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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?

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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!

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  • $\begingroup$ @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. $\endgroup$
    – Andrea Mori
    Commented Mar 16, 2010 at 17:51
  • $\begingroup$ 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. $\endgroup$
    – Kevin Buzzard
    Commented Mar 16, 2010 at 18:57
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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!]

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