Alexander Dahl, York University
"Distribution of class numbers in continued fraction families of real quadratic fields"
Dirichlet's
class
number
formula
for
real
quadratic
fields
shows that
for
a
fundamental
discriminant
$d
>
0$,
the
class
number
of $\mathbb{Q}(\sqrt{d})$
denoted
by
$h(d)$
depends
on
the
ratio
of $L(1,\chi_d)$
and
the
logarithm
of
the
fundamental
unit.
In
order to
study
the
distribution
of
class
numbers,
one
must
therefore control
these
two
values.
One
strategy
is
to
consider
a
family
of discriminants
for
which
the
fundamental
unit
is
explicitly
given in
terms
of
$d$,
and
then
study
the
distribution
for
$L(1,\chi_d)$ for
$d$
in
that
family.
Some
examples
of
such
families
were
studied
by
Chowla
and
Yokoi.
The
fundamental
unit
in
these
cases was
as
small
as
possible,
and
it
was
shown
that
(eventually)
all of
the
$d$
in
these
families
have
$h(d)>1$.
In
a
recent
preprint, A.
Dahl
and
Y.
Lamzouri
take
the
above
approach
to
studying
the distribution
of
class
numbers
in
Chowla's
family,
constructing
a random
model
for
$L(1,\chi_d)$
for
$d$
in
the
family.
In
a
joint work
with
V.
Kala,
we
observe
that
Chowla's
and Yokoi's
families
belong
to
a
larger
class
of
families
whose
fundamental
units
are as
small
as
possible
and
that
arise
from
continued
fractions. These
families
are
defined
by
solutions
to
the
equation
$ \sqrt{d}
=
[\lfloor
\sqrt{d}
\rfloor,\overline{u_1,u_2,\dots,u_{s-1},2\lfloor
\sqrt{d}
\rfloor}]$
with
fixed
coefficients
$u_1,u_2,\dots,u_{s-1}$.
M3 3103