Bounds for the Dimensions of pp-Adic Multiple LL-Value Spaces

  • Go Yamashita

    School of Mathematical Sciences University of Nottingham University Park Nottingham NG7 2RD, United Kingdom

Abstract

First, we will define pp-adic multiple LL-values (pp-adic MLV's), which are generalizations of Furusho's pp-adic multiple zeta values (pp-adic MZV's) in Section 2. Next, we prove bounds for the dimensions of pp-adic MLV-spaces in Section 3, assuming results in Section 4, and make a conjecture about a special element in the motivic Galois group of the category of mixed Tate motives, which is a pp-adic analogue of Grothendieck's conjecture about a special element in the motivic Galois group. The bounds come from the rank of KK-groups of ring of SS-integers of cyclotomic fields, and these are pp-adic analogues of Goncharov-Terasoma's bounds for the dimensions of (complex) MZV-spaces and Deligne-Goncharov's bounds for the dimensions of (complex) MLV-spaces. In the case of pp-adic MLV-spaces, the gap between the dimensions and the bounds is related to spaces of modular forms similarly as the complex case. In Section 4, we define the crystalline realization of mixed Tate motives and show a comparison isomorphism, by using pp-adic Hodge theory.