Re: Introduction, Curve Algorithm

[Brad Klee <bradklee@gmail.com>]

\\ Genus 2,  parity symmetric sextic ( Cf. OEIS: A006480 ).
\\ 2F1 parameters: (a,b,c) = (1/3,2/3,1).
PFData = HyperellipticPicardFuchs(-(1/4)*(18*x^2+48*x^4+32*x^6));
[-PFData[3],CheckCertificate(PFData)]
T=sum(n=0,100,(3*n)!/(n!)^3*(z/27)^n);
[T,T',T'']*PFData[3]
\\ [ [2, 18*z - 9, 9*z^2 - 9*z]~, 0]
\\ BigInt*z^100
\\ time delta = 8ms, fast!
On Tue, Oct 9, 2018 at 7:11 AM Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
>
> On Mon, Oct 08, 2018 at 10:44:22PM -0500, Brad Klee wrote:
> > Hi Bill Allombert and Pari Users,
> >
> > 2b. EllipticCurves.gp
> > The easiest examples are elliptic curves. Each torus S_z has a real and
> > complex period, which are related by a second order differential equation.
> > Vector A has three components. In two special cases vector A determines
> > a hypergeometric differential equation, solved by T(z) = 2F1(a,b;c;k*z).
>
> Let see if I understand the example (Examples/EllipticCurves.gp)
>
> \\ Hypergeometric Quartic
> \\ We define
> T(z)=hypergeom([1/4,3/4],[1],z);
> \\ (hypergeom is available in the GIT master branch)
> PFData = HyperellipticPicardFuchs((1/2)*(x^2-(1/4)*x^4));
> Eqn=PFData[3]
> \\ [3,32*z-16,16*z^2-16*z]~
> \\ So we set
> h(z)=[T(z),T'(z),T''(z)]*subst(Eqn,'z,z);
> \\ which should be identically 0 and indeed
> sum(i=1,100,abs(h(i/1000)))
> \\ 8.717140089686723290E-37
> \\ OK!
>
> \\ Hypergeometric Cubics
> \\ We define
> T(z)=hypergeom([1/6,5/6],[1],z);
> PFData = HyperellipticPicardFuchs((1/2)*(x^2-c*x^3));
> Eqn=-vector(3,j,9/4*(polcoef(PFData[3][j],2,c)*(4/27)+polcoef(PFData[3][j],0,c)))~;
> h(z)=[T(z),T'(z),T''(z)]*subst(Eqn,'z,z);
> sum(i=1,100,abs(h(i/1000)))
> \\ 1.5733945968008239299E-36
> \\ OK too!
>
> Cheers,
> Bill.
>