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.

11G55, 11R42, 14F42, 14F30

p-adic multiple zeta values, mixed Tate motives, algebraic K-theory, p-adic Hodge theory