Type of Thesis
Katherine E. Stange
We begin with background in algebraic number theory, specifically studying quadratic fields K and rings of integers inside those fields O_K. From there, we study properties of the matrix group PSL_2(O_K) and its congruence subgroup PSL_2(p) for some prime ideal p. The Schmidt arrangement is the orbit of the real line under Mobius transformations described by the matrices in PSL_2(O_K). We then examine how these arrangements are affected when the transformations are limited to the matrices in PSL_2(p); this forms what we will call a congruence-p subarrangement of the full arrangement. We prove some tangency properties of the congruence subarrangement and finally conjecture that the congruence-(2) subarrangement of the Schmidt arrangement of the Eisenstein integers has an Apollonian circle packing structure.
Oliver, Evan, "Congruence Subarragements of the Schmidt Arrangement" (2016). Undergraduate Honors Theses. 1025.