;
vector<bigfloat> ans; ans.reserve(nlambda);
ans.push_back(to_bigfloat(0));
long nn = (kcode<5? kcode: kcode+3);
ans.push_back( -(nn*logp)/6 );
return ans;
}
vector<bigfloat> lambda_bad(const CurveRed& C, long& nlambda, int verbose)
{
vector<bigfloat> ans;
nlambda = 1;
ans.push_back(to_bigfloat(0));
bigint discr = getdiscr(C);
vector<bigint> plist = getbad_primes(C);
long i, j, nl, nnl;
vector<bigint>::const_iterator pr=plist.begin();
while(pr!=plist.end())
{
bigint p = *pr++;
if (ndiv(p*p,discr))
{
if(verbose)
cout<<"Lambda_bad("<<p<<") has only one element, 0.\n";
continue;
}
// else do some real work
long kcode = getKodaira_code(C,p).code;
long npd = getord_p_discr(C,p);
vector<bigfloat> list = lambda_bad_1(p,kcode,npd,nl);
if(verbose)
{
cout << "Lambda_bad("<<p<<") has " << nl << " element(s): ";
cout << list << endl;
}
nnl = nlambda*nl;
ans.reserve(nnl);
for(i=0; i<nlambda; i++)
for(j=0; j<nl; j++)
ans.push_back(ans[i]+list[j]);
nlambda = nnl;
} // end of loop on p
return ans;
}
int make_point_from_x(Curvedata* CD, const bigint& xa, const bigint& xd, Point* P)
{
bigint a(xa),b,c,d(xd);
if(d<0) {a=-a; d=-d;}
if(!isqrt(d,c)) return 0;
bigint b2,b4,b6,b8;
CD->getbi(b2,b4,b6,b8);
bigint d2=d*d;
bigint d3=d2*d;
bigint e,e2 = ((4*a+b2*d)*a + 2*b4*d2)*a + b6*d3;
if(!isqrt(e2,e)) return 0;
bigint a1,a2,a3,a4,a6;
CD->getai(a1,a2,a3,a4,a6);
b = (e-a1*a*c-a3*c*d)/2;
*P = Point(CD,a*c,b,c*d);
return 1;
}
int make_point_from_x(Curvedata* CD, const bigfloat& x, long maxdd, Point* P)
{
bigint a,b,c,d;
//cout<<"In ratapprox2 with x = " << x << endl;
bigint x0, x1, x2, y0, y1, y2;
bigfloat xx, diff, xc;
xx = x; x0 = 0; x1 = 1; y0 = 1; y1 = 0;
diff = 1;
bigint maxdenom = pow(BIGINT(10),maxdd);
while ( !is_approx_zero(diff) && (y2<maxdenom))
{ c = Iround( xx ); xc=I2bigfloat(c);
x2 = x0 + c*x1; x0 = x1; x1 = x2;
y2 = y0 + c*y1; y0 = y1; y1 = y2;
if(make_point_from_x(CD,x2,y2,P)) return 1;
diff = abs( x - (I2bigfloat(x2)/I2bigfloat(y2)) );
//cout<<"x2,y2,diff = " << x2 << ", " << y2 << ", " << diff << endl;
if ( is_approx_zero(abs(xx - xc)) ) diff = 0;
else xx = to_bigfloat(1)/(xx - xc);
}
a = x2; d = y2;
if ( d < 0 )
{::negate(a); ::negate(d); }
if(!isqrt(d,c)) return 0;
bigint b2,b4,b6,b8;
CD->getbi(b2,b4,b6,b8);
bigint d2=d*d;
bigint d3=d2*d;
bigint e,e2 = ((4*a+b2*d)*a + 2*b4*d2)*a + b6*d3;
if(!isqrt(e2,e)) return 0;
bigint a1,a2,a3,a4,a6;
CD->getai(a1,a2,a3,a4,a6);
b = (e-a1*a*c-a3*c*d)/2;
*P = Point(CD,a*c,b,c*d);
return 1;
}
int make_point_from_x_and_ht(Curvedata* CD, vector<bigfloat> lambdas, const bigfloat& xp, const bigfloat& ht, Point* P)
{
bigfloat rh = realheight(xp,CD);
vector<bigfloat>::const_iterator lam = lambdas.begin();
while(lam!=lambdas.end())
{
bigfloat logd = (ht-rh-(*lam++))/2;
bigfloat approxd = exp(logd);
bigint xa, xd2, xd = Iround(approxd);
if(xd>0)
{
xd2 = xd*xd;
xa = Iround(xp*I2bigfloat(xd2));
if(make_point_from_x(CD,xa,xd2,P)) return 1;
}
bigint xdx; long id, delta=10;
for(id=1;id<=delta;id++)
{
xdx=xd+delta;
if(xdx>0)
{
xd2 = xd*xd;
xa = Iround(xp*I2bigfloat(xd2));
if(make_point_from_x(CD,xa,xd2,P)) return 1;
}
xdx=xd-delta;
if(xdx>0)
{
xd2 = xd*xd;
xa = Iround(xp*I2bigfloat(xd2));
if(make_point_from_x(CD,xa,xd2,P)) return 1;
}
}
}
return 0;
}
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