Lemma 5.2

Let and be two reduced definite integral binary quadratic forms such that there exists with . Then . Furthermore

these cases being inclusive.


Since H and are equivalent, they have the same discriminant and represent the same integers. To say that H is reduced implies that P is the minimum of H on and that R is the second minimum, hence and . Equality of discriminants implies , hence since they are both nonnegative.

Let as usual and be the generators of the modular group , let be the usual compact fundamental domain for the modular group in the upper half plane .

Let be a definite binary quadratic form, and the roots of . Since H is reduced in the usual sense, we have .

Now let be an automorphism of H.

If , then it fixes , hence by the usual theory, is either , (if ), or (if ).

If , then it swaps and , so that

Taking imaginary parts, we get a=-d. Taking real parts, then using and , we obtain bP=aQ+cR. Finally, the determinant condition gives . One easily checks that the three conditions a=-d, and bP=aQ+cR are necessary and sufficient conditions for M to be an automorphism of H.

Now we clearly have and , and since H is reduced we have . Thus we obtain the following:

If , then and P=Q=R, so b=0 hence c=-a, d=-a and .

If a=0, then bc=1 and so R=P and , hence b=c and d=0.

If c=0, then and bP=aQ. This implies that either b=0 and Q=0, or b=a and P=Q.

This finishes the proof of the lemma. Note that it follows from this result that the group G of automorphisms of H is always isomorphic to a group of the form , with m=1, m=2 (when Q=0 or P=R or P=Q are the only equalities), m=4 (when P=R and Q=0), or m=6 (when P=Q=R).