Is there any deep relation between lens space and number theory?

Question

It writes on Wiki that "...(lens space) were the first known examples of 3-manifolds which were not determined by their homology and fundamental group alone." And actually we have known the complete classfication of the lens space $L(p,q)$ w.r.s to the prime p, and it is obtained by the different operation of surgery. My question is that since the classification rely highly on the prime $p$ and a kind of reverseble operation due to the primeness of $p$, is there any deep relation between lens space and number theory except for the technical 3-manifold operation? Or I wonder how a number theoriest will treat the lens space?

Answer 1

Despite the letter used, there is no reason to require $p$ prime. The requirement is that $p$ and $q$ are coprime. (This ensures that the action of $\Bbb Z/p$ on $S^3 \subset \Bbb C^2$ via $$k \cdot (z,w) = (e^{2\pi i k/p} z, e^{2\pi i kq/p})$$ is free, so that the quotient is a manifold. Sometimes people refer to $L(p,q)$ for (p,q) not coprime, referring to the quotient considered as an orbifold.

The only clear relationship is that the classification of lens spaces up to homeomorphism or homotopy equivalence is a modular arithmetic statement. I don't think there's much of number theoretic interest to say here.