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By Mazurs Torsion Theorem, there are fourteen possibilities for the non-trivial torsion subgroup $T$ of a rational elliptic curve. For each $T$, 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 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~$1$.
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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 constructive proofs of special cases 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.
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.
A family $mathcal{F}$ of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups $E(F)[$tors$]$ of those elliptic curves $E_{/F}in mathcal{F}$ can be made uniformly bounded after removing from $mathcal{F}$ those whose number field degrees lie in a subset of $mathbb{Z}^+$ with arbitrarily small upper density. For every number field $F$, we prove unconditionally that the family $mathcal{E}_F$ of elliptic curves over number fields with $F$-rational $j$-invariants is typically bounded in torsion. For any integer $dinmathbb{Z}^+$, we also strengthen a result on typically bounding torsion for the family $mathcal{E}_d$ of elliptic curves over number fields with degree $d$ $j$-invariants.
We present rank-record breaking elliptic curves having torsion subgroups Z/2Z, Z/3Z, Z/4Z, Z/6Z, and Z/7Z.
An elliptic curve $E$ over $mathbb{Q}$ is said to be good if $N_{E}^{6}<max!left{ leftvert c_{4}^{3}rightvert ,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 Massers 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 Mazurs Torsion Theorem, there are infinitely many good elliptic curves $E$ with $E!left(mathbb{Q}right) _{text{tors}}cong T$.
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