How important is Differential Geometry for Number Theory?

Solution 1:

I think that the ways one uses differential geometry in number theory are just not as directly visible as some of the other areas NT utilizes--such as abstract algebra--which makes it more difficult to see its usefulness. I find the most use of the subject in the application of Lie groups to the subject.

Of course, there are degenerate cases: Minkowski theory is a rich area of number theory that uses geometry extensively to study solutions to Diophantine problems. Naturally some of the convex bodies involved can (though in practice aren't always too bad) get somewhat complicated or defy simple computations of volume, which of course is a classical question in differential topology. I call it "degenerate" because much of it is linear algebra, but in practice some of the convex bodies actually require the topology, so including it seemed apropos since the question is pretty open-ended.

Grigory Margulis is a genius who has used methods from the analysis on groups like the special linear group to prove results on quadratic forms. He also has proven important results on arithmeticity by analyzing these groups from a topological and dynamical point of view.

Techniques in ergodic theory are used in Diophantine approximation regularly. Classical results include the uniform distribution of points $[n\alpha]$ in the interval $[0,1]$ iff $\alpha$ is irrational. Horocycle flow is intimately related to hyperbolic geometry in a way which explicitly utilizes the differential structure on the space and the metric on the upper half-plane.

The structure on $SL_n(\Bbb R)$ especially is used to establish subgroups which generate the entire group, which comes from general facts about Lie groups, established using differential topology rules. These structure theorems are used over and over in the proofs. Amir Mohammadi is a student of the aforementioned Margulis who is doing excellent work in this area using such ideas.

And of course, I would be remiss not to mention the amazingly-named ping-pong lemma. It neatly plays the group theory and geometry off of one another to prove results about each using information from the other.

I think if you're expecting anything particularly complex, you'll be disappointed as is common in most cross-uses in areas, it's not usually the cutting edge in both fields working on the same problems, but certainly they show up and are important.