Abstract:
For a given group $G$ and an elliptic curve $E$ defined over a number field $K$, I discuss the problem of finding $G$-extensions of $K$ over which $E$ gains rank. I prove the following theorem, extending a result of Fearnley, Kisilevsky, and Kuwata: Let $n = 3,4,$ or $6$. If $K$ contains its $n^{th}$-roots of unity then, for any elliptic curve $E$ over $K$, there are infinitely many $\mathbb{Z}/n\mathbb{Z}$-extensions of $K$ over which $E$ gains rank.

Abstract:
For an elliptic curve E over a number field K, we prove that the algebraic rank of E goes up in infinitely many extensions of K obtained by adjoining a cube root of an element of K. As an example, we briefly discuss E=X_1(11) over Q, and how the result relates to Iwasawa theory.

Abstract:
Let $E$ be an elliptic defined over a number field $K$. Then its Mordell-Weil group $E(K)$ is finitely generated: $E(K)\cong E(K)_{tor}\times\mathbb{Z}^r$. In this paper, we will discuss the cyclic torsion subgroup of elliptic curves over cubic number fields.

Abstract:
Let $\mathcal{X}$ be a Riemann surface of genus $g>0$ defined over a number field $K$ which is a degree $d$-covering of $\mathbb{P}^1_K$. In this paper we show the existence of infinitely many linearly disjoint degree $d$-extensions $L/K$ over which the Jacobian of $\mathcal{X}$ gains rank. In the case where 0, 1 and $\infty$ are the only branch points, and there is an automorphism $\sigma$ of $\mathcal{X}$ which cyclically permutes these branch points, we obtain the same result for the Jacobian of $\mathcal{X}/\sigma$. In particular if $\mathcal{X}$ is the Klein quartic, then the construction provides an elliptic curve which gains rank over infinitely many degree $7$-extensions of $\mathbb{Q}$. As an application, we show the existence of infinitely many elliptic curves that gain rank over infinitely many cyclic cubic extensions of $\mathbb{Q}$.

Abstract:
Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(3^\infty)$ be the compositum of all cubic extensions of $\mathbb{Q}$. In this article we show that the torsion subgroup of $E(\mathbb{Q}(3^\infty))$ is finite and determine 20 possibilities for its structure, along with a complete description of the $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves, and a complete list of $j$-invariants for each of the 4 that do not.

Abstract:
Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as $S$. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves that appear over the field with smallest absolute value of its discriminant and having Galois group $S_3$ and over the field with smallest absolute value of its discriminant and having Galois group $\Z/3\Z$. Secondly, for all except two groups $G\in S$, we find the field $K$ with smallest absolute value of its discriminant such that there exists an elliptic curve over $K$ having $G$ as torsion. Finally, for every $G\in S$ and every cubic field $K$ we determine whether there exists infinitely many non-isomorphic elliptic curves with torsion $G$.

Abstract:
In this paper, we study the theories of analytic and arithmetic local constants of elliptic curves, with the work of Rohrlich, for the former, and the work of Mazur and Rubin, for the latter, as a basis. With the Parity Conjecture as motivation, one expects that the arithmetic local constants should be the algebraic additive counterparts to ratios of local analytic root numbers. We calculate the constants on both sides in various cases, establishing this connection for a substantial class of elliptic curves. By calculating the arithmetic constants in some new cases, we also extend the class of elliptic curves for which one can determine lower bounds for the growth of p-Selmer rank in dihedral extensions of number fields.

Abstract:
Cyclic codes with two zeros and their dual codes as a practically and theoretically interesting class of linear codes, have been studied for many years. However, the weight distributions of cyclic codes are difficult to determine. From elliptic curves, this paper determines the weight distributions of dual codes of cyclic codes with two zeros for a few more cases.

Abstract:
Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p odd and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity conjecture for such curves. We prove the analogous results for p=2 under the additional assumption that E is not supersingular at primes above 2.

Abstract:
Let F be the cubic field of discriminant -23 and let O be its ring of integers. By explicitly computing cohomology of congruence subgroups of GL(2,O), we computationally investigate modularity of elliptic curves over F.