User simon - MathOverflow

Answer by Simon for Statistics of Number fields

2011-07-07T21:45:16Z

A conjecture of Malle (which is known to be false in general, but probably still provides a good heuristic in many cases) implies that if $G$ is a transitive subgroup of $S_n$, then a positive proportion of number fields (ordered by discriminant) of degree $n$ have Galois group $G$ if and only if $G$ contains a transposition. The relevant constant here is $a$ from unknown (google)'s answer: it's the minimum value of $n$- the number of cycles in $g$, where $g$ runs over all non-identity elements of $G$. For example, when $n=4$, the dihedral group $D_4$ contains a transposition, which explains why a positive proportion (roughly 11%) of quartic fields have Galois group $D_4$.

See this paper by Jürgen Klüners for the precise statement of the conjecture, together with a counterexample.

Statements in group theory which imply deep results in number theory

2009-12-03T20:26:26Z

Can we name some examples of theorems in group theory which imply (in a relatively straight-forward way) interesting theorems or phenomena in number theory?

Here are two examples I thought of:

The existence of Golod-Shafarevich towers of Hilbert class fields follows from an inequality on the dimensions of the first two cohomology groups of the ground field.

Iwasawa's theorem on the size of the $p$ part of the class groups in $\mathbb{Z}_p$ -extensions follows from studying the structure of $\mathbb{Z}_p[\![T]\!]$ -modules.

Can you name some others?

Answer by Simon for Is there a simple relationship between K-theory and Galois theory?

2010-01-09T16:27:01Z

There are lots of maps from $K$ groups to various other things, including Galois groups. Here's an example, due to Kato, which generalizes local class field theory.

Classical local class field theory says that if $F$ is a local field (by which I mean, in this case, a field complete with respect to a discrete valuation and with a finite residue field), then there is a natural map $F^\times\to Gal(F^{ab}/F)$ which behaves well on the finite levels: if $L/F$ is a finite extension, then $F^\times/N_{L/F}L^\times\to Gal(L/F)$ is an isomorphism. But $F^\times=K_1(F)$ , so we have a map $K_1(F)\to Gal(F^{ab}/F)$ that behaves in the same way on finite levels.

We'll generalize this by defining an $r$-dimensional local field inductively. A 0-dimensional local field is a finite field, and an $r$-dimensional local field is a field complete with respect to a discrete valuation whose residue field is an $(r-1)$-dimensional local field. Thus a classical local field is a 1-dimensional local field.

Kato found maps $K_r^M(F)\to Gal(F^{ab}/F)$ which are isomorphisms on the finite levels, where $K_r^M$ are the Milnor (not Quillen!) $K$-groups. They also behave similarly well on the finite levels.

S-unit equation and small sets of places

2009-12-03T15:46:24Z

Let $K$ be a number field, and let $S_x$ denote the set of primes of norm at most $x$. Is it possible to find a smaller set of places $T_x\subset S_x$ so that a lot of the solutions of the $S_x$ -unit equation $a+b=1$ for $a,b\in S_x$ are solutions of the $T_x$ -unit equation?

Here's a possible precise statement (although I'd be interested in other formulations as well): Does there exist a constant $0<c<1$ , depending on $K$ (but not $x$), so that for each $x$, there is a $T_x\subset S_x$ with $|T_x|\le\sqrt{|S_x|}$ so that the number of solutions to the $T_x$ -unit equation is at least $c$ times the number of solutions of the $S_x$ -unit equation?

I'm interested in this mostly by analogy: Bellabas and Gangl have a bound for the set of places of a number field one must check in order to compute $K_2$ of the ring of integers. It would be interesting to know if one could at least get a pretty good approximation for $K_2$ by looking at a much smaller set of places.

Answer by Simon for Any reason why K_23(Z) has order 65520?

2009-12-31T05:48:18Z

More generally, if $F$ is a number field with ring of integers $\mathfrak{o}$, and $\zeta_F^\ast(m)$ is the first nonzero coefficient in the Taylor expansion of $\zeta_F$ at $m$, then Lichtenbaum (and Quillen) conjectured that $|\zeta_F^\ast(1-i)|=\frac{\# K_{2i-2}(\mathfrak{o})_{\text{tors}}}{\# K_{2i-1}(\mathfrak{o})_{\text{tors}}}$ , times a regulator and some power of 2 (which I believe is not understood in general, although some progress was made on this in Ion Rada's PhD thesis). Hence, odd $K$ groups are related to the denominators of the Bernoulli numbers, and the even ones are related to the numerators. Also, not much cancellation occurs; I think the two $K$-groups can only share factors of 2.

