Colloquium publications.

36

THE MADISON COLLOQUIUM.

Hence the special points invariant under transformations
whose characteristic congruences have no integral roots are all
of the form (rp + s, 1), where r and s are integers, r not divisible
by p, while p is a fixed root of a particular one of these congruences (5).
We next show that these p2 - p imaginary special points are
all conjugate under the group G. It suffices to prove that they
are all conjugate with (p, 1), which is invariant under
X ax - y, y' = x.
Now transformation (1) replaces (p, 1) by (R, 1), where
R bp+ d
cp + e
We are to prove that there exist integers b, c, d, e satisfying
(9) be - cd - 1 (mod p),
such that R rp + s, where r and s are any assigned integers
for which r is not divisible by p. Denote the second root of (5)
by p' and multiply the numerator and denominator of R by
cp' + e. Using (9), we get
R =+ - N N = be+ de + ddca, q = c2+ ace e2.
q
We first show* that we can choose integers c and e such that
q - i (mod p), where i is any assigned integer not divisible by p.
If i is a quadratic residue of p, we may take c = 0. Next, let
i be a quadratic non-residue of p. Taking c 4 0, e = kc, we
have
q- cf(k), f(k) = 1 + ak + k2.
Now f(k) - f(K) if and only if K k or K - - a-k. Hence
the p - 1 values of k other than - a/2 give by pairs the same
value off(k). Thus for k = 0, * *, p - 1,f(k) takes 1 + (p-1)
incongruent values, no one a multiple of p [since (5) has no
*If p = 2, then a 0; taking c = 1, e =0, we have q = 1- i (mod 2).