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Let $E_{1}$ and $E_{2}$ be elliptic curves defined over a number field $K$. We say that $E_{1}$ and $E_{2}$ are discriminant ideal twins if they are not $K$-isomorphic and have the same minimal discriminant ideal and conductor. Such curves are said to be discriminant twins if, for each prime $\mathfrak{p}$ of $K$, there are $\mathfrak{p}$-minimal models for $E_{1}$ and $E_{2}$ whose discriminants are equal. This article explicitly classifies all prime-isogenous discriminant (ideal) twins over $\mathbb{Q}$. We obtain this classification as a consequence of our main results, which constructively gives all $p$-isogenous discriminant ideal twins over number fields where $p\in\left\{ 2,3,5,7,13\right\}$, i.e., where $X_0(p)$ has genus $0$. In particular, we find that up to twist, there are finitely many $p$-isogenous discriminant ideal twins if and only if $K$ is $\mathbb{Q}$ or an imaginary quadratic field. In the latter case, we provide instructions for finding the finitely many pairs of $j$-invariants that result in $p$-isogenous discriminant ideal twins. We prove our results by considering the local data of parameterized $p$-isogenous elliptic curves., Comment: 35 pages

The symmetric tensor power of graphs is introduced and its fundamental properties are explored. A wide range of intriguing phenomena occur when one considers symmetric tensor powers of familiar graphs. A host of open questions are presented, hoping to spur future research.

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 pages

By an $abc$ triple, we mean a triple $(a,b,c)$ of relatively prime positive integers $a,b,$ and $c$ such that $a+b=c$ and $\operatorname{rad}(abc)0$, there are finitely many $abc$ triples $(a,b,c)$ such that $\operatorname{rad}(abc)^{1+\epsilon}

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 Theory

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: 17 pages; incorporates referee's suggestions; a small correction to the statement of Theorem 3.3; final version to appear in Bulletin of the London Mathematical Society

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 Arithmetica

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 Mathematics

An elliptic curve $E$ over $\mathbb{Q}$ is said to be good if $N_{E}^{6}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert ,c_{6}^{2}\right\} $ where $N_{E}$ is the conductor of $E$ and $c_{4}$ and $c_{6}$ are the invariants associated to a global minimal model of $E$. In this article, we generalize Masser's Theorem on the existence of infinitely many good elliptic curves with full $2$-torsion. Specifically, we prove via constructive methods that for each of the fifteen torsion subgroups $T$ allowed by Mazur's Torsion Theorem, there are infinitely many good elliptic curves $E$ with $E\!\left(\mathbb{Q}\right) _{\text{tors}}\cong T$., Comment: 19 pages; incorporates referee's suggestions; final version to appear in Journal of Number Theory

In this paper, we explicitly classify the minimal discriminants of all elliptic curves $E/\mathbb{Q}$ with a non-trivial torsion subgroup. This is done by considering various parameterized families of elliptic curves with the property that they parameterize all elliptic curves $E/\mathbb{Q}$ with a non-trivial torsion point. We follow this by giving admissible change of variables, which give a global minimal model for $E$. We also provide necessary and sufficient conditions on the parameters of these families to determine the primes at which $E$ has additive reduction. In addition, we use these parameterized families to give new proofs of results due to Frey and Flexor-Oesterl\'{e} pertaining to the primes at which an elliptic curve over a number field $K$ with a non-trivial $K$-torsion point can have additive reduction., Comment: 34 pages; incorporates suggestions by referees; results in section 3 have been strengthened; final version to appear in Research in Number Theory

The Modified Szpiro Conjecture, equivalent to the $abc$ Conjecture, states that for each $\epsilon>0$, there are finitely many rational elliptic curves satisfying $N_{E}^{6+\epsilon}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert,c_{6}^{2}\right\} $ where $c_{4}$ and $c_{6}$ are the invariants associated to a minimal model of $E$ and $N_{E}$ is the conductor of $E$. We say $E$ is a good elliptic curve if $N_{E}^{6}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert,c_{6}^{2}\right\} $. Masser showed that there are infinitely many good Frey curves. Here we give a constructive proof of this assertion., Comment: 9 pages