Timeline for "Space" of L-functions
Current License: CC BY-SA 3.0
33 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Sep 16, 2011 at 1:48 | comment | added | Emerton | Dear Daniel, Okay, now we're moving from an MO question to something much more interesting, which is your mathematics! If/when you can explain what you mean by deforming a Galois representation, I would very much like to know! In particular, I'm very surprised to hear that you're doing this, while not thinking about the $p$-adic viewpoint, for which there is a very well established deformation theory. I am easily contactable via the information on my user page, if/when you're willing to share more. Best wishes, Matthew | |
| Sep 15, 2011 at 15:52 | comment | added | Daniel Larsson | Matthew, the $p$-adic theory sounds extremely interesting and I would be very happy to know more. This is very much in tune with my question (although I wasn't thinking $p$-adically). /Daniel | |
| Sep 15, 2011 at 15:47 | comment | added | Daniel Larsson | Dear Matthew, actually you can deform the Galois representation as a module, easily (in principle) when it factors through a finite quotient, harder when pro-finite, but maybe doable. Actually, me and a colleague are doing a computation on a variant of this now, and this is the (deeply) underlying reason for my question. | |
| Sep 15, 2011 at 1:53 | comment | added | Emerton | Dear Daniel, At the moment the only known ways to deform Galois reps. are $p$-adically, and one then expects that the $L$-functions also deform as $p$-adic $L$-functions (in some sense), as Rob said. The reason for the parenthetical qualification is that the theory of $p$-adic $L$-functions is still very much a theory in progress, and it's not yet clear (at least to me) what form the general theory will take. Regards, Matthew | |
| Sep 14, 2011 at 22:06 | answer | added | B R | timeline score: 2 | |
| Sep 14, 2011 at 19:45 | comment | added | Rob Harron | the last sentence of David Hansen's comment is about I guess. Anyway, for $p$-adic things you have more interesting families, so you can get more interesting deformations of $p$-adic $L$-functions. | |
| Sep 14, 2011 at 19:44 | comment | added | Rob Harron | @BR: Oh, yeah sorry, I wasn't thinking right when I said that was deforming at infinity. I guess I started out writing my comment as a reply to Daniel's question about being able to continuously deform $L$-functions, I certainly view these as deformations of $L$-functions albeit rather trivial ones. Re Eisenstein series: If I take the induced representation $\mathrm{Ind}(|\cdot|^a,1)$, then for any $a\neq1$ (or $-1$,...) I get an Eisenstein series $E_a$ whose $L$-function is basically $\zeta(s-a)\zeta(s)$, which is more interesting, though still not that interesting. And this is what... | |
| Sep 14, 2011 at 19:18 | comment | added | B R | (cont) Eisenstein series are another example, but again, shifting the parameter just shifts all the local data by the same amount. This might be what the OP wanted, but it doesn't contradict my point (which was more in the spirit of Strong Multiplicity One). The idea of interpreting deformations of L-functions as p-adic L-functions is intriguing, but I don't know enough about them to see the content. Maybe worth it as an answer? | |
| Sep 14, 2011 at 19:18 | comment | added | B R | Rob, I wouldn't call that deforming the data at infinity, because $|\cdot|_\infty^a$ is not a Hecke character (thinking adelically). In order to shift the L-function, you need to use $\prod_{p\le\infty}|\cdot|_p^a$, in which case you are shifting everything by the same amount. I wasn't thinking about it, but it is fair to say that any automorphic representation $\pi$ (on $GL_2$ for simplicity) lies in the family of automorphic representations $\pi \otimes|\cdot|^s$ (using the Converse Theorem to get back to $GL_2$), but I think this just amounts to changing the central character. (cont) | |
| Sep 14, 2011 at 18:46 | comment | added | David Hansen | @Rob Harron: You can deform any L-function by $L(s,\pi)\mapsto L(s+it,\pi)$ for $\pi$ cuspidal on $GL_n$ and a fixed real $t$, and it's a conjecture of Sarnak that these are the only possible deformations (of standard L-functions of cusp forms anyway; anything reasonable should be a product of such). | |
| Sep 14, 2011 at 18:45 | comment | added | Gerhard Paseman | I also appreciate "out-of-curiosity" questions. The issue I have is similar to focus or precision. If this were a reference-request or something not warranting a blog-like discussion, I would say it was a better fit for MathOverflow. As a question by itself for some other forum, I see no problem with your initial post. Gerhard "Ask Me About System Design" Paseman, 2011.09.14 | |
| Sep 14, 2011 at 18:41 | comment | added | Gerhard Paseman | MathOverflow is for asking questions which have an answer (that is essentially focussed, easy to answer or redirect, and does not require writing Wikipedia articles, I believe). Your first question did not fit the above well, and could have as a potential (but silly and time-wasting) answer "well, map them into this set with these operations and get a hyperassociative semigroup, aren't you happy? I am." Motivation precludes some time-wasting; I'm glad you gave it; thank you. I'm still not sure of the question's fit; let's see. Gerhard "Does Appreciate The New Edits" Paseman, 2011.09.14 | |
| Sep 14, 2011 at 18:02 | history | edited | Daniel Larsson | CC BY-SA 3.0 |
added 6 characters in body
|
| Sep 14, 2011 at 17:31 | history | edited | Daniel Larsson | CC BY-SA 3.0 |
added 634 characters in body; added 19 characters in body
|
