Citation
Hanlon, Philip James (1981) Applications of the Quaternions to the Study of Imaginary Quadratic Ring Class Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/2G8AY604. https://resolver.caltech.edu/CaltechETD:etd04122004144125
Abstract
Let m = m_{1}f^{2} where m_{1} is a squarefree 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(√m_{1}). The proof given by Gauss involves the arithmetic of binary quadratic forms; Venkow (see [12]) obtained an alternative proof by embedding the field ℚ(√m_{1}) 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 B_{1}(m) be the set of quaternions in E with trace 0 and norm m. The group U acts on B_{0}(m) by conjugation; let B_{1}(m) denote the set of orbits of B_{0}(m) under the action of U. For µ = ui_{1} + vi_{2} + wi_{3}εB_{1}(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(√m_{1}) and its prime divisor ℙ_{2} is a regular ideal with respect to the discriminant 4m. It is shown that the class containing ℙ_{2} 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 ℂ^{2} contains ℙ_{2}. 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.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  15 May 1981 
Record Number:  CaltechETD:etd04122004144125 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd04122004144125 
DOI:  10.7907/2G8AY604 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  1351 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  14 Apr 2004 
Last Modified:  14 Jun 2023 23:07 
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