Timeline for Locally constant functions with compact support = smooth ?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 9, 2011 at 13:50 | comment | added | TaQ | I am not aware of such applications, and generally group representations and alike matters are beyond my intrests and expertise. By accident, I just wanted to point out the misleading terminology. This kind of inconsistencies are not rare in mathematics, and I did not really imagine to be able to change the convention. | |
| May 9, 2011 at 12:47 | comment | added | KConrad | TaQ: where has the notion of differentiability of maps from Q_p to C which you are advertising been used in applications (i.e., not in papers just about definitions)? Continuous homomorphisms from the additive group Q_p to C* are locally constant, and this illustrates why continuous locally constant functions are the basic functions of interest from p-adic domains to archimedean domains. Look at Tate's thesis for a nontrivial application of these concepts. You're not going to get people to change terminology at this point. | |
| May 9, 2011 at 12:02 | comment | added | TaQ | Already 70 years ago, A. D. Michal considered differentiability of maps between Abelian topological groups. So smoothness in this kind of general setting is not a new thing. See Proc. Nat. Acad. Sci. USA, Vol. 26, No. 5 (May 15, 1940), pp. 356-359, or jstor.org/pss/87250 . | |
| May 9, 2011 at 11:24 | comment | added | TaQ | In fact, if we consider BGN−smoothness associated with the setting where one has continuous maps, defined on open subsets, between topological modules over the discrete topological ring $\mathbb Z$ , or put otherwise, Abelian topological groups, then $f$ being smooth $\mathbb Q_p\supseteq U\to\mathbb C$ is equivalent to it being continuous. This is easily verified by directly appealing to the definition. If one wished to use a less misleading term in the context of complex representations of $p$−adic groups, one might call those maps for example nice. That would be even shorter! | |
| May 9, 2011 at 7:55 | comment | added | TaQ | I edited my answer to explain that it is meaningfull to speak of differentiability of maps between $p$-adic and the complex field. (Unfortunately, the latex server did not manage to do its job properly although I checked that the code is correct!) So the language "one is used to" really is confusing. | |
| May 9, 2011 at 7:44 | history | edited | TaQ | CC BY-SA 3.0 |
added 1206 characters in body
|
| May 8, 2011 at 23:31 | comment | added | KConrad | TaQ: The word "smooth" does not mean anything about differentiability in the context of functions from p-adic fields to the complex numbers. Why use such a misleading term? The functions one cares about in the classical setting (complex domain, complex codomain) are genuinely smooth, so we simply adopt the word smooth to refer to the relevant class of functions in the new setting (p-adic domain, complex codomain) even though it really is not as calculus-like as you might wish. As von Neumann would say, you just get used to it. | |
| May 8, 2011 at 17:56 | comment | added | TaQ | "So there's no calculus involved when studying smooth complex (or $l$-adic) representations of a $p$-adic group." Why then use such misleading phrases as $C^\infty$, "smooth"? You already referred to this in you answer. $C^\infty$ and "smooth" in the sense of some kind of differentiability are meaningfull only when basically speaking of a map $f:X\supseteq U\to Y$ where $X,Y$ are (topological or otherwise suitably structured) modules over the same (structured) ring, or more specifically, a field. | |
| May 8, 2011 at 17:51 | history | edited | TaQ | CC BY-SA 3.0 |
added 18 characters in body
|
| May 8, 2011 at 12:46 | comment | added | Joël Cohen | But the point here is that $\mathbb{C}$ is not an $F$-algebra. So there's no calculus involved when studying smooth complex (or $l$-adic) representations of a $p$-adic group. Now when you're interested in $p$-adic representations of $p$-adic groups, that a whole different story (and your paper is indeed useful for that case). | |
| May 8, 2011 at 11:37 | history | answered | TaQ | CC BY-SA 3.0 |