Explicit classification of isogeny graphs of rational elliptic curves

Let $n>1$ be an integer such that $X_{0}\!\left( n\right)$ has genus $0$, and let $K$ be a field of characteristic $0$ or relatively prime to $6n$. In this article, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a non-trivial isogeny over $\mathbb{Q}$. We achieve this by introducing $56$ parameterized families of elliptic curves $\mathcal{C}_{n,i}(t,d)$ defined over $K(t,d)$, which have the following two properties for a fixed $n$: the elliptic curves $\mathcal{C}_{n,i}(t,d)$ are isogenous over $K(t,d)$, and there are integers $k_{1}$ and $k_{2}$ such that the $j$-invariants of $\mathcal{C}_{n,k_{1}}(t,d)$ and $\mathcal{C}_{n,k_{2}}(t,d)$ are given by the Fricke parameterizations. As a consequence, we show that if $E$ is an elliptic curve over a number field $K$ with isogeny class degree divisible by $n\in\left\{4,6,9\right\}$, then there is a quadratic twist of $E$ that is semistable at all primes $\mathfrak{p}$ of $K$ such that $\mathfrak{p}\nmid n$.Comment: 22 pages; incorporates referee's suggestions; final version to appear in International Journal of Number Theor

Reduced minimal models and torsion

Let $E/\mathbb{Q}$ be an elliptic curve. The reduced minimal model of $E$ is a global minimal model $y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$ which satisfies the additional conditions that $a_{1},a_{3}\in \{0,1\}$ and $a_{2}\in\{0,\pm1\}$. The reduced minimal model of $E$ is unique, and in this article, we explicitly classify the reduced minimal model of an elliptic curve $E/\mathbb{Q}$ with a non-trivial torsion point. We obtain this classification by first showing that the reduced minimal model of $E$ is uniquely determined by a congruence on $c_6$ modulo $24$. We then apply this result to parameterized families of elliptic curves to deduce our main result. We also show that the reduction at $2$ and $3$ of $E$ affects the reduced minimal model of $E$.Comment: 14 page

Lower bounds for the modified Szpiro ratio

Let $E/\mathbb{Q}$ be an elliptic curve. The modified Szpiro ratio of $E$ is the quantity $\sigma_{m}(E) =\log\max\left\{ \left\vert c_{4}^{3}\right\vert ,c_{6}^{2}\right\} /\log N_{E}$ where $c_{4}$ and $c_{6}$ are the invariants associated to a global minimal model of $E$, and $N_{E}$ denotes the conductor of $E$. In this article, we show that for each of the fifteen torsion subgroups $T$ allowed by Mazur's Torsion Theorem, there is a rational number $l_{T}$ such that if $T\hookrightarrow E(\mathbb{Q}) _{\text{tors}}$, then $\sigma_{m}(E) >l_{T}$. We also show that this bound is sharp.Comment: 15 pages; incorporates referee's suggestions; sharpness of lower bounds is no longer conditional on the abc conjecture; final version to appear in Acta Arithmetic

Minimal Models of Rational Elliptic Curves with non-Trivial Torsio

This dissertation concerns the formulation of an explicit modified Szpirobconjecture and the classification of minimal discriminants of rational elliptic curves with non-trivial torsion subgroup. The Frey curve y2=x( x+a) ( x-b) is a two-parameter family of elliptic curves which comes equipped with an easily computable minimal discriminant which helped pave the mathematical bridge that led to the proof of Fermat\u27s Last Theorem. In this dissertation, we extend the ideas of the Frey curve by considering two- and three- parameter families of elliptic curves which parameterize all rational elliptic curves with non-trivial torsion. First, we use these families to give a new proof of a classic result of Frey, Flexor, and Oesterlé which pertains to the primes at which an elliptic curve over a number field can have additive reduction. While our proof gives a weaker variant of the original statement, it is explicit and does not require the Néron model of an elliptic curve. As a consequence of this new proof, we attain our classification of minimal discriminants of rational elliptic curves with non-trivial torsion. In addition, we give necessary and sufficient conditions for when a rational elliptic curve with non-trivial torsion has additive reduction at a given prime. We also study the connection between torsion structure of a rational elliptic curve and the possible reduced minimal models The second theme of this dissertation concerns the modified Szpiro conjecture, which is equivalent to the ABC Conjecture. Roughly speaking, the modified Szpiro conjecture states that certain elliptic curves, known as good elliptic curves, are rare in nature. Masser gave a non-constructive proof which showed that there were infinitely many good Frey curves. In this dissertation, we give a constructive proof of Masser\u27s assertion. We then extend this result by proving that for each of the fifteen torsion subgroups T allowed by Mazur\u27s Torsion Theorem, there are infinitely many good elliptic curves E with torsion subgroup isomorphic to T. This proof is also constructive and allows for the construction of a database which consists of 13870964 good elliptic curves. We provide an analysis of these good elliptic curves to parallel the work done by the ABC@Home project concerning the ABC Conjecture and good ABC triples. The data obtained is then used to formulate an explicit version of the modified Szpiro conjecture. We then show that this explicit formulation allows for the construction of databases of elliptic curves which are exhaustive up to a given conductor. Lastly, we use the classification of minimal discriminants to study the local data of rational elliptic curves at a given prime via Tate\u27s Algorithm. These results and a study of the naive height of an elliptic curve allow us to prove that there is a lower bound on the modified Szpiro ratio which depends only on the torsion structure of an elliptic curve

Representations attached to elliptic curves with a non-trivial odd torsion point

We give a classification of the cuspidal automorphic representations attached to rational elliptic curves with a non-trivial torsion point of odd order. Such elliptic curves are parameterizable, and in this paper, we find the necessary and sufficient conditions on the parameters to determine when split or non-split multiplicative reduction occurs. Using this and the known results on when additive reduction occurs for these parametrized curves, we classify the automorphic representations in terms of the parameters.Comment: 16 page

Local data of rational elliptic curves with non-trivial torsion

By Mazur's Torsion Theorem, there are fourteen possibilities for the non-trivial torsion subgroup $T$ of a rational elliptic curve. For each $T$, such that $E$ may have additive reduction at a prime $p$, we consider a parameterized family $E_T$ of elliptic curves with the property that they parameterize all elliptic curves $E/\mathbb{Q}$ which contain $T$ in their torsion subgroup. Using these parameterized families, we explicitly classify the Kodaira-N\'{e}ron type, the conductor exponent, and the local Tamagawa number at each prime $p$ where $E/\mathbb{Q}$ has additive reduction. As a consequence, we find all rational elliptic curves with a $2$-torsion or a $3$-torsion point that have global Tamagawa number $1$.Comment: 36 pages; incorporates referee's suggestions; final version to appear in Pacific Journal of Mathematic

The FIFA World Cup: Analyses and Interpretations of the World’s Biggest Sporting Spectacle

This special session assembles scholars from various countries with keen interests in the marketing and societal dynamics of the world’s most popular sporting spectacle, the FIFA World Cup. It builds on a stream of research in, for example, ethology (e.g., Morris 1981) economics (e.g., Kuper and Szymanski 2014) consumer behavior (e.g., Shultz 1998; Tavassoli, Shultz and Fitzimons 1995), marketing (e.g., Shultz et al. 2010) and macromarketing (Shultz and Burgess 2011) that explores the systemic complexities of this enormous, global extravaganza and the extent to which it benefits and/or harms myriad players, fans, local and global consumers, companies, governments, societies and the environment