I forwarded this to sage-nt to increase that chances of a helpful response.
John
On Wed, 3 Oct 2018 at 12:26, Simon Brandhorst <sbrandho...@web.de> wrote:
> 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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