That all sounds very sensible to me.
Why don't you make a trac with most of your previous email in it?
I'll be busy with other matters but am happy to have more ideas
bounced off me.
John
On 10/01/2008, David Kohel <[EMAIL PROTECTED]> wrote:
>
> Dear John (et
> al.),
>
> I think the inner product should be the same irrespective of the
> field.
>
> The inner product as dot product is relevant to the study of
> quadratic
> forms, conics, and orthogonal groups. For instance finding a
> rational
> point on the conic x^2 + y^2 + z^2 = 0 over CC is equivalent to
> a
> representation of zero of the quadratic form, but not if the form
> is
> replaced by x\bar{x} + y\bar{y} + z\bar{z}, and I don't think we
> want
> to break this correspondence between points on conics and
> isotropic
> vectors.
>
> What is missing is a class for Hermitian modules, which would
> have
> a ring with involution as base ring. Such a class would be useful
> for
> studying Riemannian lattices such as the period lattice of an
> abelian
> variety, and unimodular groups. This class would allow one to
> define
> the Hermitian inner product x\bar{x} + y\bar{y} + z\bar{z} on a
> ring
> with inner product x |->
> \bar{x}.
>
> One choice of inner product is not sufficient to represent all such
> forms,
> and defining the default inner product to be the Hermitian product
> for
> complex fields would require making similar definition for
> imaginary
> quadratic fields, cyclotomic fields, and other CM fields for
> consistency.
> Even recognition of a CM field is nontrivial and this would lead
> to
> arbitrary choices. Worse, for other number fields, a Hermitian
> product
> would depend on a particular choice of
> embedding.
>
> So if this is entered into a trac, I think it should be for creation
> of a class
> of Hermitian forms together with a class of rings with involution
> (from
> which CC, quadratic rings, cyclotomic rings, and a class of CM
> fields
> would
> inherit).
>
> --
> David
> >
>
--
John Cremona
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