User jeff h - MathOverflow

Digital Copy of Kato's Paper

2013-01-24T16:23:43Z

Kato's celebrated paper "p-adic Hodge theory and values of zeta functions of modular forms" is published in the bound Asterisque collection "Cohomologies p-adiques et applications arithmétiques. Volume III".

I have a copy of this book, but I'm searching (to no avail) for a digital copy of Kato's paper. This would be nice for at least two reasons:

(1) I wouldn't need to carry the book around if I had a laptop/tablet handy.

(2) I could print out portions of the paper and write notes all over them.

Does anyone know where a digital copy of the paper can be found, or whether there are legal reasons preventing a digital copy from existing?

Number Fields Arising from Newforms

2012-07-30T20:30:33Z

It is well-known that, given a normalized eigenform $f=\sum a_n q^n$, its coefficients $a_n$ generate a number field $K_f$.

In their 1995 paper "Fermat's Last Theorem", Darmon, Diamond, and Taylor remark that, at the time of writing, very little was known about what sort of number fields could arise as some $K_f$. They do claim, however, that $K_f$ must be totally real or CM. This claim is made just before Lemma 1.37, on page 40 of the copy I linked to.

This is probably standard knowledge among experts, but I'm having trouble finding a reference, so my questions are:

1) Can someone please provide a reference for this claim?

2) Is this still the state of the art, or do we now know more about what types of fields can appear as $K_f$ for some $f$? What if we restrict our attention to weight $k=2$?

Thank you!

Edit: In my question, I originally just wrote "modular form" instead of "normalized eigenform". Thanks to @Stopple for pointing this out! Also, I originally claimed the paper was published in 2007, but Kevin Buzzard pointed out it was published in 1995. Thanks Kevin!

Answer by Jeff H for Computational number theory

2012-06-29T14:43:28Z

You might be interested in Project Euler (http://projecteuler.net/), which guides you through solving progressively more difficult math problems using computer programming. This is not exactly what you asked about, but is rather a complement to quid's excellent answer.

Answer by Jeff H for Blackbox Theorems

2012-06-28T03:09:21Z

Deligne's construction of Galois representations corresponding to modular forms.

Answer by Jeff H for Suggestions for good books on class field theory

2011-12-11T04:38:12Z

If all you need is the major statements from CFT with a few examples, check out the appendix in Lawrence Washington's "Introduction to Cyclotomic Fields" for a speedy overview of both local and global class field theory. It's not big on exposition, and you won't learn the proofs, but it's short and not time-consuming to read. You can then see some applications of the theory in chapter 11 of the same book.

When are roots of power series algebraic?

2011-10-21T21:30:25Z

Let $K$ be a field and consider a power series $f(T) \in K[[T]]$. Under what conditions (on $K$ and/or on $f$) can we conclude that if $\alpha$ is a root of $f(T)$ then $\alpha$ is in fact algebraic over $K$?

This question is inspired by the following: In this paper of R. Pollack, in the proof of his Lemma 3.2, the author states that if $K$ is a finite extension of $\mathbb{Q}_p$ and $f(T)$ converges on the open unit disc of $\mathbb{C}_p$ and has finitely many roots, then each of these roots is algebraic over $K$.

I haven't yet come up with a proof of this statement (and any proofs offered would be appreciated), and perhaps having a proof in hand would help me to answer my question on my own, but I'm now curious about other situations in which this can occur.

Prime Decomposition in Cyclotomic Z_p-extensions

2011-09-15T04:16:21Z

In their classic paper "Class fields of abelian extensions of $\mathbf{Q}$", Mazur and Wiles assert that

"in a cyclotomic $\mathbf{Z}_p$-extension only finitely many primes lie above any prime of $\mathbf{Q}$."

My only other source in learning this material so far has been Washington's "Introduction to Cyclotomic Fields", and the only result along these lines is that such extensions are unramified outside of $p$.

So apparently, all primes lying above $l \neq p$ stop splitting at some finite level $K_n$, after which they remain inert. I've been unable to make much progress is proving this.

How can we see that this statement is true, and what other, more general results do we have about prime decomposition in cyclotomic extensions?

Answer by Jeff H for Changing field of study post-PhD

2011-07-12T14:37:06Z

I don't have the necessary reputation to comment, so I'll leave this as an answer. You asked if we knew of anyone who has made this switch, so I'll chime in: Several well-known mathematicians have done this, and surely many more not-so-well-known mathematicians have as well. The first one that comes to mind is Barry Mazur, who started off working in algebraic topology before switching to number theory. Serre did something similar.

