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Register a new userTypically bounding torsion on elliptic curves with rational $j$-invariant
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Tyler Genao
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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.
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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.
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~$1$.
We present rank-record breaking elliptic curves having torsion subgroups Z/2Z, Z/3Z, Z/4Z, Z/6Z, and Z/7Z.
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