Bill Allombert on Thu, 18 Sep 2014 16:06:11 +0200

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hyperelliptic curve zeta function

Dear PARI developers,

I have added a GP function to compute the Zeta function
of an hyperelliptic curve over a finite field
(with some limitation), following Gaury-Gürel version of
Kedlaya algorithm.
The function actually return the characteristic polynomial
of the Frobenius.

Some examples:

The elliptic curve y^2=x^3+x over GF(101)
? hyperellzeta((x^3+x)*Mod(1,101))
%1 = x^2-2*x+101
? ellap(ellinit([1,0]),101)
%2 = 2

The curve y^2=x^5+g*x^3+2*x+5 over GF(79^3), 
where g is a root of x^3+x^2-2*x-1:
? g=ffgen(ffinit(79,3),'g);
? hyperellzeta(x^5+g*x^3+2*x+5)
%5 = x^4+819*x^3+945075*x^2+403798941*x+243087455521