Bill Allombert on Fri, 19 Feb 2016 17:20:39 +0100

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L function associated to genus 2 curves

Dear PARI developers,

I have added a new GP function lfungenus2 to build L-functions
associated to curves of genus 2, with an interface similar to

There are some limitations, namely the model needs to be minimal at
2, the valuation at 2 of the conductor might be incorrect, and the
root number is not computed.

(If the model is not minimal at 2, the Euler factor at 2 might be

Since computing coefficients of such series is slow, we recommend
using a low bit precision (using \pb of localbitprec()).

Some examples:
? C=[x^2+x,x^3+x^2+1];
? genus2red(C)
%2 =

[2,-1] does not appear in the factorization, so the conductor is
correct. Indeed:

? L=lfungenus2(C);
? lfuncheckfeq(L)
%3 = -124

? C=[x^6+3*x^5+6*x^4+7*x^3+6*x^2+3*x+1,x^2+x];
? genus2red(C)
%2 = [49,[2,-1;7,2],...
Here the conductor is wrong by some power of 2.
? L=lfungenus2(%5);
  *** lfungenus2: Warning: unknown valuation of conductor at 2.
? L[5]*=4; \\ found by trial and error
? lfuncheckfeq(L)
%5 = -124