Timeline for Why is image of prime ideal under Hecke (Grossencharacter) character is prime element of the local field?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 30, 2022 at 12:31 | comment | added | Duality | Is that something to do with relation between $ψ(I)$ and $I$ ? | |
| Aug 30, 2022 at 6:15 | comment | added | Duality | @David Loeffler I didn't know that concept. | |
| Aug 29, 2022 at 19:07 | comment | added | David Loeffler | @dandelion: do you know what the infinity-type of $\psi$ is (and do you know what those words mean)? | |
| Aug 29, 2022 at 19:06 | comment | added | David Loeffler | @bouyant: the Hecke character associated to an elliptic curve over K with CM by K takes values in $K^\times$. If you don't understand this, please ask a new question, rather than jumping in on somebody else's. | |
| Aug 29, 2022 at 13:09 | comment | added | Duality | Ok, thank you. Then $1/ord_I(a)$ should be $1$ because it takes value group $ \Bbb{Z}$. Using your argument, can we relate $ψ(I)$ and $I$ ? $ψ(I)＝λ^{1/h}$, and because $p^h＝λO_K$, can we describe some relation between $ψ(I)$ and $I$ ? | |
| Aug 29, 2022 at 12:50 | comment | added | Duality | @David Leffler By the way, codomain of Hecke character is $ \Bbb{C}^{×}$. ' $ψ(I)$ is prime of $K_I$ ' makes no sense to me because the codomain is not $K_I$ which is not embedded to $ \Bbb{C}^{×}$, where am I missing ? | |
| Aug 29, 2022 at 8:19 | comment | added | David Loeffler | You're trying to prove $\psi(I)$ is a prime element, so you definitely don't want its valuation to be 0, you want it to be 1. | |
| Aug 29, 2022 at 8:11 | comment | added | Duality | I want to prove is $ord_I(ψ(I))＝0$. $ord_I(ψ(I)^h)＝ord_I(λ)$ so $h・ord_I(ψ(I))＝ord_I(λ)$. On the other hand, $h＝ord_I(λ)ord_I(a)$ for some ideal $a$ thus $ord_I(ψ(I))＝1/ord_I(a)$ , but this cannot be $0$ so I'm confused. | |
| Aug 26, 2022 at 6:08 | comment | added | David Loeffler | Sorry, I meant to let $h$ be the order of the ray class group modulo the conductor $f$ of $\psi$. Then $I^h$ is trivial in the ray class group, so $I^h = (\lambda)$ for some $\lambda \in K^\times$ with $\lambda = 1 \bmod f$. Hence $\psi(I)^h = \lambda$. So what can you say about the valuation of $\psi(I)$ at the prime $I$? Or at any other prime? | |
| Aug 26, 2022 at 3:06 | comment | added | Duality | Could you give me one more hint ? | |
| Aug 25, 2022 at 15:25 | comment | added | Duality | In this situation, automatically $h＝1$, so it is just $ψ(I)$. | |
| Aug 25, 2022 at 6:13 | comment | added | David Loeffler | Hit: if $h$ is the class number of $K$, then what can you say about $\psi(I)^h$? | |
| Aug 25, 2022 at 3:59 | history | asked | Duality | CC BY-SA 4.0 |