#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Andrei A. Suslin's Sixtieth Birthday (2010), 687-723

** Go Yamashita **
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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