Timeline for Rankin-Selberg convolutions with mixed integral and half-integral weights
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Apr 25 at 15:58 | comment | added | Peter Humphries | If $f$ is the automorphic induction of a Hecke Grossencharakter $\psi$ of $\mathbb{Q}(\sqrt{D})$ with $D$ a positive fundamental discriminant, it should be possible to use the theta correspondence to directly show that $\langle y^{1/4}f\overline{\theta},g\rangle$ is equal to the product of two period integrals: one period integral of $G \psi^2$ over the closed geodesics of discriminant $D$, and one period integral given by the $D$-th Fourier coefficient of $g$. But I don't think this kind of identity appears anywhere in the literature... | |
| Apr 25 at 15:56 | comment | added | Peter Humphries | @awwalker Whoops, yes, it sholud be $\Lambda(1/2,\operatorname{ad} f \otimes G)$. | |
| Apr 25 at 15:23 | history | edited | YCor | CC BY-SA 4.0 |
formatting, added tag
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| Apr 25 at 15:16 | comment | added | awwalker | @Joseph Yes, conjugation maps weight $\kappa$ Maass forms to weight $-\kappa$ Maass forms. | |
| Apr 25 at 15:09 | comment | added | Joseph | To clarify the definitions: Does $g$ being a Maass form of weight $-1/2$ mean that $g$ is the complex conjugate of a Maass form of weight $1/2$? | |
| Apr 25 at 14:56 | comment | added | awwalker | Thanks for the reference! Presumably this should be $\Lambda(1/2, \mathrm{ad} f \otimes G)$? | |
| Apr 25 at 14:48 | comment | added | Peter Humphries | Also note that if $f$ is dihedral, then $\operatorname{ad} f$ factorises further, as in my answer here: mathoverflow.net/a/285653/3803. | |
| Apr 25 at 14:39 | comment | added | Peter Humphries | In general, it is nonvanishing. Let $G$ be the Shimura lift of $g$ to a Maass form of weight $0$. Then $|\langle y^{1/4} f\overline{\theta},g\rangle|^2$ is equal, up to local constants and normalising factors, to $\Lambda(1/2,\operatorname{ad}f \otimes g)$. See Theorem 4.5 of the following paper: doi.org/10.1093/imrn/rnt179 | |
| S Apr 25 at 14:03 | review | First questions | |||
| Apr 25 at 14:42 | |||||
| S Apr 25 at 14:03 | history | asked | awwalker | CC BY-SA 4.0 |