Timeline for Fontaine-Mazur for GL_1
Current License: CC BY-SA 2.5
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 27 at 2:27 | comment | added | Phanpu | @Aurel Here is the pdf version webusers.imj-prg.fr/~laurent.fargues/Motifs_abeliens.pdf | |
| Oct 3, 2018 at 4:45 | comment | added | Aurel | New link to Laurent Fargues's paper: webusers.imj-prg.fr/~laurent.fargues/Motifs_abeliens.ps | |
| Apr 17, 2011 at 12:28 | comment | added | SGP | @Laurent F.: Thanks for the very nice and very readable paper. | |
| Aug 14, 2010 at 12:28 | comment | added | Peter Scholze | Indeed, a great reference! Thanks a lot! | |
| Jul 30, 2010 at 11:01 | comment | added | BCnrd | Laurent, great reference -- I will print it out and take a look at it, but in the meantime a question comes to mind: do you plan to publish it somewhere, say in a journal devoted to expository articles (or elsewhere)? | |
| Jul 29, 2010 at 8:53 | comment | added | Laurent F. | Peter, see math.u-psud.fr/~fargues/Motifs_abeliens.ps section 8 where you can find the references for FM conjecture for GL_1 (in particular you have to use a transcendence result in general). | |
| Jul 28, 2010 at 1:34 | comment | added | Rob Harron | Indeed BCnrd, it would be great to see the proof somewhere! And I encourage you to put it in the Hawaii notes, though that's not necessarily the first place I'd think to look for a proof of Fontaine–Mazur for GL(1)... Good luck finishing up the notes, the preliminary version is already great!! | |
| Jul 27, 2010 at 23:00 | comment | added | BCnrd | Dear Rob H. & Junkie: It is kind of embarrassing for the field that with all of the recent activity around the F-M conjecture there isn't a place in the recent (= last 15 years) literature where the 1-dimensional case is specifically addressed and proved (even if the "proof" amounts to gathering appropriate earlier references and explaining why they are sufficient for the desired conclusion). Hmm, maybe I should put this in the Hawaii lecture notes (oy vey, just 2 more months till that's due to be done...). | |
| Jul 27, 2010 at 16:29 | comment | added | Rob Harron | Yes, I was hoping Blasius' paper attached motives to algebraic Hecke characters in the process, but I see from this excellent paper of Harder–Schappacher that you've found that he instead uses a trick. Thanks for that paper it's quite nice. | |
| Jul 27, 2010 at 10:44 | comment | added | Junkie | Blasius does the CM case of Deligne's conjecture, and Harder/Schappacher (Section 5) then show this for all Hecke L-functions of totally imaginary fields (not just CM fields) using this. Siegel (as noted by Serre in an Annals paper of Coates/Lichtenbaum) did the totally real case, and the other cases are trivial. Not exactly Fontaine/Mazur, but.... $$ $$ Harder/Schappacher dx.doi.org/10.1007/BFb0084583 $$ $$ Goldstein and Schappacher (Crelle paper) had already done this in many cases, essentially following the Damerell methods as developed by Weil and Shimura and others. | |
| Jul 26, 2010 at 14:56 | comment | added | Rob Harron | Quick comment (more later): I didn't mean to imply the CM case was easy to handle or had been, I just wanted to reduce to the CM case. My reading of the question hadn't noticed the parenthetical emphasizing the fact that the OP didn't know how to deal with ALL CM alg Hecke characters. I would've guessed that either "Motifs et groupes de Taniyama" in Deligne–Milne–Ogus–Shih" or Blasius' 1986 article "On the critical values of Hecke L-series" would address the question of obtaining all algebraic Hecke characters over a CM field, though perhaps neither makes sure they get ALL Hecke characters. | |
| Jul 26, 2010 at 8:47 | comment | added | Junkie | OK, I see the problem now. I can add that, At the bottom of section 1 of chapter 1, Schappacher also emphasizes "all" when he says $$ $$ In order to find "geometric" objects over $K$ whose $L$-functions include ALL $L$-functions of algebraic Hecke characters of $K$ we have to pass from abelian varieties (with complex multiplication) to motives. | |
| Jul 26, 2010 at 8:20 | comment | added | BCnrd | Dear Junkie: I had in mind Casselman's thm (though I didn't know such a result was proved by Casselman, and unfortunately I find Shimura's writing hard to unravel...pre-modern to me), but it doesn't fit what Rob H. said since it entails some extra hypotheses on the alg. Hecke character whereas Rob H. he makes a statement on the structure of all algebraic Hecke characters which I had not heard before. So I wanted to know if Rob H. has a "modern" reference to justify what he said. The Schappacher reference seems more suitable (ultimately relying on Casselman's thm); thanks. | |
| Jul 26, 2010 at 7:16 | comment | added | Junkie | "but to conversely relate all alg. Hecke characters to CM abelian varieties is totally not obvious (& there's no "modern" published ref.), right?" $$ $$ Do you just mean Casselman's theorem as in Shimura's paper? That is, Theorem 6 of jstor.org/stable/1970768 $$ $$ Maybe section 4 of chapter 1 of Schappacher is also of interest, for it deals with motives. springer.com/mathematics/numbers/book/978-3-540-18915-2 | |
| Jul 25, 2010 at 20:26 | comment | added | BCnrd | Rob, should mention refs, such as Ch. III (esp. App. A) in Serre's book on abelian $\ell$-adic repn's, for why pst (or HT) at $p$-adic places implies global algebraicity up to finite-order, with finite-order part coming from ramification away from $p$ (finite, by global CFT) and finiteness of Hilbert class fields. Main Thm of CM relates Tate mods of CM abelian var. to certain alg. Hecke characters factoring through reflex norm of a CM type, but to conversely relate all alg. Hecke characters to CM abelian varieties is totally not obvious (& there's no "modern" published ref.), right? | |
| Jul 25, 2010 at 20:21 | vote | accept | Peter Scholze | ||
| Jul 25, 2010 at 20:21 | comment | added | Peter Scholze | Thanks for your answer! I wasn't aware that algebraic characters have this special form. | |
| Jul 25, 2010 at 17:24 | history | edited | Rob Harron | CC BY-SA 2.5 |
change notation to avoid double-subscripts
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| Jul 25, 2010 at 16:49 | history | answered | Rob Harron | CC BY-SA 2.5 |