## Yen-Liang Kuan, Department of Mathematics, University of Waterloo

### "On the distribution of torsion points modulo primes"

Let
A
be
a
commutative
algebraic
group
dened
over
a
number
field
K.
For
a
prime
}
in
K
where
A
has
good
reduction,
let
N};n
be
the
number
of
n-torsion
F}-rational
points
of
the
reduction
of
A
modulo
}
where
F}
is
the
residue
field
of
}
and
n
is
a
positive
integer.
When
A
is
of
dimension
one

and
n
is
relative
prime
to
a
fixed
finite
set
of
primes
depending
on
A=K,
we
determine
the
average
values
of
N};n
as
the
prime
}
varies.
This
average
value
as
function
of
n
agrees
with
a
divisor
function
for
almost
all
n.