Timeline for p-adic L-functions
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Apr 30, 2010 at 4:24 | comment | added | schur | @Emmerton: Thanks so much for your answer! So has GL_n been done over Q, or is this also part of your paper? | |
| Apr 23, 2010 at 16:45 | comment | added | Rob Harron | Katz's result, as well as what the first Harris–Li–Skinner paper states, restricts to the "ordinary" case (for Katz, this is that every prime above $p$ splits in the CM field). Has there been any work generalizing to the non-ordinary case (for Größencharaktere/automorphic forms on unitary groups)? The symmmetric square $p$-adic $L$-functions have been constructed for non-ordinary newforms, for example (Dabrowski–Delbourgo (ams.org/mathscinet-getitem?mr=1434442)). | |
| Apr 23, 2010 at 12:35 | comment | added | Junkie | "Comment on your answer: for Grossencharacters of any field, one can consider the job done now, because for silly reasons they vanish identically if the field isn't totally real or CM." Is your terminology like mine? Grossencharacters for totally imaginary but not CM fields exist, but factor through the norm down to the CM-subfield. With an example, the Deligne conjecture was proved by Blasius for CM fields and Harder for totally imaginary. The paper of Harder and Schappacher discusses this, page 36 and on to 43. dx.doi.org/10.1007/BFb0084583 | |
| Apr 23, 2010 at 7:23 | comment | added | David Loeffler | See Olivier's answer to my question [mathoverflow.net/questions/18884/… here]. The verdict there seems to be that there are examples where interpolating values of classical L-functions gives you zero, but if you assume a whole constellation of conjectures you might still hope to be able to interpolate leading terms of classical L-functions. | |
| Apr 23, 2010 at 6:47 | comment | added | Kevin Buzzard | Comment on your answer: for Grossencharacters of any field, one can consider the job done now, because for silly reasons they vanish identically if the field isn't totally real or CM. Presumably there is a conjectural analogue of this statement for GL_n, so, in particular, just sticking to GL_n over a CM field isn't as restrictive as it might sound perhaps? | |
| Apr 23, 2010 at 6:45 | comment | added | Kevin Buzzard | @Emerton: let me ask a related question. If G is connected reductive over a global field, and pi is an automorphic representation for G, then it's not pi that has an L-function, but the pair pi,rho, where rho is a representation of the L-group of G. If we believe functoriality for a second, this seems to reduce us to the case of GL_n (because rho_*(pi) will be an auto rep of GL_n). Can you see how anticyclotomic p-adic L-functions for ell curves fit into this story? Is it somehow one part of a conjectural 2-variable p-adic L-function? | |
| Apr 23, 2010 at 5:41 | history | answered | Emerton | CC BY-SA 2.5 |