---------- Forwarded message ---------
From: Simon Brandhorst <[email protected]>
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 <[email protected]>
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.
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