| Sep 14, 2011 at 17:16 | comment | added | Daniel Larsson | @BR: you have a very valid point and one that I can appreciate and fits nicely with my moduli analogy in my comment on Gerhard P's comment. | |
| Sep 14, 2011 at 17:14 | comment | added | Daniel Larsson | @Gerhard: Well, curiosity should be reason and motivation enough I would think. Also, I'm not asking for a definitive answer. I'm asking for how or if experts think about this. I mean it can't possibly be passed of as a particular original thought: some kind of "moduli" or classifying object for L-functions. Therefore there must be people with knowledge if this can be done in any sense of the word. Generally, I think people here are very quick to dismiss questions of the "out-of-curiosity"-type, which I always thoughy of as kind of strange as this is exactly what research is about. | |
| Sep 14, 2011 at 16:57 | comment | added | Rob Harron | @BR: you can deform the local data at infinity: For any complex $a$, $|\cdot|^a$ is a Hecke character, its $L$-function is $\zeta(s-a)$. Similarly, you should be able to deform $L$-functions of Eisenstein series, no? (I've enlarged the class of motivic $L$-functions to automorphic $L$-functions, but I assume the OP won't mind). If you want to deform at a finite place, then you can look at $p$-adic $L$-functions, which aren't quite usual $L$-functions, but are just as fun. | |
| Sep 14, 2011 at 16:22 | comment | added | user9072 | ...mean structure in an algebraic sense, but for example there is a notion of rank and conjectures that certain ranks cannot exist and this can be proved for certain ranges and so on. Perhaps I will try to write something more coherent (but as said I am not a good person to do so). In any case, in the nearest future I can't for external reasons. | |
| Sep 14, 2011 at 16:18 | comment | added | B R | the resulting $\chi'(x)=|x|_p^{s'_p-s_p}$ for all $x\in \mathbb Q$, so it would not be a Hecke character. Generally, you can see that deforming the local data would destroy the functional equation of the resulting "L-function" (and add an extraneous pole), so it can not be attached to anything that should have a nice L-function. | |
| Sep 14, 2011 at 16:18 | comment | added | B R | Daniel, if you deform the (local) data in an L-function, you almost certainly wind up with something that is not attached to anything global. Let's look at the $GL_1$ over $\mathbb Q$ case as an example. The Hecke L-function attached to a Hecke character $\chi=\prod_p\chi_p$ is $L(s,\chi)=\prod_p(1-\chi_p(p)p^{-s})^{-1}$, where the product is over places where $\chi_p$ is unramified, meaning $\chi_p=|\cdot|_p^{s_p}$ for some complex number $s_p$. Since $\chi$ is a Hecke character, $\chi(x)=1$ for all $x\in \mathbb Q$. If we were to perturb $\chi$ by changing $s_p$ to $s'_p$ at some prime ... | |
| Sep 14, 2011 at 16:15 | comment | added | user9072 | Disclaimer I am not the person to explain this: There is some structure on this class and/or an extension of it (extended Selberg class, S#); a semigroup structure and one can study unique factorization for example. See eg mathoverflow.net/questions/70518/… And there is various recent work to understand this class and/or an extension of it (extended Seleberg class, S#) in particular by Perelli and Kaczorowski (On the structure of the Selberg class, I-VII, perhaps menawhile more); though I believe 'structure' does not only or mainly ... | |
| Sep 14, 2011 at 16:15 | comment | added | Gerhard Paseman | If your question had some more motivation (why you are thinking of this, what would you do with such a space, what you hoped you could do with such a space), it might slip in under the wire as a reasonable MathOverflow question that has an answer. In its current form it falls way short, in my view. Please edit and refine. Gerhard "Ask Me About System Design" Paseman, 2011.09.14 | |
| Sep 14, 2011 at 16:04 | history | edited | Daniel Larsson | CC BY-SA 3.0 |
added 167 characters in body; added 1 characters in body
|
| Sep 14, 2011 at 15:52 | comment | added | Daniel Larsson | @quid: Well, almost. But it would be very nice to have some structure on this class (you know, vector space, blabla) not just a rather non-friendly enitity called "class". | |
| Sep 14, 2011 at 15:45 | comment | added | user9072 | I am not sure this is at all what you are looking for but perhaps the notion of Selberg class (Selberg beginning 90s) is of interest en.wikipedia.org/wiki/Selberg_class It gives an axiomatic definition of functions that conjecturally contains many L-functions. | |
| Sep 14, 2011 at 15:35 | comment | added | Daniel Larsson | Seriously, what you're saying is that there is no "continuous deformation" of an L-function (of something) that stays an L-function (of something, perhaps, else) under this deformation. Right? | |
| Sep 14, 2011 at 15:18 | history | edited | Daniel Larsson | CC BY-SA 3.0 |
edited title
|
| Sep 14, 2011 at 15:18 | comment | added | Daniel Larsson | But, say, vector spaces over finite fields are discrete... (hoping, hoping, hoping; I know I'm being naïve) | |
| Sep 14, 2011 at 15:12 | history | edited | Daniel Larsson | CC BY-SA 3.0 |
added 219 characters in body; added 2 characters in body
|
| Sep 14, 2011 at 15:10 | comment | added | David Hansen | Nothing that simple, unfortunately. In our current understanding they are discretely occuring in every sense of the word. However, they should conjecturally form a kind of "tensor category" (or at least automorphic representations should). | |
| Sep 14, 2011 at 15:06 | history | edited | Daniel Larsson | CC BY-SA 3.0 |
deleted 16 characters in body
|
| Sep 14, 2011 at 15:00 | history | asked | Daniel Larsson | CC BY-SA 3.0 |