Timeline for Basis for modular forms of half-integral weight
Current License: CC BY-SA 2.5
4 events
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| Apr 7, 2010 at 18:03 | comment | added | Emerton | Dear Kevin, I believe you. I've used the algorithm I described to compute dimensions of spaces of half-integral weight forms, but I don't think I've used it to actually compute a basis. On the other hand, I think in his thesis Nick Ramsey used this description, or some variant of it, to $p$-adically interpolate half-integral weight forms. So it has some theoretical value, which might make up for its computational inadequacy! | |
| Apr 7, 2010 at 17:34 | comment | added | Kevin Buzzard | But it will be a pain to do at level $4N$ because you will have to isolate the subspace of $M_{k+1}(\Gamma_1(4N))$ consisting of forms which vanish at all cusps to at least the degree that $\theta$ vanishes. If $N>1$ then the degeneracy maps are typically ramified at the cusps so this will be a pain to do: the problem is that a computer algebra package typically computes modular forms as $q$-expansions at infinity, and seeing what's going on at the middle cusp will be a bit of a pain. | |
| Apr 6, 2010 at 23:08 | comment | added | David E Speyer |
I think one should add to Emerton's answer that the only zeroes of $\theta$ are at cusps, because one has $$\theta(\tau) = \prod (1-q^{2k})(1+q^{2k-1})^2,$$ with $q=e^{2 \pi i \tau}$. So the method Emerton proposes does not require any transcendental computations.
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| Apr 6, 2010 at 22:27 | history | answered | Emerton | CC BY-SA 2.5 |