The Analogue of the Gauss Class Number Problem in Motivic Cohomology

Abstract

<p>Let F be a totally real number field of degree d and let n >= 2 be an even integer. We denote by W K^M_2n-2(F) the n-th motivic wild kernel of F, which acts as an analogue to<br />the class group of F. Assuming the 2-adic Iwasawa Main Conjecture, we prove that the there are only finitely many totally real number fields F having |W K^M_2n-2(F)| = 1 for some even integer n>=2. In particular we show that there are no totally real number fields having trivial n-th motivic wild kernel for n >= 6, and that there is precisely one totally real number field having trivial 4th motivic wild kernel, namely Q(√(5)). We prove that all totally real number fields having trivial 2nd motivic wild kernel must be of degree d <= 117 (respectively d <= 46 under the assumption of the Generalized Riemann Hypothesis). Using Sage mathematical software, we enumerate all totally real fields of degree d < 10 having trivial 2nd motivic wild kernel, finding 21 such fields. Under restrictions on the local properties of F, we enumerate all relevant fields having trivial 2nd motivic wild kernel.</p>