User anonymous - MathOverflow

Answer by anonymous for Fontaine's rings of periods

2011-04-20T23:36:36Z

I would say that Bloch-Kato conjecture (giving the exact value of complex $L$ functions of motives) was a rather unexpected application.

Answer by anonymous for What is the high-concept explanation on why real numbers are useful in number theory?

2011-04-14T19:33:31Z

Two of the most basic facts of algebraic number theory, namely the finiteness of the class number and the structure of units in rings of integers of number fields do not seem to be provable without the use of real numbers (or some use of the archimedean nature of real numbers).

Added: In fact, most finiteness results in arithmetic geometry seem to use this archimedean prime; this is the case of Mordell-Weil theorem, Mordell conjecture, finite generation of Galois cohomology groups of number fields, etc.

Afterthoughts: There are many branches of number theory and in some of them you can't even state the results without using real numbers as was pointed out elsewhere. Now, in algebraic number theory or arithmetic geometry, which does not suffer from this problem, the analogy with function fields is a quite powerful tool to try to guess what can be true, and if you look at the problem from this angle you realise that real numbers are more of a nuisance than a help : all the statements above can be proven for function fields where real numbers play no role, and many others like the Riemann hypothesis or the global Langlands correspondence still elude us in the number field setting. The fact that you have to use them to prove the above results seem to indicate that you cannot ignore this nuisance so easily... (despite the product formula that makes you believe that the information that you can extract from the archimedean prime should be readable from the others).

Answer by anonymous for What are some correct results discovered with incorrect (or no) proofs?

2011-04-07T17:23:32Z

In 1983 or 84, Frey announced that he could prove that Taniyama-Weil conjecture implies Fermat's last theorem. The proof was flawed but this announcement had spectacular consequences:

$\bullet$ Serre pulled out an unpublished conjecture of his and strengthened it so that Taniyama-Weil + $\varepsilon$ would imply FLT,

$\bullet$ Ribet proved enough of $\varepsilon$ so ensure that TW would imply FLT,

$\bullet$ Wiles realized that FLT would be proved as TW could not be ignored and so decided that it had to be by him (in doing so, he completely changed the way people thought about the field and this has led to impressive results including the proof of TW or of Sato-Tate conjecture),

$\bullet$ Shimura decided that he wanted his name attached to the conjecture and Lang made a campaign to remove Weil's...

Answer by anonymous for Elementary+Short+Useful

2011-04-04T18:38:29Z

Lagrange's theorem (order of a sugroup divides the order of the group).

Answer by anonymous for What are some correct results discovered with incorrect (or no) proofs?

2011-04-04T18:30:44Z

The Kronecker-Weber theorem needed 3 proofs spanned upon 30 years before being completely proved (it states that all abelian extensions of ${\mathbb Q}$ can be found inside cyclotomic fields). It lead to class field theory.

Answer by anonymous for Experimental Mathematics

2011-03-16T19:47:31Z

I think that Zagier's conjectures on polylogarithms were based on considerable amount of numerical evidence (this has led to impressive work by Beilinson-Deligne, Goncharov...).

Serre's conjecture on modularity of mod $p$ $2$-dimensional Galois representations was also made precise thanks to simultaneous theoretical advances and numerical computations (this has paved the way to the proof of Fermat's last theorem...).

To come back to the prime number theorem, I think that Euler already "proved" it long before Gauss conjectured it by differentiating $\sum_{p\leq x}\frac{1}{p}\sim \log\log x$.

Answer by anonymous for Which mathematical ideas have done most to change history?

2011-03-15T17:32:58Z

Probability theory and statistics have changed the way we think about many things and they are used in a lot of aspects of everyday life.

Answer by anonymous for Thinking and Explaining

2011-03-11T18:23:10Z

The ring ${\mathbb Z}/N{\mathbb Z}$ is usually defined in a rather cumbersome way, and it takes some time (infinite in most cases) before students realize that you can think of it as ${\bf Z}$ with just one added relation $N=0$ to do the computations (and the problem that $xy=0$ does not necessarily imply that $x=0$ or $y=0$), so that it is indeed a very simple object and not some horribly abstract invention.

Answer by anonymous for Exotic principal ideal domains

2011-02-24T19:30:56Z

Fontaine's ring $B_{cris}^{\varphi=1}$ is a PID, and no expert in the field would have bet on it in the first place (this led to some very nice recent developments by Fargues and Fontaine).

http://www.math.u-psud.fr/~fargues/Courbe.pdf

Comment by anonymous

2011-04-20T22:59:08Z

umpa.ens-lyon.fr/~lberger/article05/article05.pdf math.jussieu.fr/~colmez/897-asterisque.pdf

Comment by anonymous

2011-04-06T00:01:31Z

You can use it to prove Fermat's little theorem, mix it with some group action to prove combinatorial results or Cauchy's theorem on the existence of elements of order $p$, etc.