# Locally profinite fields ?

This is a question about terminology and should be taken lightly.

The expression local field is used in at least three different senses :

1) For a locally compact totally disconnected field. These are the finite extensions of $\mathbb{Q}_p$ or of $\mathbb{F}_p((T))$, where $p$ is a prime number and $T$ is an indeterminate.

2) For a field complete with respect to a discrete valuation whose residue field is merely perfect, not necessarily finite as in 1). This is how Fontaine uses the expression.

3) For locally compact fields which are not discrete. These are the fields in 1), but also $\mathbb{R}$ and $\mathbb{C}$ in addition. People who adopt this definition refer to the fields in 1) as non-Archimedean local fields.

(Nobody insists that a local field is a local ring which happens to be a field, but the "logic" is impeccable.)

Question. Would locally profinite field be a good piece of terminology for the fields in 1) ?

This would certainly avoid the confusion with the other fields in 2) and 3).

The expression locally profinite group is already in use (for example in the book Bushnell-Henniart). The additive and the multiplicative groups of a locally profinite field would be locally profinite groups in their sense.

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Hausdorffness is a global condition, so it's not included in "locally compact", is it? – Harry Gindi Mar 4 '10 at 7:38
It is. For most of the world, compact spaces are separated by definition. If they had not been, fpqc would have been fpc... – Chandan Singh Dalawat Mar 4 '10 at 8:03
Please, let's not start this argument up again! [@fpqc: you're right that locally Hausdorff does not imply Hausdorff, since otherwise non-Hausdorff manifolds wouldn't exist, and they do. Chandan is using the convention "locally compact" = "Hausdorff and locally quasi-compact", which is again a standard one, especially among Europeans.] – Pete L. Clark Mar 4 '10 at 8:08
@Chandan: sorry, I don't like the term "locally profinite field", although it is perfectly correct and self-evident. I usually just say "locally compact field", with the belief that the context will make clear whether I mean to include R and C or not. [Personal anecdote: I used this terminology at the beginning of a 2-hour talk on WC-groups to a very distinguished audience at MSRI. Bjorn Poonen immediately asked, "Do you mean to allow the field to be discrete?" Sigh. Yet another hard lesson on saying exactly what you mean.] – Pete L. Clark Mar 4 '10 at 8:15
I like letters to stand for mathematical objects. A word such as $l$-group'' leads me to think of a prime $l$ and a group of order $l^n$ for some $n\in\mathbb{N}$. – Chandan Singh Dalawat Mar 4 '10 at 8:58

I was going to leave this as a comment, but I have a firm conviction about this, so here's my answer. The fields in your (1) are called (a) locally compact non-archimedean fields, or (b) non-archimedean local fields, and I fear that attempts to use other terminology (however logical) might lead to confusion. (A "locally profinite field" sounds like it should admit an open profinite subfield, which doesn't make sense.) Which is not to say that authors vary in how they refer to such objects, but always there is complete precision. Eg:

Lubin and Tate, Formal Complex Multiplication in Local Fields: "Let $k$ be a field complete with respect to a discrete valuation, with finite residue field..."

Harris, On the Local Langlands Correspondence: "The local Langlands correspondence for GL(n) of a non-Archimedean local field $F$ parametrizes irreducible admissible representations..."

Bushnell and Henniart, Calculs De Facteurs Epsilon De Paires Pour GL(n) Sur Un Corps Local, I: "Soit $F$ un corps commutatif localement compact non archimédien; notons $p$ sa caractéristique résiduelle et $q$ le cardinal de son corps résiduel."

Whether you want to capitalize "archimedean" is another story. But the moral is: defer to tradition and leave no doubt in the reader's mind as to what you mean.

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A "locally profinite field" sounds like it should admit an open profinite subfield, A locally compact field does not admit an open compact subfield... – Chandan Singh Dalawat Mar 4 '10 at 9:17
I'm not advocating the use of the expression "locally profinite field" without any explanation. You will need to define the notion and prove that fields in 1) are the only ones which qualify. My question is about the linguistic merits of the expression. – Chandan Singh Dalawat Mar 4 '10 at 9:25
Point well taken. By no means is it necessarily a bad phrase. I only meant that there could be some ambiguity in how the phrase is interpreted (whereas there is much precedent for "locally compact group"). – Jared Weinstein Mar 4 '10 at 9:33