Let m = m1f2 where m1 is a square-free positive integer and m is congruent to 1 or 2 mod 4. A theorem of Gauss (see [5]) states that the number of ways to write m as a sum of 3 squares is 12 times the size of the ring class group with discriminant -4m in the field ℚ(√-m1). The proof given by Gauss involves the arithmetic of binary quadratic forms; Venkow (see [12]) obtained an alternative proof by embedding the field ℚ(√-m1) in the quaternion algebra over ℚ. This thesis takes Venkow's proof as its starting point. We prove several further facts about the correspondence established by Venkow and apply these results to the study of imaginary quadratic ring class groups.
Let H denote the quaternion algebra over ℚ, let E denote the maximal order in H and let U denote the group of 24 units in E. Let B1(m) be the set of quaternions in E with trace 0 and norm m. The group U acts on B0(m) by conjugation; let B1(m) denote the set of orbits of B0(m) under the action of U. For µ = ui1 + vi2 + wi3εB1(m) we let [u,v,w] denote the orbit containing µ.
Venkow proved Gauss's result by defining a sharply transitive action of Γ(m), the ring class group with discriminant -4m, on B(m). In chapter 2 we establish some more subtle properties of this action. The prime 2 ramifies in the extension ℚ(√-m1) and its prime divisor ℙ2 is a regular ideal with respect to the discriminant -4m. It is shown that the class containing ℙ2 maps [u,v,w] to [-u,-w,-v]. It is shown that if an ideal class ℂ maps [r,s,t] to [u,v,w] then the class ℂ-1 maps [-r,-s,-t] to [-u,-v,-w]. From these two facts, several results follow. If ℂ maps [r,s,o] to [u,v,w] then ℂ has order 2 if one of u, v or w is 0. If ℂ maps [r,s,o] to [u,v,v] then ℂ has order 4 and the class ℂ2 contains ℙ2. If ℂ maps [r,s,o] to [u,v,w] then ℂ-1 maps [r,s,o] to [-u,-v,-w]. If m can be written as a sum of two squares then a class ℂ is the square of another class (i.e. ℂ is in the principal genus) if ℂ maps some bundle [u,v,w] to [-u,-v,-w].
We apply these results to the following problem; given an odd prime p and an odd integer n, in which ring class groups are the prime divisors of p regular ideals in classes of order n? It is shown that the number of such ring class groups having discriminant -4m where m is a sum of two squares is related to the class number h(-4p) of the field ℚ(√-p). For n = 3 the number is given by
1/16 f(p)h(-4p) - 6h(-4p) + 2 if p ≡ 1 mod 4
1/8 f(p)h(-4p) - 6h(-4p) if p ≡ 3 mod 8
0 if p ≡ 7 mod 8
Here f(p) is the number of ways to write p as a sum of 4 squares plus the number of ways to write 4p as a sum of 4 odd squares. A simple algorithm for producing the discriminants of all such ring class groups is given. Similar, but more complicated formulas hold for odd numbers n greater than 3.
