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 genus2red. 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 incorrect). 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 = [169,Mat([13,2]),... [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 Cheers, Bill