The Voevodsky-Rost theorem might prove the Lichtenbaum conjecture, but I haven't seen anyone come out and say definitely that this is the case.

I don't have much intuition for this, except that the $K$-groups seem to be objects that like to map into étale cohomology groups. In this paper (link to MathSciNet), Soulé constructs Chern class maps from certain $K$-groups to étale cohomology groups. Furthermore, these maps frequently have small (or trivial) kernels and cokernels. I suppose the idea, then, is that $K$-theory is supposed to be a slightly better behaved version of étale cohomology, at least for the purpose of understanding zeta functions.

The rank of $K$-groups of rings of integers was computed by Quillen in the early 70's: it's rank 1 in dimension 0, rank $r_1+r_2-1$ in dimension 1 (Dirichlet's unit theorem), rank 0 in even dimensions $>0$, rank $r_1+r_2$ in dimensions $1\pmod 4$ except 1, and rank $r_2$ in dimensions $3\pmod 4$.

Answer by Simon for Order of the Tate-Shafarevich group

2009-12-28T01:41:31Z

Brian Conrad told me that this is not always the case; Tate's paper where he claimed this was misunderstood until some counterexamples were found. William Stein has a paper "Shafarevich-Tate groups of nonsquare order'' with counterexamples; it's available online at http://modular.fas.harvard.edu/papers/nonsquaresha/final2.ps.

Answer by Simon for Where can I find the text of Weyl's Fields Medal speech for Serre?

2009-12-18T03:43:39Z

Here are a few snippets from International Mathematical Congresses: An Illustrated History

... I hope the Congress as a whole will approve our choice. In justification of it let me say this: by study and information we became convinced that Serre and Kodaira had not only made highly original and important contributions to mathematics in recent years, but that these hold our great promise for future fruitful non-analytic (will say: non-foreseeable) continuation.

... I realize how difficult it is for a man of my age to keep abreast of the rapid development ... which that young generation forces upon our old science ... [The burden] rests more heavily on my than on my predecessors' shoulders; for while they reported on things within the circle of classical analysis, where every mathematician is at home, I must speak on achievements that have a less familiar conceptual basis ... Be prepared then to have to listen now to a short lecture on cohomology, linear differential forms, faisceaux or sheaves, Kählder manifolds and complex line bundles ...

... If I omitted essential parts of misrepresented others, I ask your pardon, Dr. Serre and Dr. Kodaira; it is not easy for an older man to follows your striding paces.

... The mathematical community is proud of the work you both have done. It shows that the old gnarled tree of mathematics is still full of sap and life. Carry on as you began!

Answer by Simon for Just starting with [combinatorial] game theory

2009-12-09T05:09:57Z

Winning Ways for your Mathematical Plays (in four volumes) has an enormous amount of stuff about combinatorial games. But most of it you probably won't be interested in for a while. There are a few quickly diverging directions one could study in combinatorial games. Here are some that come to mind immediately, and a possible list of topics to study in each:

1) Impartial games. Read a bit of On Numbers and Games so that you know how to read the notation and understand game equivalence and addition. Then learn the winning strategy for Nim, read the relevant bits of Chapter 3 and all of Chapter 4 of Winning Ways. After that, if you like the infinite theory, Lenstra has a paper called "On the algebraic closure of two" which is really nice. If you like the finite theory, learn about nim multiplication from ONAG, and then read Conway and Sloane's paper "Lexicographic codes: error-correcting codes from game theory." I think this part of the theory is the most interesting.

2) (Surreal) numbers. Again, learn how to read the notation and about game equivalence and addition. (You will need this for everything.) Then read the first part of ONAG. Then, perhaps learn about real-closed fields in general; you can make most of real analysis work over the Field of surreal numbers. (A Field is something like a field, but it has a proper class of objects instead of a set.)

3) Weird games, for example from Hackenbush and Domineering. Read Volume 1 of Winning Ways. The stuff on thermography and all-small games is quite interesting.

Answer by Simon for Factorization of elements vs. of ideals, and is being a UFD equivalent to any property which can be stated entirely without reference to ring elements?

