Number Theory and Geometry/Several Complex Variables

Question

This is a question for all you number theorists out there...based on my skimming of number theory textbooks and survey articles, it seems like most of the applications of geometry and complex variables to number theory are restricted to surfaces and the theory of a single complex variable. My questions are

1) Is this impression indeed accurate?

and, if so

2) Why is this? Is it because the theories of surfaces and of a single complex variable are "easier"(in the sense that they're simple enough to have a single unified theory,) or is there some deeper reason? And if the former is the case, are there some deep conjectures in number theory that could be solved using higher-dimensional geometry/complex variables?

Answer 1

I have heard algebraic number theory called "algebraic geometry in one dimension". (Or maybe you could call it arithmetic geometry in one dimension.) There is a natural emphasis in algebraic number theory on elliptic curves, function fields, etc. The reason is that algebraic geometry in one dimension is relatively well controlled, so that you can instead focus on arithmetic issues. I might describe the philosophy this way, even though it is not my area and not my philosophy: If number theory is like playing music, and algebraic geometry is like juggling, then general arithmetic geometry is like playing music while juggling the instruments. I.e., it's a great thing to do, but it combines two problems that already hard enough separately.

To the extent that you adopt this emphasis, it makes sense that you would learn more from complex curves than from higher-dimensional complex geometry. But without the one-dimensional emphasis, it is not true. For instance, the Weil conjectures are a deep result in higher-dimensional number theory, if you can call finite fields number theory, and they are both motivated by and informed by higher-dimensional complex geometry. (If number theory is music, finite fields could be music mainly in one note. If I had to juggle instruments, I might rather juggle tambourines or triangles.)

I should add that restricting to one dimension isn't a completely consistent vision of geometry. To give two examples, if $C$ is a complex curve of genus $g > 1$, it comes from a higher-dimensional moduli space, and it has a higher-dimensional Jacobian variety. These issues also arise in positive characteristic. The philosophy that algebraic number theory is one-dimensional is there, but it is not a completely serious philosophy, and not just because higher-dimensional generalizations exist. The Langlands program eventually leads you to the opposite philosophy.

Finally there are also connections between number theory and manifolds with three real dimensions, because of the fact that the boundary of hyperbolic space $\mathbb{H}^3$ is a Riemann sphere and $\text{Isom}(\mathbb{H}^3) = \text{PSL}(2,\mathbb{C})$.

Answer 2

No, I wouldn't say that "most" applications of algebraic and complex geometry to number theory are limited to the case of spaces of complex dimension 1. A more detailed answer follows:

1) Classical algebraic number theory and classical algebraic geometry both fit under the aegis of scheme theory. In this regard, there is an analogy between the ring of integers Z_K of a number field K and an affine algebraic curve C over a field k: both are one-dimensional, normal (implies regular, here) integral affine schemes of finite type. (The analogy is especially close if the field k is finite.) My colleague Dino Lorenzini has written a very nice textbook An Invitation to Arithmetic Geometry, which focuses on this analogy. I might argue that it could be pushed even further, e.g. that students and researchers should be as familiar with non-maximal orders in K as they are with singular curves...

2) Algebraic number theory is closely related to arithmetic geometry: the latter studies rational points on geometrically connected varieties. To do so it is essential to understand the "underlying" complex analytic space, and it is undeniable that by far the best understood case thus far is when this space has dimension one: then the theorems of Mordell-Weil and Faltings are available. Greg Kuperberg's remark about mixing two things which are in themselves nontrivial is apt here: it is certainly advantageous in the arithmetic study of curves that the complex picture is so well understood: by now the algebraic geometers / Riemann surface theorists understand a single complex Riemann surface (as opposed to moduli spaces of Riemann surfaces) rather well, and this firm knowledge is very useful in the arithmetic study.

3) In considering a scheme X over a number field K, one often "gains a dimension" in thinking about its geometry because key questions require one to understand models of X over the ring of integers Z_K of K. For instance, the study of algebraic number fields as fields is the study of zero-dimensional objects, but algebraic number theory proper (e.g. ramification, splitting of primes) begins when one looks at properties not primarily of the field K but of its Dedekind ring of integers Z_K.

A consequence of this is that in the modern study of curves over a number field, one makes critical use of the theory of algebraic surfaces, or rather of arithmetic surfaces, but the latter is certainly modeled on the former and would be hopeless if we didn't know, e.g. the classical theory of complex surfaces.

4) On the automorphic side of number theory we are very concerned with a large class of Hermitian symmetric domains and their quotients by discrete subgroups. For instance, Hilbert and Siegel modular forms come up naturally when studying quadratic forms over a general number field. More generally the theory of Shimura varieties is playing an increasingly important role in modern number theory.

5) Also classical Hodge theory (a certain additional structure on the complex cohomology groups of a projective complex variety) is important to number theorists via Galois representations, Mumford-Tate groups of abelian varieties, etc.

And so forth!

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Addendum: An (only a few years) older and (ever so much) wiser colleague of mine who does not yet MO has contacted me and asked me to mention the following paper of Bombieri:

MR0306201 (46 #5328) Bombieri, Enrico Algebraic values of meromorphic maps. Invent. Math. 10 (1970), 267--287. 32A20 (10F35 14E99 32F05)

He says it is "an extreme counterexample to the premise of the question." Because my august institution does not give me electronic access to this volume of Inventiones, I'm afraid I haven't even looked at the paper myself, but I believe my colleague that it's relevant and well worth reading.

Edit: He is now a MO regular: Emerton.

Answer 3

"Higher dimensional geometry" traditionally doesn't mean this I think, so this is only partially relevant to your question (and this isn't really related to complex analysis side); if you broaden your definition and include "algebraic geometry in high dimension" or something like that, then perhaps this is relevant.

Studying higher-dimensional varieties/schemes are important in number theory - for instance, etale cohomology, study of Shimura varieties, these are used very heavily in the aspects of number theory involving the $p$-adic Langlands correspondence, and $p$-adic Hodge theory. As I understand, Taylor's proof of $p$-adic local Langlands involves, in part, studying the $l$-adic cohomology of Shimura varieties. From what I've read, $p$-adic Hodge theory involves studying etale cohomology of certain schemes and relating them to Hodge structures coming from de Rham cohomology & crystalline cohomology of the same schemes. Calabi-Yau manifolds also occassionally pop up as a natural generalization of elliptic curves - for example in Taylor's recent work on Sato-Tate conjecture. I know this is probably not quite what you meant, but these areas of number theory do involve studying ''higher dimensional" structures in depth.