Thesis (Ph. D.)--University of Wisconsin--Madison, 1971. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Undergraduate honors paper--Mount Holyoke College, 2006. Dept. of Mathematics. / Includes bibliographical references (leaves 66-67).

For definite quadratic lattices over the rings of integers of algebraic number fields, it is shown that lattices are determined up to isometry by their local structure and sublattices of codimension 1. In particular, a theorem of Yoshiyuki Kitaoka for $\mathbb{Z}$-lattices is generalized to definite lattices over algebraic number fields.

Doctor of Philosophy / Department of Mathematics / Christopher G. Pinner / For a fixed integer n ≥ 2, we show that a permutation of the least residues mod p of the form f(x) = Ax[superscript k] mod p cannot map a residue class mod n to just one residue class mod n once p is sufficiently large, other than the maps f(x) = ±x mod p when n is even and f(x) = ±x or ±x [superscript (p+1)/2] mod p when n is odd. We also show that for fixed n the image of each residue class mod n contains every residue class mod n, except for a bounded number of maps for each p, namely those with (k −1, p−1) > (p−1)/1.6n⁴ and A from a readily described set of size less than 1.6n⁴. For n > 2 we give O(n²) examples of f(x) where the image of one of the residue classes mod n does miss at least one residue class mod n.

Master of Science / Department of Mathematics / Christopher Pinner / In 1988, Masser and Oesterlé conjectured that if A,B,C are co-prime integers satisfying A + B = C, then for any ε > 0, max{|A|,|B|,|C|}≤ K(ε)Rad(ABC)[superscript]1+ε, where Rad(n) denotes the product of the distinct primes dividing n. This is known as the ABC Conjecture. Versions with the ε dependence made explicit have also been conjectured. For example in 2004 A. Baker suggested that max{|A|,|B|,|C|}≤6/5Rad(ABC) (logRad(ABC))ω [over] ω! where ω = ω(ABC), denotes the number of distinct primes dividing A, B, and C. For example this would lead to max{|A|,|B|,|C|} < Rad(ABC)[superscript]7/4. The ABC Conjecture really is deep. Its truth would have a wide variety of applications to many different aspects in Number Theory, which we will see in this report. These include Fermat’s Last Theorem, Wieferich Primes, gaps between primes, Erdős-Woods Conjecture, Roth’s Theorem, Mordell’s Conjecture/Faltings’ Theorem, and Baker’s Theorem to name a few. For instance, it could be used to prove Fermat’s Last Theorem in only a couple of lines. That is truly fascinating in the world of Number Theory because it took over 300 years before Andrew Wiles came up with a lengthy proof of Fermat’s Last Theorem. We are far from proving this conjecture. The best we can do is Stewart and Yu’s 2001 result max{log|A|,log|B|,log|C|}≤ K(ε)Rad(ABC)[superscript]1/3+ε. (1) However, a polynomial version was proved by Mason in 1982.

We provide first a purely VOA-theoretic guide to the theory of coordinate transformations for a VOA in direct accordance with its first appearance in a paper of Zhu. Among these results, we are able to obtain new closed-form expressions for the square-bracket Heisenberg modes. We then elaborate on the connection to p-adic modular forms which arise as characters of states in p-adic VOAs. In particular, we show that the image of the p-adic character map for the p-adic Heisenberg VOA contains infinitely-many p-adic modular forms of level one which are not quasi-modular. Finally, we introduce a new VOA structure obtained from the Artin-Hasse exponential, and serving as the p-adic analogue of the square-bracket formalism. / Thesis / Master of Science (MSc)

In the 1970's Serre conjectured a correspondence between modular forms and two-dimensional Galois representations. Ash, Doud, and Pollack have extended this conjecture to a correspondence between Hecke eigenclasses in arithmetic cohomology and n-dimensional Galois representations. We present some of the first examples of proven cases of this generalized conjecture.

This paper examines the theory of an adelization of Shintani's zeta function, especially as it relates to density theorems for discriminants of cubic extensions of number fields. / Mathematics

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