Timeline for Do L-functions exist for Half-integral weight modular forms?
Current License: CC BY-SA 3.0
11 events
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| May 20, 2022 at 20:03 | comment | added | Alison Miller | The link in this answer no longer works: I'm assuming it was meant to go to the article Hansen, D., Naqvi, Y. Shimura lifts of half-integral weight modular forms arising from theta functions. Ramanujan J 17, 343–354 (2008). doi.org/10.1007/s11139-007-9020-1 | |
| Apr 1, 2016 at 18:16 | history | edited | Peter Humphries | CC BY-SA 3.0 |
added 24 characters in body
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| Jan 5, 2013 at 6:15 | vote | accept | N. Kumar | ||
| Jan 5, 2013 at 6:15 | |||||
| Nov 21, 2012 at 14:24 | vote | accept | N. Kumar | ||
| Jan 5, 2013 at 6:14 | |||||
| Nov 19, 2012 at 17:42 | comment | added | David Loeffler | Marty: You should write your sequence of comments as a new answer -- it's a far better answer than mine was! | |
| Nov 19, 2012 at 17:02 | comment | added | Marty | So -- long story short -- I share Nick's intuition that the L-function is the entire family of quadratic twists, and I went so far as to cook up a theory of L-groups for (split, for now) metaplectic groups in order to understand this better. | |
| Nov 19, 2012 at 17:01 | comment | added | Marty | When $G$ is the simplest metaplectic group $\widetilde{SL}_2$, I argue that the L-group ${}^L G$ is a group scheme over $\mathbb{Z}$, which is isomorphic to $SL_2 \times \Gamma$ ($\Gamma$ the Galois group), but noncanonically and not until base change to $\mathbb{ZZ}[i]$. However, each additive character of ${\mathbb A} / {\mathbb Q}$ gives such an isomorphism, and this realizes a bijection between quadratic twists and (L-)isomorphisms of the L-group to $SL_2 \times \Gamma$. Taking the standard representation of $SL_2$, you get the L-function of a quadratic twist. | |
| Nov 19, 2012 at 16:56 | comment | added | Marty | Following up on Nick's question -- the Langlands correspondence for metaplectic groups is something I've thought about a lot in the past few years. I have a paper on L-groups for Metaplectic Groups on the ArXiv, for example. Since that paper is hard to read (I'm working hard to get rid of all the Hopf algebra machinery used there), I'll summarize here. An L-function should come from two pieces of data: an automorphic representation of $G$ (for half-integral weight eigenforms, $G$ is a metaplectic group) and an algebraic (finite-dimensional) representation of the L-group ${}^L G$. | |
| Nov 19, 2012 at 14:40 | comment | added | N. Kumar | @David: Thanks for your reply. I agree that there is a relation between half-integral and integral weight modular forms by Shimura. However, the way the $L$-functions are related in the "Shimura lifting" it only involves $n^2$ terms of the half-integral weight modular forms. What about the non-square free terms? They dont matter much in defining the $L$-function, unlike the classical case? | |
| Nov 19, 2012 at 13:51 | comment | added | Ramsey | Just to emphasize something touched on in the last sentence here: I've always thought of the natural "$L$-function type object" of a half-integral weight form as the entire family of quadratic twists of its Shimura image form. I don't know enough about the metaplectic/automorphic picture to understand if there's something intrinsic about this point of view, but I've been idly curious about this. For that matter: is there a Langlands correspondence for metaplectic groups? Is the Galois side just the family of quadratic twists of the Galois representation of the Shimura lifting in this case? | |
| Nov 19, 2012 at 11:10 | history | answered | David Loeffler | CC BY-SA 3.0 |