2009/4/16 Steve Finch <sfin...@hotmail.com>:
>
> Hi again,
>
> Let M_{3/2}(N) be the space of modular forms of weight 3/2, level N
> and trivial character.
>
> It seems that the Cohen-Oesterle (CO) dimensions are too small. For
> example, let
>
> f(z) = 1 + 6*q + 12*q^2 + ...
>
> be the (unique) basis element of M_{3/2}(4) and
>
> g(z) = 1 + 2*q + 4*q^2 + ...
>
> be a basis element of M_{3/2}(8). Clearly
>
> f(2*z) = 1 + 6*q^2 + 12*q^4 + ...
>
> and note that the set
>
> {f(z), f(2*z), g(z)}
>
> is linearly independent in M_{3/2}(8). Hence while dim(M_{3/2}^{CO}
> (8))=2 according to Magma/Sage, in truth we have dim(M_{3/2}(8))=3.
>
> This observation is not new; please see the undergraduate research
> paper
>
> http://www.math.clemson.edu/~kevja/REU/2004/YaraChelsea.pdf
>
> for more details. Also, it seems that, when 4|N,
>
> dim( M_{3/2}(N)) = sum_{d|N} dim( M_{3/2}^{CO}(d) )
>
> but I don't know how to prove this. In short, the half-integer
> formulas in Cohen-Oesterle need to be revisited (unless I am making a
> mistake). The implementations in both Magma & Sage would need to be
> changed or, at least, the documentation would require revision.
> Comments? Thank you,
My understanding is that there is no implementation of the
half-integer Cohen-Oesterle formulas in Sage. Isn't that right?
Magma has an implementation of some dimension formulas, evidently, but
I've never seen the code in this case.
>
> Steve
>
> P.S. Note the important word "exactly" on the third line of page 13
> in the undergraduate writeup. I'm unsure whether Cohen-Oesterle
> actually specified this and would appreciate some expert opinions!
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