Timeline for Galois representations attached to Maass form
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 29, 2018 at 21:22 | vote | accept | Dror Speiser | ||
| Mar 2, 2012 at 20:13 | comment | added | GH from MO | So I think it is fair to say that the Maass forms that are automorphically induced from a Hecke character are precisely the ones constructed by Maass and described by Bump (apart from level considerations). | |
| Mar 2, 2012 at 20:13 | comment | added | GH from MO | In my previous message I meant that in the first display of Page 113 in Bump one has $\nu\in i\mathbb{R}$, at least for a weight zero Maass forms as in (9.18) and under the assumption of the Ramanujan conjecture. As explained before, the case of even weight does not yield any more $L$-functions. In addition, the mentioned display does not represent the $L$-function of any odd Maass form (as seen by the shape of gamma factors). | |
| Mar 2, 2012 at 19:43 | comment | added | GH from MO | @Rob: Thank you. Note that if a Maass form is induced form a character as in Bump's (9.18), then we must have $m_1=-m_2\in i\mathbb{R}$. This is because the $L$-function of a Maass form is rather restricted, it must be of the form as in the first display of Page 113 in Bump. Of course by shifting $s$ you can get other $\mathrm{GL}_2$ $L$-functions, but these will not correspond to Maass forms but to archimedean $\mathrm{GL}_1$ twists of the corresponding cuspidal representation on $\mathrm{GL}_2$. | |
| Mar 2, 2012 at 18:49 | comment | added | Rob Harron | (cont'd) an integrality statement on the eigenvalues of some natural differential operator on the infinite ideles. So, basically it's saying that the infinite component is integral in some sense. So, depending on your normalizations, you're saying the $L$-parameter at infinity is given by integral (or half-integral) data. This was generalized to automorphic representations of $GL(n)$ by Clozel in the Ann Arbor proceedings, and by Buzzard–Gee in their recent preprint for general groups. | |
| Mar 2, 2012 at 18:45 | comment | added | Rob Harron | @GH: Weil used the terminology "Hecke character of type $A_0$" for what we now call "algebraic Hecke characters". One way to define these is to view the Hecke character as a complex (quasi-)character $\chi$ of the ideles $\mathbf{A}^\times_K$ of the number field $K$. Then, on the connected component of the identity of the infinite ideles, we have $\phi((x_v)_v)=\prod x_v^{m_v}$ ($v$ varying over the infinite places). Then, χ is algebraic if $m_v\in\mathbf{Z}$ for all $v$. One can also phrase this in terms of the infinitesimal character of the $(\mathfrak{g},K)$-module of $\chi$, or as ... | |
| Mar 2, 2012 at 17:32 | comment | added | GH from MO | @Rob: Yes, the Maass forms found by Maass are the CM forms. These are sparse among all Maass forms like squares among all integers (this follows from the Weyl law for the counting function of Maass forms). Also, indeed, for an analytic guy like me, the coefficient usually means the normalized one (which produces the $L$-function centered at $1/2$). Thanks for mentioning Clozel, Buzzard-Gee, motivic. Can you tell me, just for the record, when is a Hecke character algebraic (more precisely a quasi-character as Weil would call it)? | |
| Mar 2, 2012 at 16:07 | comment | added | Rob Harron | (cont) (and more generally, you need the Buzzard–Gee notion of $C$-algebraic, as well), hence why it's also called "motivic". | |
| Mar 2, 2012 at 16:07 | comment | added | Rob Harron | (cont) Otherwise, it's clear what their associated Galois representations are. I'm not so silly as to think all $L$-functions of GL(2) come from GL(1). Furthermore, it is quite simple to obtain Hecke characters with transcendental $L$-parameters using my broader sense, since you can twist by any power of the norm character. More importantly, transcendence isn't really the right notion, rather Clozel's notion of algebraic automorphic representation, which is encoded in the $L$-parameter. But "algebraic" here doesn't mean "non-transcendental", but rather "integral" in a specific sense... | |
