Timeline for Local root numbers of the Hecke character associated with some specific CM elliptic curves, should they be some roots of unity?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 9, 2018 at 7:44 | comment | added | Taekyung Kim | @ChrisWuthrich For example once Cesnavicius used this argument before (Proof of Proposition 6.3). Since the root number of $E/K$ is $\pm 1$, we expect that the root number of $\psi_v$ must be in $\mu_4$. I suspect that we surely make mistakes somewhere though. | |
| Dec 9, 2018 at 7:38 | comment | added | Taekyung Kim | @ChrisWuthrich Thank you very much for your comment. In our situation, the elliptic curve and the corresponding character $\psi$ is defined over $K$. The property $V_\ell E \cong \psi_v \oplus \psi_v$ remains valid in this case, because the $\ell^n$-torsion points of $E$ decompose into $E[L^n] \oplus E[L'^n]$ where $L, L'$ are the two primes in $K$ lying over $\ell$ ([Rubin] Proposition 5.4 and the Chinese Remainder Theorem) and the Galois action on these components are given by $\psi_v$ ([Rubin] Theorem 5.15(ii)). | |
| Dec 7, 2018 at 14:24 | comment | added | Chris Wuthrich | Oh sorry, my bad. Ignore my first comment. I think $V_{\ell} E$ is $\psi \oplus \bar\psi$ as a $G_{\mathbb{Q}}$ module. Your reference is about $G_F$-modules where $E$ has cm defined over $F$. Now the root numbers will be complex conjugates that multipy to the root number of $E$, which is $\pm 1$, but I don't think there is a reason that they are in $\mu_4$. | |
| Dec 7, 2018 at 13:09 | comment | added | Chris Wuthrich | Why do you expect the root number of $\psi$ to be $\pm1$? The $L$-function of $E$ is a product of two $L$-functions, one of $\psi$ and its conjugate. | |
| Dec 7, 2018 at 11:17 | answer | added | literature-searcher | timeline score: 1 | |
| Dec 7, 2018 at 11:05 | review | First posts | |||
| Dec 7, 2018 at 11:31 | |||||
| Dec 7, 2018 at 11:02 | history | asked | Taekyung Kim | CC BY-SA 4.0 |