Is the Leopoldt conjecture almost always true?

Question

The famous Leopoldt conjecture asserts that for any number field $F$ and any prime $p$, the $p$-adic regulator of $F$ is nonzero. This is known to be equivalent to the vanishing of $H^2(G_{F'/F},\mathbf{Q}_p/\mathbf{Z}_p)$, where $F'$ is the maximal pro-$p$ extension of $F$ unramified outside $p$. When $F/\mathbf{Q}$ is abelian, the conjecture was proven by Brumer.

My question: is there any reasonable sense in which the Leopoldt conjecture is "usually" true - e.g., is it known for any fixed $F$ at almost all primes $p$, or (say) for almost all quartic extensions of $\mathbf{Q}$ with $p$ fixed? A glance through the mathscinet reviews of all the papers with "Leopoldt conjecture" in their title didn't reveal anything, but perhaps this is well-known to experts.

Answer 1

Olivier and all,

If you trust your own minds, you should better try directly and read version 2 of the proof for only CM fields, which I posted in June this years. The rest is hot wind - people may bet, in front of the list of names who failed at Leopoldt you may put the odds for my breakthrough around 1% - but it is only reading which can provide your own judgment of whether this 1 has to be more likely than the complementary 99. I teach the proof in class since 3 weeks and it works quite fluidly and the students can grab the construction very well - useless to say, it is enriched by many details, since it is a 3-d year course (guess something like first graduate year). I gave up the construction of techniques for non CM fields, the Iwasawa skew symmetric pairing, and reduced to the skeletton of the principal ideas, exactly in order to respond to the loud whispers about my expressivity.

As for the Cambridge seminar mentioned, it was a great experience - but it happened during a week loaded with important other seminars, and in spite of the particular attention offered, we did not have more than 3 or 4 meetings of two hours, this was certainly not enough for completing a proof with all the details, just the time for gathering some important questions and find out on what particular issue people would like to know more. This is taken into account in the present version.

It is also true that Minhyong Kim, this friendly and enthusiastic fellow, asked for my allowance to put the draft on the blog, exactly in the expectation that more students and young researchers would just try and bite at it, and raise questions, which were very welcome. The expected impact did not happen. Therefore I friendly invite you to simply read. Anyone having concrete questions is gladly invited to write me a mail, if I understand the question his chances are one in a thousand that I will not respond.

Sorry if I intruded your discussion

Preda Mihailescu

Answer 2

I think that not much is known. For example I don't think that we are any closer to prove Leopoldt's conjecture for a given $F$ for infinitely many $p$ than to prove it for all $p$.

Here is a result though: for $K_n=$ cyclotomic fields generated over $\bf Q$ (variante: over a fixed quadrqtic imqginary field; over a fixed totally real field) by the roots of unity of order $p^n$, the "defect" of Leopoldt's conjecture (the dimension of $H^2(K_n'/K_n,{\bf Q}_p)$) stays bounded as $n$ goes to infinity. This is a consequence of the main conjecture known in this case and in the variantes. This is already a very useful result (used for example by Minhyong Kim in his beautifull new proofs of old Diophantine results, such as Siegel theorem for a CM elliptic curve).

Not really in the same spirit, but somehow similar to the brummer proof of the abelian case; it is important to mention Waldschmidt's beautiful result, that the defect in Leopoldt's conjecture is at most half of the degree of $F$.

Since both Brumer's result and Waldschmidt's have proof using fundamentally the theory of transcendence, and for other reasons as well, many people (including myself) think that proving Leopoldt's conjecture will require some transcendence methods (as opposed as methods of algebraic number theory, automorphic forms, etc.). But "generic results" as asked by the question might be more accessible, if by no means simple.

Answer 3

Leopoldt's conjecture seems to have been proved now by Mihailescu:

http://arxiv.org/abs/0905.1274

In fact before that, I think Fujiwara had done significant work on it (maybe the case of totally real fields?).

Answer 4

Fix $p$. Leopoldt's conjecture is that the $p$-adic regulator does not vanish. Since that is an open condition in the $p$-adic numbers, the set of fields on which the conjecture is true ought to be open. That sounds like a pretty good avatar of your question, although it doesn’t give density like the Zariski topology. Also, this is only a conjecture.

What does that even mean? Define the $p$-adic topology on the set of number fields by way of test curves, which are parameterized by finite extensions of $\mathbb Q(t)$. Each unramified rational value of $t$ gives an extension of $\mathbb Q$. So each such test curve defines a set of number fields and identifies it with a cofinite subset of $\mathbb Q$. We can transfer the $p$-adic topology on $\mathbb Q$ to the set of number fields. A subset of fields is $p$-adically open if its preimage in each test curve is open. A function of number fields is $p$-adically continuous if its pullback to each test curve is.

Is the $p$-adic regulator a continuous function of number fields? I doubt it. Consider the test curve $x^2=t$. For negative $t$ there are no units, so the regulator is 1. For positive $t$, it isn’t. So probably the test curves are too big and we should make the finite partition according to Archimedean behavior. But even after restricting to real quadratics, it is still unclear because the fundamental unit is hard to control because of the class group.

But there are some test curves for which we can control specific units to make a continuous proxy for the regulator, and thus prove that the set of points satisfying Leopoldt’s conjecture is open (though maybe empty). Specifically, the Ankeny-Brauer-Chowla family is the extension of $\mathbb Q(a_1,\ldots,a_n)$ defined by $\prod (x-a_i)=1$. Each specialization is a degree $n$ totally real field with $n-1$ independent units easily described in terms of the generator. The behavior of the extension of $\mathbb Q_p$ is locally constant, so we can locally think of the $n-1$ units as varying continuously through a single $p$-adic algebra. Thus the logarithms of the units vary continuously and so do their minors. This isn’t quite the regulator, since these units don’t necessarily generate all units, but a maximal minor avoiding zero is equivalent to Leopoldt’s conjecture, so the conjecture is open on this test variety.

I got this all from here, which also discusses a generalization (indeed, a deformation) of ABC fields to fields with imaginary places.