Arithmetic Dijkgraaf-Witten invariants for real quadratic fields, quadratic residue graphs, and density formulas

Abstract

We compute Hirano's formula for the mod 2 arithmetic Dijkgraaf-Witten invariant ${Z}_k$ for the ring of integers of the quadratic field $k=\mathbb{Q}(\sqrt{p_1\cdots p_r})$, where ${p_i}$'s are distinct prime numbers with $p_i \equiv 1 \pmod{4}$, and give a simple formula for $Z_k$ in terms of the graph obtained from quadratic residues among $p_1,\cdots, p_r$. Our result answers the question posed by Ken Ono. We also give a density formula for mod 2 arithmetic Dijkgraaf-Witten invariants.