Timeline for Katz Modular Functions and Emerton's Completed Cohomology
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Jun 24, 2011 at 23:23 | vote | accept | ndk | ||
| Jun 24, 2011 at 23:23 | vote | accept | ndk | ||
| S Jun 24, 2011 at 23:23 | |||||
| Jun 24, 2011 at 12:01 | comment | added | Emerton | ... computing modular symbols for $X_0(11)$ (say) by hand, directly computing a matrix of $T_2, T_3$, etc., finding the eigenvalues, and then discovering the $q$-expansion of $q\prod_n (1-q^n)^2(1-q^{11n})2$ coming out, one coefficient at a time! It makes the objects in question appear much more concrete (or at least, it had that effect with me). Best wishes, and good luck, Matthew | |
| Jun 24, 2011 at 11:58 | comment | added | Emerton | Dear ndk, While Langlands's Antwerp paper was pivotal, it's not something one should read as a beginner. Have you read Scholl's paper in Inventiones? This follows Deligne's 1969 paper quite closely, but is more detailed and (I think) easier to understand for a newcomer. Another possibility is to learn the theory of modular symbols: this makes the cohomology of modular curves a quite concrete object, and so dispells some of the mystery. Of course, you won't see the Galois action concretely, but you will see Eichler--Shimura theory. I think there is something very satisfying about ... | |
| Jun 24, 2011 at 6:04 | comment | added | ndk | Dear Matt. Thanks for your answer. I'm not very clear with all the cohomology theories that people are using. A few months ago, I tried to read Deligne's paper (in 1969) that constructed l-adic representation from eigenforms, and then Langlands' one in the Antwerp II. Both attempts failed miserably... | |
| Jun 24, 2011 at 5:58 | vote | accept | ndk | ||
| Jun 24, 2011 at 23:23 | |||||
| Jun 23, 2011 at 21:16 | comment | added | Emerton | Dear ndk, There is not terribly much difference between the different flavours of cohomology (usual, compactly supported, open curves, closed curves). The main difference between completed cohomology and $p$-adic modular forms is that the former have an action of $GL_2(\mathbb Q_p)$, while the latter just have an action of $U_p$. Regards, Matthew | |
| Jun 23, 2011 at 7:17 | answer | added | Emerton | timeline score: 30 | |
| Jun 22, 2011 at 17:12 | comment | added | ndk | Dear David, I take it back, because: 1) Emerton used the non-compactified (instead of the compactified) modular curve to define his completed cohomology group. 2) As Joel and Kevin pointed below, even we somehow have a natural map of C_p vector spaces, it's still not a good reason to believe there's a natural map between the Z_p modules. | |
| Jun 22, 2011 at 16:51 | answer | added | Joël | timeline score: 8 | |
| Jun 22, 2011 at 6:08 | vote | accept | ndk | ||
| Jun 22, 2011 at 6:08 | |||||
| Jun 22, 2011 at 6:07 | comment | added | David Hansen | Agh, sorry, Iovita's map is not an isomorphism; I misremembered his title. | |
| Jun 22, 2011 at 5:36 | answer | added | Kevin Buzzard | timeline score: 13 | |
| Jun 22, 2011 at 5:13 | comment | added | ndk | Here's a brief reason for my "belief", hope there's no mistake: Let N and p be nice, and we define the space V as in Gouvea's book. Instead of taking the whole completed cohomology as in p.45 of Emerton's paper, we choose the "smaller piece" H^1(K^p)_A in p.44, where K^p is the Gamma_1(N)-problem, and A is Z_p. I think there's an embedding after tensoring both of these Z_p-modules by C_p. | |
| Jun 22, 2011 at 5:10 | comment | added | David Hansen | Dear ndk: Fine, but why do you believe this? Take a look at the final section of Emerton's 2006 Inventiones paper, and the rather circuitous route by which he compares his construction for $GL2/\mathbf{Q}$ to the Coleman-Mazur eigencurve. Anyway, here is the video for Iovita's talk - video.ias.edu/graf/2011/03/21/iovita - which I for one found quite clear and enjoyable. | |
| Jun 22, 2011 at 4:18 | comment | added | ndk | Dear David. Somehow, I believe the answer is yes. | |
| Jun 22, 2011 at 3:57 | comment | added | David Hansen | You should check out Iovita's talk, "An overconvergent Eichler-Shimura isomorphism", at the IAS Galois representations and modular forms conference this past March, for some ideas of what results of this flavor might be true (and are provably true!). But I am fairly sure the answer to your actual question is "no", or at the very least "the actual relation between overconvergent modular forms and Emerton's completed cohomology groups is rather mysterious, aside from the fact that they seem to be producing the same Hecke eigenpackets in certain situations". | |
| Jun 22, 2011 at 0:48 | history | asked | ndk | CC BY-SA 3.0 |