Colloquium publications.

34

THE MADISON COLLOQUIUM.

We do not restrict the coordinates to be integers, but permit
their ratio to be a root of any congruence with integral coefficients
modulo p. A point is called real if the ratio of its coordinates is
rational.
A point (x, y) is invariant under a transformation (1) if
- px, y' = py, or
(2) (b - p)x + dy — 0, cx + (e- p)y- 0 (mod p).
If these congruences hold identically as to x, y, then
d- c - 0, b- e - I1 (modp)
and the transformation is one of the transformations
(3) x' -= ~ x, y' -= - y (mod p),
which leave every point invariant.
A special point is one invariant under at least one transformation (1) not of the form (3). There are p(p2 - 1) transformations (1). We shall assume in the text that p > 2 (relegating to foot-notes the modifications to be made when p = 2).
Then there are two transformations (3). Hence any non-special
point is one of exactly*
(4) c = lp(p2 1)
conjugate points under the group G, while a special point is one
of fewer than o conjugates.
Let (x, y) be a special point and let (1) be a transformation,
not of the form (3), which leaves it invariant. Thus the congruences (2) are not both identities. The determinant of their
coefficients must therefore be divisible by p. Hence p is a root
of the characteristic congruence (in which a = b + e)
(5) p2 - ap + 1 0 (mod p).
First, suppose that (5) has an integral root p. For this value
of p, one of the congruences (2) is a consequence of the other,
and the ratio x: y is uniquely determined as an integer modulo p.
* For p = 2, w is to be replaced by 2(22 - 1) = 6.