Fwd: [sage-devel] Special values of a dirichlet L-series are needed to rescue the mass formula in sage

[John Cremona]

---------- Forwarded message ---------
From: Simon Brandhorst <sbran...@web.de>
Date: Wed, 3 Oct 2018 at 12:26
Subject: [sage-devel] Special values of a dirichlet L-series are needed to rescue the mass formula in sage
To: sage-devel <sage-...@googlegroups.com>

The mass of a positive definite quadratic form is defined as the
sum over 1/#Aut(L') where L' runs over the representatives of the genus of L.

It can be calculated by combining informations at the different completions of L at all primes by analytic methods. It is a particularly beautiful and useful piece of mathematics. But unfortunately it is broken in sage :-(. I would love to fix it! But I need your help with L functions. #26378

I believe the reason for our bug is that Hanke used the

quadratic_L_function__exact

This function seems to work fine. However I believe it is the wrong L-function.
It is defined as follows:

L(\chi_D, s) =\sum_{m \in \NN} \chi_D(m) / m^s
here \chi_D(m) = \left( \frac{ D }{m} \right)
Now the tricky part is what D  over m means. Hanke does not write this but in

quadratic_L_function__numerical

he means by D over m the Kronecker symbol.

Now if one carefully reads

Low-Dimensional Lattices. IV. The Mass Formula
Author(s): J. H. Conway and N. J. A. Sloane
Source: Proceedings of the Royal Society of London. Series A, Mathematical and Physical
Sciences, Vol. 419, No. 1857 (Oct. 8, 1988), pp. 259-286
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/2398465

in equation (8) the authors define what they mean by D over m:
It is zero if m is even and the jacobi symbol else.
( The jacobi symbol is defined only for odd integers  https://en.wikipedia.org/wiki/Jacobi_symbol)
So the L series we really need is
\sum_{k \in \NN} (\chi_D(2k -1) /(2k-1)^s

Now we do not need the L series but infinitely many special values.
In (13) Conway and Sloane cite a (complicated) formula for that involving Bernulli numbers and a Gaussian sum. Now here is where I am stuck. I am not an expert on Dirichlet characters
(The easy way out for me is to just use the magma interface and type mass(L) - that is what I am doing right now - so basically I am fixing this bug on my free time which is why I would appreciate help.) The formula involves the conductor
k_1 and the modulus k and we have to decompose our dirichlet character as \chi_1 * \psi where
\chi_1 is the principal character modulo k and \psi is a primitive character modulo k_1.
Finally Conway and Sloane did their calculations probably by hand - and I do not know if their description of the formula is computationally friendly.

O.K. I can probably figure out the modulus and conductor, but I do not know how to figure out the decomposition. Well there seems to be a sage package about Dirichlet characters but then again I do not know how to input our character - probably because I do not understand enough about them.

--
You received this message because you are subscribed to the Google Groups "sage-devel" group.
To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+...@googlegroups.com.
To post to this group, send email to sage-...@googlegroups.com.
Visit this group at https://groups.google.com/group/sage-devel.
For more options, visit https://groups.google.com/d/optout.