| Mar 2, 2012 at 16:06 | comment | added | Rob Harron | (cont) It's also natural to take the automorphic induction of any of these Hecke characters and obtain some automorphic representation of GL(2). I'm perfectly willing to believe this doesn't produce all Maass forms, just like if I induce a Hecke character from an imaginary quadratic field I get a CM holomorphic modular form, not an abitrary holomorphic modular form. To be more precise, I should have said that those Maass forms that are automorphically induced, if any of them are not known to have associated Galois representations, then they must be induced from non-algebraic Hecke characters. | |
| Mar 2, 2012 at 16:05 | comment | added | Rob Harron | @GH: I consider my Hecke characters to be homomorphisms from the idele class group to $\mathbf{C}^\times$ (as many people do, including Weil who was the first to phrase Hecke's Grössencharaktere in terms of ideles). Of course, the difference between the two notions is just a power of the norm, which if you are considering $L$-functions amounts to a shift, but it is something completely natural to study (e.g. CM elliptic curves naturally correspond to Hecke characters in the broader sense that I use), especially when discussing algebraicity and Galois representations. | |
| Mar 2, 2012 at 12:39 | comment | added | GH from MO | Note also that: (1) most Maass forms are expected to have transcendental Langlands parameters, I doubt characters of any sort would produce that; (2) a Maass form is essentially either weight zero or weight one. More precisely, the weight of any Maass form can be changed by $\pm 2$ without altering its $L$-function, so Maass forms of weight 100 (say) produce the same $L$-functions as Maass forms of weight zero. | |
| Mar 2, 2012 at 12:39 | comment | added | GH from MO | @Rob: For me a Hecke character is a homomorphism of the idele class group to the unit circle of the complex plane, these are the ones considered by Hecke. I don't know what are the Hecke characters you talk about, but I doubt general Maass forms would correspond to these. You see, that would mean that all $L$-functions coming from $\mathrm{GL}_2$ also come from $\mathrm{GL}_1$ which is nonsense. | |
| Mar 2, 2012 at 5:21 | comment | added | Rob Harron | Also, "automorphic induction" refers to the part of functoriality that translates induction of Galois representations. So, I would think that if you had a Hecke character that had an attached Galois representation (of the absolute Galois group of the real quadratic field), then the associated Maass form would have an associated Galois representation (of the absolute Galois group of Q), namely the induced representation. I assume that in general Maass forms are automorphically induced from non-algebraic Hecke characters. Again, I'm not sure about this as I've never thought about Maass forms. | |
| Mar 2, 2012 at 5:08 | comment | added | Rob Harron | @GH: From what I can tell, Bump is only considering Hecke characters of weight 0, so it's not surprising that his Maass forms have weight 0. | |
| Mar 1, 2012 at 9:04 | comment | added | GH from MO | @Rob: I am not too familiar with $p$-adic Galois representations. At any rate, I had complex representations in mind. Note also that Hecke characters give weight zero Maass forms regardless of their archimedean component, see Theorem 1.9.1 in Bump: Automorphic forms and representations. And yes, this is a special case of automorphic induction. | |
| Mar 1, 2012 at 1:08 | comment | added | Rob Harron | @GH: algebraic Hecke characters of infinite order can be identified with 1-dimensional ($p$-adic) Galois representations. Of course, for a totally real field, all Hecke characters are cyclotomic twists of the finite order characters, so as Katz says, I've been "speaking prose all along". Perhaps you mean to say that only finite order Hecke characters give weight zero Maass forms? I'm not really familiar with the construction, though I imagine it's just an automorphic induction. | |
| Feb 29, 2012 at 15:50 | comment | added | GH from MO | @Dror: They are relevant, but you need to assume the Hecke character is of finite order so that it can be identified with a 1-dimensional Galois representation. | |
| Feb 29, 2012 at 15:06 | comment | added | Dror Speiser | I wonder, are Maass's forms for real quadratic fields as I mentioned above also not relevant? | |
| Feb 29, 2012 at 14:28 | history | answered | Joël | CC BY-SA 3.0 |