## Why are topological ideas so important in arithmetic?

For example, Wikipedia states that etale cohomology was "introduced by Grothendieck in order to prove the Weil conjectures". Why are cohomologies and other topological ideas so helpful in understanding arithmetic questions?

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Counting points over a finite field is asking for points fixed by Frobenius, and Weil's dream was to derive a formula by some (at the time) unknown analogue of the Lefschetz fixed point formula in algebraic topology. In a nutshell that is what first brought cohomology into arithmetic, for a very specific purpose. See math.ucdavis.edu/~osserman/classes/256B/notes/… or Dieudonne's "On the history of the Weil Conjectures" in Math. Intelligencer (1975). – KConrad May 27 2010 at 0:38
Other ideas which were not first expressed in cohomological terms later took on that language, e.g., class field theory was at one time expressed in terms of central simple algebras, but if you focus only on the cocycles defining the multiplication in the algebras then the theorems in class field theory turn into statements in Galois cohomology. – KConrad May 27 2010 at 0:41
I will supplement KConrad's insightful and specific answer with a vague unsubstantiated comment. Many of the important questions in number theory arise as Diophantine problems: what are the integer solutions to a family $P_i(x_1, \ldots, x_j) = 0$ of polynomial equations over $\mathbb{Z}$? If you instead look for complex solutions to the same family of polynomial equations, then you get a geometric object to which the tools of algebraic topology ought to apply. It doesn't seem so unreasonable to expect connections between the arithmetic and geometric properties of the system. – Paul Siegel May 27 2010 at 1:14
One answer which has come up on MO in the past is "because we always try to apply topological ideas to whatever we study, and often it works." – Qiaochu Yuan May 27 2010 at 1:15

Why are topological ideas so important in arithmetic? In some sense KConrad is of course spot on, but let me offer a completely different kind of answer.

Why are complex functions of one variable so important in arithmetic? (Zeta function, L-functions, Riemann hypothesis, Birch--Swinnerton-Dyer, modular forms, theta series, Eisenstein series...).

Why is geometry so important in arithmetic? (Faltings' theorem, applications of algebraic geometry, low-dimensional arithmetic of varieties (elliptic curves etc))

Why is K-theory so important in arithmetic? (Bloch-Kato, Voevodsky...)

Why is logic so important in arithmetic? (Julia Robinson, Matiyasevich, Ax-Kochen and then Hrusovski proving that "if it's true in char p for suff large p then it's true in char 0" in the context of some very deep statements)

Why is functional analysis so important in arithmetic? (L^2 functions on $\Gamma\backslash G$ with $G$ a semisimple Lie group being related to automorphic forms and hence to number theory via Langlands, with crucial analytic tools like the trace formula).

Why are dynamical systems so important in arithmetic? (3x+1 problem, work of Deninger, or of Lind/Ward and their school).

Here's the answer: it's because arithmetic is a very mature subject---it has been around literally thousands of years, and because it has been around so long, there is far more of a chance that someone will come along with an insight relating [insert arbitrary area of pure mathematics here] with arithmetic. So in some sense it's a historical fluke. If we were all born with continuum-many fingers which we could move only in real-analytic ways, and we didn't discover the positive integers until much later on, then arithmetic would be all new and we'd be waiting for Gauss, and real analysis would be as old as the hills, and people would be asking "why is [insert arbitrary thing] so important in real analysis"?

[PS (1) yeah I know, I was being facetious at the end, and (2) yeah I know, my list at the top is woefully incomplete]

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I don't find this answer very satisfying, because all it says is that if you study something long enough, you will eventually connect it to almost everything else. That may be so, but some of us still want to know whether there is something within the world of "numbers" that explains each connection. Modern number theory and its use of apparently unrelated concepts like topology remains an extremely mysterious subject to a outsider like me. – Deane Yang May 27 2010 at 12:22
I take your point Deane. I personally think you're asking for too much though. If X and Y are things in the universe and they happen to interact in some way, this might just be a fluke, or something way too complicated for a human to understand, or anything. In particular I'm not always sure it's reasonable to ask "why" they interact. My answer gives lots of evidence that lots of other ideas in mathematics impact upon arithmetic, and my premise is that if any two things are around for long enough then it might not be unreasonable to expect to see some interaction amongst them. – Kevin Buzzard May 27 2010 at 12:40
Kevin, you make good points, too. It's just that in the purely transcendental world of differential geometry and topology, we can often concoct plausible stories for why things interact. This appears more difficult to do for all these applications of transcendental ideas to number theory. I'm in awe of number theorists being able to work in such a setting . – Deane Yang May 27 2010 at 12:53
It seems to me that a reasonably honest and nonvacuous answer is: because algebraic geometry has such strong connections to both topology and number theory. – Pete L. Clark May 27 2010 at 14:06
@Deane and Kevin: I think the way analysis impacts arithmetic is more than just being a tool. Why should prime numbers be distributed in such a way that we need the logarithm function to describe it? The analytical properties are there in the integers. It is not as if smart humans applied analysis to study them. – Idoneal May 28 2010 at 4:17
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New lectures by Atiyah on physics-inspired questions in geometry mention some fascinating connections with number theory, e.g. the question about a "quantum analogue of the Weil conjectures"and an "infinite dimensional version" of them.

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If we think about Diophantine equations in general, the situation is "hopeless". That's a theorem. Nevertheless in number theory we want to study such equations, in special cases at least, so some ideas are required to sort out "Diophantine equation space" (DES) into parts that we might come to understand, and parts to leave alone.

Classically this was done by "degree of an equation". This takes you a certain way, but not really far enough, as genus already shows. There may also be simultaneous equations, for example, and all the insights of algebraic geometry then come in. The slogan of "Diophantine geometry", that the geometry of a set of equations affects the diophantine properties, by now has much credibility. The part of DES where the geometry is well understood seems to correspond quite well to the part fruitful to study by current methods. (For experts, I'm sliding from integer to rational points here.)

"Topological methods of algebraic geometry" explains what happened in the 1950s and 1960s in that subject. From the 1970s, and really with much more pain, related methods have been used in Diophantine geometry. Those working in number theory have always been grateful for any general methods they can get with traction on DES.

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 Charles, I am not sure about your last sentence, "Those working in number theory have always been grateful for any general methods they can get with traction on DES." Not always! And those working outside DEs are sometimes grateful for the methods born in this area. As standard example I could hint about influence of FLT on maths. – Wadim Zudilin May 29 2010 at 7:22 See en.wikipedia.org/wiki/Diophantine_geometry for more accurate and neutral perspective (I hope). Gauss is alleged to have said about FLT that he thought it should be developed in a broader context. Kummer initiated a theory, and, as you say, this special equation had a broad influence. General methods for Diophantine equations really started with Thue, and I wouldn't call Diophantine approximation "topological"; post-Vojta there is a new view? FLT's solution I don't understand, but deformation of Galois representations are bounding Selmer groups? There is topology on one side there. – Charles Matthews May 29 2010 at 10:05

May I suggest that we don't have to consider cohomology to see the influence of topology on arithmetic? Looking for rational points on curves leads to the question of which curves have rational parameterization, and Riemann found that the answer is topological -- the curves of genus zero. One also observes special behavior on curves of genus 1 (elliptic curves), etc.

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Well, some would consider the genus of a curve to be a cohomological concept. (But I take your point.) – Pete L. Clark May 27 2010 at 2:59