Edit: He is now a MO regular: Emerton.

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!

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!

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.