Answer by Jeff H for Free, high quality mathematical writing online?

2010-11-19T18:47:21Z

I'm glad someone mentioned Keith Conrad's notes, as they are excellent.

I would also like to point people towards Tom Weston's webpage. He has expository papers at http://www.math.umass.edu/~weston/ep.html on several topics, including cobordism theory and spectral sequences.

He also has some course notes at http://www.math.umass.edu/~weston/cn.html, including truly excellent book-length notes on introductory algebraic number theory, as well as several dozen illuminating pages on local fields and ideles.

Comment by Jeff H

2013-01-30T03:29:23Z

This is not new research in the geometry of numbers, but rather an application of classical results to another classical problem, that of determining primes of the form x^2+ny^2: tcnj.edu/~hagedorn/papers/…

Comment by Jeff H

2013-01-24T17:03:34Z

@YangMills That answers my question. Thank you very much!

Comment by Jeff H

2013-01-24T17:00:51Z

@Franz I'm not sure what you're implying with the "spamming" comment. I think this question is quite reasonable for MO, considering both the importance and scarcity of this paper. Scanning is not really an option since it is a borrowed book and I'm afraid that scanning would do considerable damage to the binding. @Serge Thanks, but I checked there and they don't have it. It's a hard paper to track down.

Comment by Jeff H

2012-07-30T22:28:24Z

@Rob: Thank you! This is perfect.

Comment by Jeff H

2012-07-30T22:25:00Z

@Kevin Buzzard: Oh! You're absolutely right -- I see on Darmon's website that it was indeed published in 1995. I guess the tex file was recompiled in 2007, leading to the date I cited from this version of the paper: www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf Thanks!

Comment by Jeff H

2012-07-30T21:15:00Z

Hi Stopple, good point. It's a bad habit, but in my mind, when I say modular form I (almost) always mean "normalized eigenform", or even just "newform", but certainly the way I stated my question is incorrect. I'll fix my question to make it more precise. Thanks for pointing this out!

Comment by Jeff H

2012-06-26T02:45:52Z

I'm going to let this question sit for a bit in the hope that someone with a strong opinion in one particular direction will come along, otherwise I'll accept this answer.

Comment by Jeff H

2012-06-26T02:44:44Z

Hmm, I was hoping for a "right answer", and I find it somewhat disturbing that there could be so many "correct" ways to pronounce one person's name (there is certainly only one correct way to pronounce /my/ name, after all...), but the comment by @Francois does suggest some ambiguity with Dirichlet.

Comment by Jeff H

2012-06-26T02:13:15Z

Yes, that is a long thread discussing this question, but I also find it incredibly hard to read (the quote boxes are broken), and I see arguments made for every one of the possibly pronunciations that I listed, plus a few more. So, to me, this long discussion you've referenced seems very far from definitive. Again, I've heard arguments for all of these pronunciations, but it's very hard for someone with no expertise in any language outside of English to get to the truth of the matter. So I'm hoping someone will answer this thread in a coherent and persuasive way.

Comment by Jeff H

2011-10-28T15:09:00Z

Rob, this is wonderful...thank you!!

Comment by Jeff H

2011-10-24T01:14:47Z

@Lubin: This is a great (counter?)example. Thanks for this alternate perspective!

Comment by Jeff H

2011-10-23T18:05:08Z

@KConrad: Thank you, this is great!

Comment by Jeff H

2011-10-23T18:04:46Z

@Robert: Thank you for explaining that with such a simple, succinct argument!

Comment by Jeff H

2011-10-22T16:07:12Z

@David: I'm not sure I understand. In the example quoted in my question (the middle paragraph), a priori $f$ does not have coefficients in the ring of integers of $K$. The only thing known about $f$ is that it is a formal power series with coefficients in $K$ which converges in the open unit disc of $\mathbb{C}_p$ and has finitely many zeros. Where (and how) is the Weierstrass Preparation Theorem being invoked here?.

Comment by Jeff H

2011-10-21T22:40:43Z

@Qiaochu: Yes, Jacques has the right idea. @KConrad: Yes, the Weierstrass Preparation Theorem is a great example of this phenomenon. Thank you.