On functional equations of p-adic polylogarithms. Bull. Soc. Math. France , 119 (1991)
Résumé
ABSTRACT
.
—
The
n-t
h
order
polylogarithm
Ln(z)
i
s
defined
by
th
e
series
E^Ll
zk
''t^
n
on
tn
e
Q?
611
un
it
disc.
This
functio
n
has
multivalued
analytic
prolon-
gation
t
o
C
\
{0,1}
.
The
same
series
^^
z
k
f^
defines
an
analytic
p-adic
functio
n
on
th
e
open
uni
t
disc
in
Cp
(a
completion
o
f
an
algebraic
closure
o
f
Q
a
t
some
place
above
p).
The
global
p-adic
analogues
o
f
th
e
function
s
L\n(z)
are
constructed
in
th
e
frame
o
f
rigid
analysis.
These
function
s
are
p-adic
polylogarithms.
In
thi
s
paper
we
give
sufficient
and
necessary
conditions
t
o
have
a
functiona
l
equation
o
f
polylogarithms
in
terms
o
f
maps
induced
by
rational
function
s
on
etale
fundamenta
l
groups
o
f
projective
lines
minus
finit
e
numbers
o
f
points.