Dedekind zeta motives for totally real number fields

Abstract: Let \(k\) be a totally real number field. For every odd integer \(n\geq 3\), I will construct a mixed Tate motive whose period is related to \(\zeta_k(n)\), the value of the Dedekind zeta function of \(k\). The construction is geometric and involves the action of an arithmetic subgroup of the orthogonal group on  products of hyperbolic spaces. A corollary is a motivic calculation of the regulator modulo rationals, which is analogous to a famous theorem due to Borel.