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,
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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