Timeline for Teichmuller modular forms and number theory
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 22, 2016 at 10:50 | comment | added | Dan Petersen | @DavidRoberts no, not yet. | |
| Aug 22, 2016 at 9:39 | comment | added | David Roberts♦ | @DanPetersen did that paper ever turn up? I can't seem to find it. | |
| Feb 17, 2011 at 19:55 | comment | added | David Ben-Zvi | @JSE: Do you get something interesting by using other natural correspondences over moduli spaces (changing the genus)? eg you could extend the classical Hecke correspondences by considering covering spaces of Riemann surfaces, and look for collections of modular forms of all genus which have some eigenbehavior for this? or maybe allow orbicurves and their covers (ie branched coverings)? one could imagine that such modular forms coming from nice CFTs have such properties ("base change"). Or more likely this is an obviously silly thing to do. | |
| Dec 12, 2010 at 20:17 | comment | added | Dan Petersen | but in genus three one finds on $M_3$ manifestly non-Tate cohomology of weight 9, IIRC. In particular it lives in odd weight hence can not be attached to any kind of modular form on $A_3$. But it matches perfectly that it should be a Galois representation somehow attached to the Teichmuller modular form $\mu_{3,9}$. | |
| Dec 12, 2010 at 20:12 | comment | added | Dan Petersen | My advisor is Carel Faber. The Teichmuller modular forms have appeared in as of yet unpublished joint work with van der Geer and Bergström, but I have heard him give a talk about it so it should not be secret. :) @David Hansen: that is part of the mystery! What they have done is to study the cohomology of local systems on $M_3$. For the corresponding local systems in genus one and two one finds plenty of Tate cohomology, and some Galois representations attached to elliptic modular forms or (in general vector-valued) Siegel modular forms... | |
| Dec 12, 2010 at 19:12 | comment | added | Pete L. Clark | @Victor: if you click through to Dan's homepage, you'll find the answer to your question. | |
| Dec 12, 2010 at 18:40 | comment | added | David Hansen | You are saying some very provocative things! :) How do these objects have l-adic Galois representations attached to them? In what sense does the Galois representation "match" the Teichmuller modular form, if the latter doesn't have Hecke eigenvalues? | |
| Dec 12, 2010 at 18:16 | comment | added | JSE | One way to explain "why the Teichmuller modular forms don't have Hecke operators" is that an arithmetic lattice like Sp_{2g}(Z) has a massive commensurator (namely Sp_{2g}(Q)) but the abstract commensurator of the mapping class group -- which is where you might expect Hecke operators to "come from" -- is sadly not as rich as one might want. | |
| Dec 12, 2010 at 18:08 | vote | accept | David Feldman | ||
| Dec 12, 2010 at 17:21 | comment | added | Victor Protsak | Can you, please, reveal to the uninitiated who your advisor is? | |
| Dec 12, 2010 at 12:35 | history | edited | Dan Petersen | CC BY-SA 2.5 |
added 310 characters in body
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| Dec 12, 2010 at 11:43 | history | answered | Dan Petersen | CC BY-SA 2.5 |