Let k be the algebraic closure of the field with q elements. We build upon recent work of Ulmer and Berger to give examples of elliptic curves and higher dimensional abelian varieties over the field K=k(t) with the property that their ranks become arbitrarily large when dth roots of the variable t are adjoined to K for d varying across the integers relatively prime to q. We also give a first example of an elliptic curve whose rank under such extensions grows linearly in d, for those d prime to q.

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This thesis presents the following: (i) A detailed exposition of Rankin's classical work on the convulsion of two modular L-functions is given; (ii) Let S be the calss Dirichlet series with Euler product on Re(s) > 1 that can be continued analytically to Re(s) = 1 with a possible pole at s = 1. For F,G E S, let F X G be the Euler product convolution of F and G. Assuming the existence of analytic continuation for certain Dirichlet series and some other conditions, it is proved that F x G is non-vanishing on the line Re(s) = 1; (iii) Let Fn be the set of newforms of weight 2 and level N. For f E Fn, let L(sym2f,s) be the associated symmetric square L-function. Let s0=0o + ito with 1 - 1/46 < 0o <1. It is proved that Cs0,EN1-E<#{f E Fn; L (sym2 f, so)=0} for prime N large enough. Here E>0 and Cso,E is a constant depending only on So and E. / vii, 78 leaves ; 29 cm.

We consider various infinite series and examine their arithmetic nature. Series of interest are of the form $$\sum_{n =0}^{\infty} \frac{f(n)A(n)}{B(n)}, \;\;\;\; \sum_{n \in \mathbb{Z}}\frac{f(n)A(n)}{B(n)}, \;\;\;\; \sum_{n=0}^{\infty} \frac{z^n A(n)}{B(n)}$$ where $f$ is algebraic valued periodic function, $A(x), B(x) \in \overline{\mathbb{Q}}[x]$ and $z$ is an algebraic number with $|z| \leq 1$. We also examine multivariable extensions $$\sum_{n_1, \ldots, n_k = 0}^{\infty} \frac{f(n_1, \ldots, n_k)A_1(n_1) \cdots A_k(n_k)}{B_1(n_1) \cdots B_k(n_k)}$$ and $$\sum_{n_1, \ldots, n_k \in \mathbb{Z}} \frac{f(n_1, \ldots, n_k)A_1(n_1) \cdots A_k(n_k)}{B_1(n_1) \cdots B_k(n_k)}.$$ These series are all very natural things to write down and we would like to understand them better. We calculate closed forms using various techniques. For example, we use relations between Hurwitz zeta functions, digamma functions, polygamma functions, Fourier analysis, discrete Fourier transforms, among other objects and techniques. Once closed forms are found, we make use of some of the well-known transcendental number theory including the theorem of Baker regarding linear forms in logarithms of algebraic numbers to determine their arithmetic nature. In one particular setting, we extend the work of Bundschuh \cite{bundschuh} by proving the following series are all transcendental for positive $c \in \mathbb{Q} \setminus \mathbb{Z}$ and $k$ a positive integer: $$\sum_{n \in \mathbb{Z}} \frac{1}{(n^2 + c)^k}, \;\;\; \sum_{n \in \mathbb{Z}} \frac{1}{(n^4 - c^4)^{2k}}, \;\;\; \sum_{n \in \mathbb{Z}} \frac{1}{(n^6 - c^6)^{2k}}, \;\;\; \sum_{n \in \mathbb{Z}} \frac{1}{(n^3 \pm c^3)^{2k}}$$ $$\sum_{n \in \mathbb{Z}} \frac{1}{n^3 \pm c^3}, \;\;\; \sum_{|n| \geq 2} \frac{1}{n^3 -1}, \;\;\; \sum_{|n| \geq 2} \frac{1}{n^4 -1}, \;\;\; \sum_{|n| \geq 2} \frac{1}{n^6 -1}.$$ Bundschuh conjectured that the last three series are transcendental, but we offer the first unconditional proofs of transcendence. We also show some conditional results under the assumption of some well-known conjectures. In particular, for $A_i(x), B_i(x) \in \overline{\mathbb{Q}}[x]$ with each $B_i(x)$ has only simple rational roots, if Schanuel's conjecture is true, the series (avoiding roots of the denominator) $$\mathop{{\sum}}_{n_1, \ldots, n_k =0}^{\infty} \frac{f(n_1, \ldots, n_k)A_1(n_1) \cdots A_k(n_k)}{B_1(n_1) \cdots B_k(n_k)}$$ is either an effectively computable algebraic number or transcendental. We also show that Schanuel's conjecture implies that the series $$\sum_{n \in \mathbb{Z}} \frac{A(n)}{B(n)}$$ is either zero or transcendental, when $B(x)$ has non-integral roots. We develop a general theory, analyzing various infinite series throughout. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-04-23 10:45:19.735

