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Register a new userOn Class Numbers, Torsion Subgroups, and Quadratic Twists of Elliptic Curves
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The Mordell-Weil groups $E(mathbb{Q})$ of elliptic curves influence the structures of their quadratic twists $E_{-D}(mathbb{Q})$ and the ideal class groups $mathrm{CL}(-D)$ of imaginary quadratic fields. For appropriate $(u,v) in mathbb{Z}^2$, we define a family of homomorphisms $Phi_{u,v}: E(mathbb{Q}) rightarrow mathrm{CL}(-D)$ for particular negative fundamental discriminants $-D:=-D_E(u,v)$, which we use to simultaneously address questions related to lower bounds for class numbers, the structures of class groups, and ranks of quadratic twists. Specifically, given an elliptic curve $E$ of rank $r$, let $Psi_E$ be the set of suitable fundamental discriminants $-D<0$ satisfying the following three conditions: the quadratic twist $E_{-D}$ has rank at least 1; $E_{text{tor}}(mathbb{Q})$ is a subgroup of $mathrm{CL}(-D)$; and $h(-D)$ satisfies an effective lower bound which grows asymptotically like $c(E) log (D)^{frac{r}{2}}$ as $D to infty$. Then for any $varepsilon > 0$, we show that as $X to infty$, we have $$#, left{-X < -D < 0: -D in Psi_E right } , gg_{varepsilon} X^{frac{1}{2}-varepsilon}.$$ In particular, if $ell in {3,5,7}$ and $ell mid |E_{mathrm{tor}}(mathbb{Q})|$, then the number of such discriminants $-D$ for which $ell mid h(-D)$ is $gg_{varepsilon} X^{frac{1}{2}-varepsilon}.$ Moreover, assuming the Parity Conjecture, our results hold with the additional condition that the quadratic twist $E_{-D}$ has rank at least 2.
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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.
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 $E!left(mathbb{Q}right) _{text{tors}}cong T$.
Analogously to primes in arithmetic progressions to large moduli, we can study primes that are totally split in extensions of $mathbb{Q}$ of high degree. Motivated by a question of Kowalski we focus on the extensions $mathbb{Q}(E[d])$ obtained by adjoining the coordinates of $d$-torsion points of a non-CM elliptic curve $E/mathbb{Q}$. A prime $p$ is said to be an outside prime of $E$ if it is totally split in $mathbb{Q}(E[d])$ for some $d$ with $p<|text{Gal}(mathbb{Q}(E[d])/mathbb{Q})| = d^{4-o(1)}$ (so that $p$ is not accounted for by the expected main term in the Chebotarev Density Theorem). We show that for almost all integers $d$ there exists a non-CM elliptic curve $E/mathbb{Q}$ and a prime $p<|text{Gal}(mathbb{Q}(E[d])/mathbb{Q})|$ which is totally split in $mathbb{Q}(E[d])$. Furthermore, we prove that for almost all $d$ that factorize suitably there exists a non-CM elliptic curve $E/mathbb{Q}$ and a prime $p$ with $p^{0.2694} < d$ which is totally split in $mathbb{Q}(E[d])$. To show this we use work of Kowalski to relate the question to the distribution of primes in certain residue classes modulo $d^2$. Hence, the barrier $p < d^4$ is related to the limit in the classical Bombieri-Vinogradov Theorem. To break past this we make use of the assumption that $d$ factorizes conveniently, similarly as in the works on primes in arithmetic progression to large moduli by Bombieri, Friedlander, Fouvry, and Iwaniec, and in the more recent works of Zhang, Polymath, and the author. In contrast to these works we do not require any of the deep exponential sum bounds (ie. sums of Kloosterman sums or Weil/Deligne bound). Instead, we only require the classical large sieve for multiplicative characters. We use Harmans sieve method to obtain a combinatorial decomposition for primes.
Let $q$ be a prime with $q equiv 7 mod 8$, and let $K=mathbb{Q}(sqrt{-q})$. Then $2$ splits in $K$, and we write $mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_infty$ be the unique $mathbb{Z}_2$-extension of $K$ unramified outside $mathfrak{p}$ with $n$-th layer $K_n$. For certain quadratic extensions $F/K$, we prove a simple exact formula for the $lambda$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $mathfrak{p}$ of the field $F_infty = FK_infty$. Equivalently, our result determines the exact $mathbb{Z}_2$-corank of certain Selmer groups over $F_infty$ of a large family of quadratic twists of the higher dimensional abelian variety with complex multiplication, which is the restriction of scalars to $K$ of the Gross curve with complex multiplication defined over the Hilbert class field of $K$. We also discuss computations of the associated Selmer groups over $K_n$ in the case when the $lambda$-invariant is equal to $1$.
We present a method for constructing optimized equations for the modular curve X_1(N) using a local search algorithm on a suitably defined graph of birationally equivalent plane curves. We then apply these equations over a finite field F_q to efficiently generate elliptic curves with nontrivial N-torsion by searching for affine points on X_1(N)(F_q), and we give a fast method for generating curves with (or without) a point of order 4N using X_1(2N).
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