2009-12-05T02:33:33Z

First of all, the ubiquity of category theory in algebra is fairly recent, at least given how long people have been working on algebra (not even including elementary number theory). Much of algebraic number theory was developed in the mid-19th century in attempts to prove Fermat's Last Theorem. Since category theory would not show up for another century, mathematicians like Kummer and Dedekind had little reason to think in those terms. The notion of class group showed up as an obstruction to Kummer's attempted proof of Fermat's Last Theorem, which assumed that all the cyclotomic fields $\mathbb{Q}(\zeta_p)$ had class number 1 (or at least prime to $p$). It's hard to see even what form Kummer's arguments would take if phrased in the language of factorization of ideals. I think that when flaws in Kummer's arguments were exposed, mathematicians realized that factorization of ideals behaves much better than factorization of elements. But for a mid-19th century mathematician, it must have felt a lot more natural to try to factor elements than ideals; they only studied the latter because the former usually fails. Now we understand that being a UFD (class number 1) is simply the nicest case, and the class group in general is the obstruction.

Answer by Simon for What's the "best" proof of quadratic reciprocity?

2009-11-26T15:29:43Z

There's a nice proof that involves the computation of $K_2(\mathbb{Q})$ and the interpretation of $K_2$ as a universal symbol (i.e. a bilinear map $\mathbb{Q}^\times\times\mathbb{Q}^\times\to A$ for some abelian group $A$, written multiplicatively, satisfying $(a,1-a)=1$) in Milnor's book on algebraic $K$ theory. The tame symbols are interpreted as Legendre symbols, and by universality of $K_2$ as a symbol Tate claims that this proof is essentially Gauss's original argument. I suspect that this argument can be generalized to other totally real number fields, but explicit computations of $K_2(F)$ aren't very easy for large discriminants.

It's definitely not the easiest to understand, but at the moment, it's my favorite.

Answer by Simon for Periods and commas in mathematical writing

2009-11-24T16:42:49Z

A math paper should follow all the usual rules of grammar, so in particular there should be subjects and verbs and the sort of punctuation you'd expect to find in a piece of nontechnical writing. I would prefer to write the following:

The formula for a circle is $$ x^2+y^2=r^2. $$

If I had to use the wording in the original question, I would write

This is the formula for a circle: $$ x^2+y^2=r^2. $$

Occasionally, the aesthetics of the page make punctuation look awkward. For example, one might write:

Therefore, the following diagram commutes:

M×N -> M⊗_RN
   \    |
    \   |
     \  | 
      v v
       A

with no punctuation after the diagram. There isn't any sensible location for a period at the end of a sentence, so I'd leave it out.

Homology of symmetric groups

2009-11-24T00:32:26Z

Let $S_n$ denote the symmetric group on $n$ letters, and let $S_n(p)$ denote a Sylow $p$-subgroup. Why is the image of $H_i(S_n(p))$ in $H_i(S_n)$ the $p$-primary part of $H_i(S_n)$ ?

Comment by Simon

2012-08-14T16:11:03Z

This product also shows up in the Cohen-Lenstra heuristics for the distribution of p-Sylow subgroups of class groups of imaginary quadratic fields. This is expected to be related to Andreas Blass's comment about the probability that an n by n matrix over F_p is nonsingular, thanks to a paper of Friedman and Washington.

Comment by Simon

2011-07-09T13:08:29Z

Are there at least interesting subsequences $n_1,n_2,\ldots$ of the natural numbers so that $b_1b2_\ldots b_{2^{n_i}}$ is a binary De Bruijn sequence for each $n_i$? For example, can this work if $n_i=2^i$?

Comment by Simon

2009-12-31T23:52:34Z

Fair enough. Thanks for keeping me honest!

Comment by Simon

2009-12-31T19:33:36Z

In the totally real case, at least, what I said seems to be true (if the size of $K_{2i-1}(\mathfrak{o})$ is infinite when $i$ is odd, and $\zeta_F(1-i)$ is also zero in this case). But as Bjorn pointed out, you can also get the leading coefficient by restricting to the torsion subgroups. Moreover, this works even if $F$ is not totally real. I have no idea what the actual $K$ groups (rather than just the sizes of the torsion subgroups) tell us. I'd love to know though!

Comment by Simon

2009-12-31T15:14:38Z

Thanks. I fixed it.