Additive characters on topological fields

Stream: maths

[David Loeffler]

Suppose K is a topological field (or even just a topological ring) and $e: K \to \mathbb{C}^\times$ an additive character (i.e. converts addition in K to multiplication in $\mathbb{C}^\times$) which is continuous and unitary, i.e. lands in the unit circle inside $\mathbb{C}$.

In all the examples I can think of, e is periodic -- this covers the ur-example of $x \mapsto exp(2\pi i x)$ on $\mathbb{R}$ and also some more exotic examples (beloved of number theorists) whenK is a p-adic field. But I can't see a clear reason why this would be true in general. What hypotheses do I need to place on K to make this be true?

(Context: I have a proof formalised for the Riemann–Lebesgue lemma for functions on $\mathbb{R}$, and I'm trying to work out how far the argument can be generalised.)

[Eric Wieser]

Isn't $f : \mathbb{Z} \to \mathbb{C}^\times \coloneqq x \mapsto \exp(ix)$ a counterexample for the rings case? Nevermind, I misread and didn't see that you assumed continuity.

[Sebastien Gouezel]

A version of Riemann-Lebesgue which is definitely useful and I am not sure fits in your scheme: if f is integrable (for Lebesgue measure) on a finite-dimensional real vector space E, then $\int e^{i u(x)} f(x) dx$ tends to zero as $u$ tends to infinity in the dual space of E.

[David Loeffler]

Sebastien Gouezel said:

A version of Riemann-Lebesgue which is definitely useful and I am not sure fits in your scheme: if f is integrable (for Lebesgue measure) on a finite-dimensional real vector space E, then $\int e^{i u(x)} f(x) dx$ tends to zero as $u$ tends to infinity in the dual space of E.

Indeed. But the same holds, with essentially the same proof, if E is a fd vector space over an arbitrary field K and we replace $e^{i u(x)}$ with $\psi(u(x))$ for a periodic additive character $\psi$. The case $K = \mathbb{Q}_p$ is something I could imagine finding useful myself. Hence I'm trying to cast the net wide enough to include both of these generalisations.

[David Loeffler]

(PS: "arbitrary field" was overstating the case a bit – we're going to want a topology on K with some reasonable sanity properties and the point of my question was to work out which properties those should be – but what I'm trying to say is that it doesn't have to be $\mathbb{R}$.)

[Kevin Buzzard]

Eric's counterexample on Z (with the discrete topology) is fine though right? So the question is how to rule it out by placing assumptions on K? All continuous additive characters on the reals are periodic, for example.

[Kevin Buzzard]

I guess Eric's counterexample can be extended to Q with eg the subspace topology coming from R or the discrete topology. Does one ever care about a situation where K isn't a complete normed ring? Is complete normed ring enough to ensure periodicity?

[David Loeffler]

Does mathlib have the classification of locally-compact fields? (i.e. the theorem that a non-discrete topological field has to be isomorphic to a finite extension of $\mathbb{R}$, $\mathbb{Q}_p$ for some p, or $\mathbb{F}_p((t))$ for some $p$)? That might be the appropriate level of generality – without local compactness there's not much hope for setting up Fourier theory since there won't be a Haar measure to integrate against.

[Patrick Massot]

No, mathlib doesn't have this result.

[Filippo A. E. Nuccio]

David Loeffler said:

Does mathlib have the classification of locally-compact fields? (i.e. the theorem that a non-discrete topological field has to be isomorphic to a finite extension of $\mathbb{R}$, $\mathbb{Q}_p$ for some p, or $\mathbb{F}_p((t))$ for some $p$)? That might be the appropriate level of generality – without local compactness there's not much hope for setting up Fourier theory since there won't be a Haar measure to integrate against.

FWIW, this is something that we have in sight in our project with @María Inés de Frutos Fernández where we are defining local fields. We are not yet there, though.

[David Loeffler]

Filippo A. E. Nuccio said:

FWIW, this is something that we have in sight in our project with María Inés de Frutos Fernández where we are defining local fields. We are not yet there, though.

OK, looking forward to that! I think I will stick to vector spaces over $\mathbb{R}$ for now (although this goes against all my instincts as a p-adician...), and maybe revisit the case of other local fields later on once your work with Maria has made it into mathlib.