Spin structures, quadratic maps, and the missing class group heuristic

I will report on joint work in progress with Akshay Venkatesh where we propose an arithmetic analogue of the association, in topology, of quadratic enhancements to spin structures on closed oriented 2- and 3-manifolds: a choice of spin structure provides, respectively, a quadratic refinement of the mod 2 intersection form and of the linking pairing on the first torsion homology. This adds an entry to the number field/3-manifold analogy of Mumford, Mazur, and Manin and furnishes a conceptual explanation of anomalous class group statistics.