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?



Number Theory and Geometry/Several Complex Variables
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 $\text{Isom}(\mathbb{H}^3) = \text{PSL}(2,\mathbb{C})$.


