Timeline for Half integral weight modular forms that reduce to a nonzero constant modulo a given prime

Current License: CC BY-SA 4.0

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May 18, 2020 at 14:12	comment	added	Jeremy Rouse		I used Magma with can compute the spaces $M_{k/2}(\Gamma_{0}(N),\chi)$ and $S_{k/2}(\Gamma_{0}(N),\chi)$. It appears that Sage actually computes bases for spaces of cusp forms. There's a trick you can use to verify the presence of this form - the function $g(z) = \eta^{2}(4z) \eta^{2}(20z) \in S_{2}(\Gamma_{0}(80))$. You can then look in the space of weight $13/2$ forms for $\Gamma_{1}(80)$ (with trivial character) for a form that is congruent to $g$ (up to $q^{102}$) modulo $5$.
May 18, 2020 at 2:28	comment	added	BHT		Could you explain how you found that example form? I tried to write some Sage code to search for these forms by generating a basis and then reducing mod p and solving a matrix equation, but when I ran it on weight $9/2$, $\Gamma_1(80)$ (Sage requires $16\|N$ for some reason), it didn't find the form you mentioned, so I must be doing something wrong.
May 15, 2020 at 22:20	history	edited	Jeremy Rouse	CC BY-SA 4.0	added 893 characters in body
May 15, 2020 at 21:24	comment	added	BHT		Thank you very much for the answer! In the Gross paper the result you state is proven in proposition $4.9$, by showing that the kernel of reduction mod $p$ is the principal ideal $(1-A)$, where $A$ is the Hasse invariant. The author does indeed assume $p$ does not divide $N$ generically throughout section $4$, but determining whether or not this assumption is actually needed for proposition $4.9$ is well beyond my capabilities.
May 15, 2020 at 17:33	comment	added	Kimball		I'm not sure if you were misled by the title (which isn't clear), but the OP seems to be looking for forms with a_n = 0 mod p for all n > 0 but a_0 not 0 mod p.
May 15, 2020 at 14:23	history	answered	Jeremy Rouse	CC BY-SA 4.0