The thesis centres around two problems in the enumeration of p-groups. Define f_φ(p^m) to be the number of (isomorphism classes of) groups of order p^m in an isoclinism class φ. We give bounds for this function as φ is fixed and m varies and as m is fixed and φ varies. In the course of obtaining these bounds, we prove the following result. We say a group is reduced if it has no non-trivial abelian direct factors. Then the rank of the centre Z(P) and the rank of the derived factor group P|P' of a reduced p-group P are bounded in terms of the orders of P|Z(P)P' and P'∩Z(P). A long standing conjecture of Charles C. Sims states that the number of groups of order p^m is<br/> p^{²andfrasl;₂₇m³+O(m²)}. (1) We show that the number of groups of nilpotency class at most 3 and order p^m satisfies (1). We prove a similar result concerning the number of graded Lie rings of order p^m generated by their first grading.

Doctor of Philosophy / The Hecke character of an abelian variety A/F is an isogeny invariant and the Galois action is such that A is isogenous to its Galois conjugate A^σ if and only if the corresponding Hecke character is fixed by σ. The quadratic twist of A by an extension L/F corresponds to multiplication of the associated Hecke characters. This leads us to investigate the Galois groups of families of quadratic extensions L/F with restricted ramification which are normal over a given subfield k of F. Our most detailed results are given for the case where k is the field of rational numbers and F is a field of definition for an elliptic curve with complex multiplication by K. In this case the groups which occur as Gal(L/K) are closely related to the 4-torsion of the class group of K. We analyze the structure of the local unit groups of quadratic fields to find conditions for the existence of curves with good reduction everywhere. After discussing the question of finding models for curves of a given Hecke character, we use twists by 3-torsion points to give an algorithm for constructing models of curves with known Hecke character and good reduction outside 3. The endomorphism algebra of the Weil restriction of an abelian variety A may be determined from the Grössencharacter of A. We describe the computation of these algebras and give examples in which A has dimension 1 or 2 and its Weil restriction has simple abelian subvarieties of dimension ranging between 2 and 24.

Doctor of Philosophy / The Hecke character of an abelian variety A/F is an isogeny invariant and the Galois action is such that A is isogenous to its Galois conjugate A^σ if and only if the corresponding Hecke character is fixed by σ. The quadratic twist of A by an extension L/F corresponds to multiplication of the associated Hecke characters. This leads us to investigate the Galois groups of families of quadratic extensions L/F with restricted ramification which are normal over a given subfield k of F. Our most detailed results are given for the case where k is the field of rational numbers and F is a field of definition for an elliptic curve with complex multiplication by K. In this case the groups which occur as Gal(L/K) are closely related to the 4-torsion of the class group of K. We analyze the structure of the local unit groups of quadratic fields to find conditions for the existence of curves with good reduction everywhere. After discussing the question of finding models for curves of a given Hecke character, we use twists by 3-torsion points to give an algorithm for constructing models of curves with known Hecke character and good reduction outside 3. The endomorphism algebra of the Weil restriction of an abelian variety A may be determined from the Grössencharacter of A. We describe the computation of these algebras and give examples in which A has dimension 1 or 2 and its Weil restriction has simple abelian subvarieties of dimension ranging between 2 and 24.

Thesis (Ph.D.)--Tufts University, 2001. / Advisers: Mauricio Gutierrez; Zbigniew Nitecki. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 73-74). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

Thesis (Ph.D.)--McMaster University (Canada), 1990. / Source: Dissertation Abstracts International, Volume: 52-10, Section: B, page: 5312. Supervisor: B. Banaschewski.

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--Florida State University, 2004. / Advisor: Dr. Wolfgang Heil, Florida State University, College of Arts and Sciences, Dept. of Mathematics. Title and description from dissertation home page (viewed Sept. 28, 2004). Includes bibliographical references.

Typescript. Includes vita and abstract. Includes bibliographical references (leaves 53-54) Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1967. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 240-243).

Thesis (MSc)--University of Stellenbosch, 2005. / Bibliography. Also available via the Internet.