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Register a new userA table of elliptic curves over the cubic field of discriminant -23
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Let F be the cubic field of discriminant -23 and O its ring of integers. Let Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of us (PG and DY) computed the cohomology of various Gamma_0 (n), along with the action of the Hecke operators. The goal of that paper was to test the modularity of elliptic curves over F. In the present paper, we complement and extend this prior work in two ways. First, we tabulate more elliptic curves than were found in our prior work by using various heuristics (old and new cohomology classes, dimensions of Eisenstein subspaces) to predict the existence of elliptic curves of various conductors, and then by using more sophisticated search techniques (for instance, torsion subgroups, twisting, and the Cremona-Lingham algorithm) to find them. We then compute further invariants of these curves, such as their rank and representatives of all isogeny classes. Our enumeration includes conjecturally the first elliptic curves of ranks 1 and 2 over this field, which occur at levels of norm 719 and 9173 respectively.
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We prove two theorems concerning isogenies of elliptic curves over function fields. The first one describes the variation of the height of the $j$-invariant in an isogeny class. The second one is an isogeny estimate, providing an explicit bound on the degree of a minimal isogeny between two isogenous elliptic curves. We also give several corollaries of these two results.
Most hypersurfaces in projective space are irreducible, and rather precise estimates are known for the probability that a random hypersurface over a finite field is reducible. This paper considers the parametrization of space curves by the appropriate Chow variety, and provides bounds on the probability that a random curve over a finite field is reducible.
We show that there is essentially a unique elliptic curve $E$ defined over a cubic Galois extension $K$ of $mathbb Q$ with a $K$-rational point of order 13 and such that $E$ is not defined over $mathbb Q$.
Let $C$ be a smooth projective curve over $mathbb{F}_q$ with function field $K$, $E/K$ a nonconstant elliptic curve and $phi:mathcal{E}to C$ its minimal regular model. For each $Pin C$ such that $E$ has good reduction at $P$, i.e., the fiber $mathcal{E}_P=phi^{-1}(P)$ is smooth, the eigenvalues of the zeta-function of $mathcal{E}_P$ over the residue field $kappa_P$ of $P$ are of the form $q_P^{1/2}e^{itheta_P},q_{P}e^{-itheta_P}$, where $q_P=q^{deg(P)}$ and $0letheta_Plepi$. The goal of this note is to determine given an integer $Bge 1$, $alpha,betain[0,pi]$ the number of $Pin C$ where the reduction of $E$ is good and such that $deg(P)le B$ and $alphaletheta_Plebeta$.
This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence for the model, and we make predictions about elliptic curves based on corresponding theorems proved about the model. In particular, the model suggests that all but finitely many elliptic curves over $mathbb{Q}$ have rank $le 21$, which would imply that the rank is uniformly bounded.
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