Timeline for Complex isomorphism class of abelian varieties and $L$-functions
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 30, 2019 at 1:55 | comment | added | Will Sawin | @reuns This is correct. | |
| Jun 29, 2019 at 22:59 | comment | added | reuns | @FrançoisBrunault So are you saying, for an elliptic curve $E/\Bbb{Q}$, omitting the finitely many bad reduction and ramified primes $L(E/K,s)=\prod_{p\text{ good}} \exp(\sum_{k \ge 1} \frac{p^{-sk}}{k}tr(\Pi(Frob_{p,L/\Bbb{Q}})^k) tr(\rho_K (Frob_{p,L/\Bbb{Q}})^k)$ with $Frob_{p,L/\Bbb{Q}}\in Gal(L/\Bbb{Q})$ the Frobenius of $O_L/P$, $L$ the normal closure of $K/\Bbb{Q}$, $\Pi(\sigma) \in GL_2(E[\ell^\infty])= GL_2(\Bbb{Z}_\ell)$ and $\rho_K$ the representation of $Gal(L/\Bbb{Q})$ permuting $Gal(L/\Bbb{Q})/Gal(L/K)$ ? | |
| Jun 29, 2019 at 21:32 | comment | added | François Brunault | If the extension is Galois then the L-function of the base change is just the product of the original L-function twisted by the Artin representations, with multiplicities equal to the dimensions (as in the decomposition of the regular representation). | |
| Jun 29, 2019 at 21:19 | comment | added | Sylvain JULIEN | Possibly related: mathoverflow.net/questions/149815/… | |
| Jun 29, 2019 at 20:28 | history | edited | Stanley Yao Xiao | CC BY-SA 4.0 |
added 113 characters in body
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| Jun 29, 2019 at 20:02 | history | asked | Stanley Yao Xiao | CC BY-SA 4.0 |