Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $K$ be a quadratic imaginary field, $\bf n$ an ideal in the ring of integers ${\cal O}_K$ and $\xi$ an algebraic Hecke character of type $(A_0)$ for the modulus $\bf n$. One knows (from Weil) that there exists a number field $E=E_\xi\supseteq K$ with the property that $\xi$ takes values in $E^\times$.

Let $p$ be a prime that splits in $K$. Consider the following condition: there exists an unramified place $v\mid p$ in $E$ with residue field $k_v={\Bbb F}_p$ such that $\xi$ takes values in the group of $v$-units in $E$.

The condition implies the existence of a $p$-adic avatar of $\xi$ with values in ${\Bbb Z}_p^\times$.

I would like to know:

1) to the best of your knowledge, has been this condition considered somewhere? does it have a "name"?

2) I'm tempted to say that $\xi$ is $p$-split if the condition is satisfied (and that $v$ splits $\xi$). Would this name conflict with other situations that I should be aware of?

share|improve this question
add comment

2 Answers

I think your $\xi$ had better be algebraic, but perhaps this implicit somehow in your terminology. If $v$ is any finite place of $E$, there is a $p$-adic avatar of $\xi$ with values in $E_v^\times$ (whether or not $p$ splits in $K$). This construction is as far as I know due to Weil. You seem to be highlighting this construction in the special case $E_v=\mathbf{Q}_p$. I guess $E$ is called the coefficient field of $\xi$ and you're just asking that $E$ contains a prime above $p$ which is unramified of degree 1. What am I saying? I'm saying that your condition above seems to me to have nothing to do with $\xi$, it's simply asking for a name for primes of a number field whose completion is $\mathbf{Q}_p$. I don't see why $\xi$ should enter into the terminology at all. Let me know if I have misunderstood!

share|improve this answer
@Kevin: yes, character of type $(A_0)$. I'll edit the question accordingly. About your other comment(s): the field of coefficients $E$ does depend on $\xi$. Hence, what $p$'s may work depends on $\xi$ as well. Also, if you fix $p$ in the first place then some $\xi$'s will satisfy the condition, some won't. Thus I feel that $\xi$ should enter into the terminology. –  Andrea Mori Mar 16 '10 at 17:51
I think my main point is that whether or not p splits in K, or what the degree of K is, is irrelevant. Your condition is purely a condition on E, as you clearly know. I don't have anything to say about the terminology I'm afraid---perhaps this should have been a comment rather than an answer! The reason I answered was that I wanted to make sure that you knew that even if E_v wasn't Q_p, everything you said was true, and the avatar takes values in the units of E_v. –  Kevin Buzzard Mar 16 '10 at 18:57
add comment

Just to second Kevin's comment: you are considering the coefficient field $E$ of $\xi$, or perhaps you could also call it the field of definition. It happens that you are considering $p$ which split in $E$. I would call them "primes that split in the coefficient field of $\xi$", or just "primes that split in $E$" (as Kevin suggests), if $E$ has already been introduced. Anything else is a little ambiguous, and non-standard, I think. (It is not uncommon in this context, and in other arguments involving coefficients of motives, to consider primes with various splitting properties in the field of coefficients, and I think it is common to just use the usual algebraic number theoretic terminology with regard to this field, as Kevin suggests.)

[This would have been a comment on Kevin's answer, but the comment box is too small!]

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.