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 to any number field, by using the modified étale cohomology groups and fundamental groups which take real primes 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 $p_i$'s are distinct prime numbers 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.
Acknowledgments
The author woulld like to thank his supervisor Masanori Morishita for suggesting the problem studied in this paper. He is also thankful to Junhyong Kim for discussion and to Yuji Terashima for communication. He would like to thank the referee for careful reading of the paper and useful comments.
Citation
Hikaru Hirano. "On mod 2 arithmetic Dijkgraaf-Witten invariants for certain real quadratic number fields." Osaka J. Math. 60 (4) 933 - 954, October 2023.
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