On mod 2 arithmetic Dijkgraaf-Witten invariants for certain real quadratic number fields

Abstract

Minhyong Kim introduced arithmetic Chern-Simons invariants for totally imaginary number fields as arithmetic analogues of the Chern-Simons invariants for 3-manifolds. In this paper, we extend Kim's definition for any number field, by using the modified \'etale cohomology groups and fundamental groups which take real places into account. We then show explicit formulas of mod 2 arithmetic Dijkgraaf-Witten invariants for real quadratic fields $\mathbb{Q} (\sqrt{p_1 p_2 \cdots p_r})$, where $p_i$ is a prime number congruent to 1 mod 4, in terms of the Legendre symbols of $p_i$'s. We also show topological analogues of our formulas for 3-manifolds.