# Is the Leopoldt conjecture almost always true?

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.

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

• Dear Preda, thanks for this post. I am going to read your version 2. – Joël Jul 4 '11 at 11:27
• Dear Joel, You are welcome and have fun. I will post within the next week also a shorter paper on the Gross conjecture, this might be interesting too - I teach it in the same course, and the result is somehow faster ... Let me know if you get stuck anywhere, I will try to help Best Preda – Preda Jul 4 '11 at 12:13
• Dear Preda, Six months have passed and unfortunately I have not yet found the time to read seriously your paper, which I still very much want to do. I wanted to ask you a question: you have a much shorter 2008 paper called "An application of Baker's theory to Leopoldt's conjecture" which proves Leopoldt's conjecture in the case of a totally real extension where $p$ splits completely (which is already a real breakthrough, and contains the cases I personally need of the conjecture). Is this paper published? Do you think I should read it as an introduction to your more recent paper? – Joël Feb 4 '12 at 16:44
• While the previous version only proved Leopoldt for CM fields that split the prime p, I have developed a new proof using CM Z_p extensions, a proof that holds for any CM field. The proof was discussed during 9 months with a German colleague and expert in Iwasawa theory, whose questions and hints lead to a definite improvement of the exposition - with the result that the colleague eventually understood and agreed with the proof, the responsibility remaining completely on my side, of course. It is now available on Arxiv and has been submitted for publication. – Preda Apr 22 '14 at 9:42
• Dear Joel, I only now see your question from Feb. 4-th, 2012. The answer is of course outdated, see above. Maybe the best is if the next time you write a mail first - you can find my email in the net. – Preda Apr 22 '14 at 9:44

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.

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?).

• I was under the impression that "the jury was still out" on this, so I figured I'd ask my question in the meantime. – David Hansen May 28 '11 at 4:11
• Given the number of "proofs" of big theorems I've seen on arXiv, I would agree that the "jury was still out" unless it has been accepted for publication, or other notables are talking about it and taking it seriously (like happened with Catalan). It has now been a preprint for 2 years and he gave talks in late 2009, and I haven't heard anyone particularly voice a confirmation. There is value in the paper, though maybe not everything he claims. Supposedly Lenstra gave up reading Mihailescu's manuscripts just when Catalan worked, so if it's serious math you should always give it some merit. – Junkie May 28 '11 at 5:20
• Fujiwara retracted its proof. As for Mihailescu's, credible people I discussed it with tended to be more critical than "the jury is still out" and at least one was much more so. But you never know... – Olivier May 28 '11 at 7:15
• I agree with your statement, Olivier (maybe derivative of the same credible persons), but I wasn't going to voice rumors without a specific flaw being noted. One problem is that Mihailescu has his own expositional style, to say the least, and it is often hard to figure out what he has really done or shown. As Minhyong Kim said when PM gave talks, he was very open to questions, and there was a lot going on, but the jury seems to say: this as-is is not a proof: but there are various new ideas which might lead to a proof. Now he claims the CM case separately, maybe this will clear matters. – Junkie May 28 '11 at 8:23
• @Dr Shello: It's my understanding that the experts have not come out and said that Mihăilescu's proof is incorrect, but that they have not said it is correct either. – Rob Harron May 28 '11 at 17:29

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.

• Minor caveat: the independence of the units is only Zariski dense in the ABC family. – Ben Wieland Nov 30 '16 at 0:52