The primary topic of this thesis is the construction of explicit projective equations for the modular curves $X_0(N)$. The techniques may also be used to obtain equations for $X_0^+(p)$ and, more generally, $X_0(N) / W_n$. The thesis contains a number of tables of results. In particular, equations are given for all curves $X_0(N)$ having genus $2 le g le 5$. Equations are also given for all $X_0^+(p)$ having genus 2 or 3, and for the genus 4 and 5 curves $X_0^+(p)$ when $p le 251$. The most successful tool used to obtain these equations is the canonical embedding, combined with the fact that the differentials on a modular curve correspond to the weight 2 cusp forms. A second method, designed specifically for hyperelliptic curves, is given. A method for obtaining equations using weight 1 theta series is also described. Heights of modular curves are studied and a discussion is given of the size of coefficients occurring in equations for $X_0(N)$. Finally, the explicit equations are used to study the rational points on $X_0^+(p)$. Exceptional rational points on $X_0^+(p)$ are exhibited for $p = 73,103,137$ and 191.

For any finite set B and a subset A⊆B, we define the density of A in B to be the value α=|A|/|B|. Roth's famous theorem, proven in 1953, states that there is a constant C>0, such that if A⊆{1,...,N} for a positive integer N and A has density α in {1,...,N} with α>C/loglog N, then A contains a non-trivial arithmetic progression of length three (3AP). The proof of this relies on the following dichotomy: either 1) A looks like a random set and the number of 3APs in A is close to the probabilistic expected value, or 2) A is more structured and consequently, there is a progression P of about length α√N on which A∩P has α(1+cα) for some c>0. If 1) occurs, then we are done. If 2) occurs, then we identify P with {1,...,|P|} and repeat the above argument, whereby the density increases at each iteration of the dichotomy. Due to the density increase in case 2), an argument of this type is called a density increment argument. The density increment is obtained by studying the Fourier transforms of the characterstic function of A and extracting a structure out of A. Improving the lower bound for α is still an active area of research and all improvements so far employ a density increment. Two of the most recent results are α>C(loglog N/log N)^{1/2} by Bourgain in 1999 and α>C(loglog N)^5/log N by Sanders in 2010. This thesis is a survey of progresses in Roth's theorem, with a focus on these last two results. Attention was given to unifying the language in which the results are discussed and simplifying the presentation.

The Erdős discrepancy problem asks, "Does there exist a sequence t = {t_i}_{1≤i<∞} with each t_i ∈ {-1,1} and a constant c such that |∑_{1≤i≤n} t_{id}| ≤ c for all n,c ∈ ℕ = {1,2,3,...}?" The discrepancy of t equals sup_{n≥1} |∑_{1≤i≤n} t_{id}|. Erdős conjectured in 1957 that no such sequence exists. We examine versions of this problem with fixed values for c and where the values of d are restricted to particular subsets of ℕ. By examining a wide variety of different subsets, we hope to learn more about the original problem. When the values of d are restricted to the set {1,2,4,8,...}, we show that there are exactly two infinite {-1,1} sequences with discrepancy bounded by 1 and an uncountable number of in nite {-1,1} sequences with discrepancy bounded by 2. We also show that the number of {-1,1} sequences of length n with discrepancy bounded by 1 is 2^{s2(n)} where s2(n) is the number of 1s in the binary representation of n. When the values of d are restricted to the set {1,b,b^2,b^3,...} for b > 2, we show there are an uncountable number of infinite sequences with discrepancy bounded by 1. We also give a recurrence for the number of sequences of length n with discrepancy bounded by 1. When the values of d are restricted to the set {1,3,5,7,..} we conjecture that there are exactly 4 in finite sequences with discrepancy bounded by 1 and give some experimental evidence for this conjecture. We give descriptions of the lexicographically least sequences with D-discrepancy c for certain values of D and c as fixed points of morphisms followed by codings. These descriptions demonstrate that these automatic sequences. We introduce the notion of discrepancy-1 maximality and prove that {1,2,4,8,...} and {1,3,5,7,...} are discrepancy-1 maximal while {1,b,b^2,...} is not for b > 2. We conclude with some open questions and directions for future work.

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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.