Timeline for Image of complex conjugation by modular representations in characteristic 2
Current License: CC BY-SA 3.0
6 events
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| Oct 20, 2011 at 22:35 | comment | added | Kevin Buzzard | Re: the $\Delta$ function mod 2: I proved this when I was a graduate student, using work of Swinnerton-Dyer which explicitly computes the image of the 2-adic representation. I don't know of a reference other than chapter 6 of my thesis: www2.imperial.ac.uk/~buzzard/maths/research/notes/phd.dvi There is a funny historical story around why I was thinking about this, but it's more than 500 characters so I'll send it to you by email. | |
| Oct 20, 2011 at 22:32 | comment | added | Joël | Kevin, Thanks a lot. I am trying to understand your example with $\Delta$. I see that the semi-simplification of $\bar \rho_\Lambda$ is the trivial representation (since $\Delta \equiv \sum q^{(2n+1)^2} \pmod{2}$ (Jacobi), all Hecke eigenvalues are $0=2$). By Ribet's lemma+epsilon, there is at least 2 lattices $\Lambda$ such that $\rhob_{\Lambda}$ is a non-trivial extension of the trivial character by itself.But how do you prove the existence of one such $\Lambda$ with $\bar \rho_\Lambda(c) \neq \Id$ ? | |
| Oct 20, 2011 at 22:21 | comment | added | Joël | Tommaso: right. If $\bar \rho_\Lambda$ is irreducible, then it is independent of $\Lambda$ by Brauer-Nesbitt, and actually but is not hard to prove that there is only one stable lattice $\Lambda$ (up to a scalar). | |
| Oct 20, 2011 at 21:05 | comment | added | Tommaso Centeleghe | Kevin, when you say that "both possibilities can occur even for the same modular form" then in this case the global, mod $2$ representation must be reducible, right? | |
| Oct 20, 2011 at 18:31 | history | edited | Kevin Buzzard | CC BY-SA 3.0 |
clarifications
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| Oct 20, 2011 at 18:13 | history | answered | Kevin Buzzard | CC BY-SA 3.0 |