Arithmetic Chern–Simons theory with real places

Abstract

The goal of this paper is two-fold: we generalize the arithmetic Chern–Simons theory over totally imaginary number fields studied in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3; M. Kim, Arithmetic Chern–Simons theory I, preprint (2015) arXiv:1510.05818v4] to arbitrary number fields (with real places) and provide new examples of non-trivial arithmetic Chern–Simons invariant with coefficient [Formula: see text] [Formula: see text] associated to a non-abelian gauge group. The main idea for the generalization is to use cohomology with compact support (see [17]) to deal with real places. Before the results of this paper, non-trivial examples were limited to some non-abelian gauge group with coefficient [Formula: see text] in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3] and the abelian cyclic gauge group with coefficient [Formula: see text] in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3]. Our non-trivial examples (with non-abelian gauge group and general coefficient [Formula: see text]) will be given by a simple twisting argument based on examples of [F. Bleher, T. Chinburg, R. Greenberg, M. Kakde, G. Pappas and M. Taylor, Cup products in the étale cohomology of number fields, New York J. Math. 24 (2018) 514–542].