Joseph H. Silverman, Brown University
"Finite Orbits of Points on Surfaces that Admit Three Non-commuting Involutions"
The
classical
Markoff-Hurwitz
equation
M
:
x^2
+
y^2
+
z^2
=
axyz
+
b
admits
three
non-commuting
involutions
coming
from
the
three
double
covers
M
-->
A^2.
There
has
recently
been
considerable
interest
in
studying
the
orbit
structure
of
the
(Z/pZ)-points
of
M
under
the
action
of
the
involutions.
In
this
talk
I
will
discuss
some
of
this
history,
and
then
describe
analogous
results
and
conjectures
on
K3
surfaces
W
in
P^1xP^1xP^1
given
by
the
vanishing
of
a
(2,2,2)
form.
Just
as
with
the
Markoff-Hurwitz
surface,
the
three
projections
W
-->
P^1xP^1
are
double
covers
that
induce
three
non-commuting
involutions
on
W.
Let
G
be
the
group
of
automorphisms
of
W
generated
by
these
involutions.
We
investigate
the
G-orbit
structure
of
the
points
of
W.
In
particular,
we
study
G-orbital
components
of
W(Z/pZ)
and
finite
G-orbits
in
W(C).
This
nice
blend
of
number
theory,
geometry,
and
dynamics,
requires
no
pre-requisites
beyond
an
undergraduate
algebra
course.
(This
is
joint
work
with
Elena
Fuchs,
Matthew
Litman,
and
Austin
Tran.)
MC 5501
