Bounds for the Dimensions of $p$-Adic Multiple $L$-Value Spaces

First, we will define $p$-adic multiple $L$-values ($p$-adic MLV's), which are generalizations of Furusho's $p$-adic multiple zeta values ($p$-adic MZV's) in Section $2$. Next, we prove bounds for the dimensions of $p$-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 $p$-adic analogue of Grothendieck's conjecture about a special element in the motivic Galois group. The bounds come from the rank of $K$-groups of ring of $S$-integers of cyclotomic fields, and these are $p$-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 $p$-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 $p$-adic Hodge theory.

2010 Mathematics Subject Classification: Primary 11R42; Secondary 11G55, 14F42, 14F30.

Keywords and Phrases: $p$-adic multiple zeta values, mixed Tate motives, algebraic $K$-theory, $p$-adic Hodge theory.

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