1,636 research outputs found
Modular forms and elliptic curves over the field of fifth roots of unity
Let F be the cyclotomic field of fifth roots of unity. We computationally
investigate modularity of elliptic curves over F.Comment: Added appendix by Mark Watkins, who found an elliptic curve missing
from our tabl
On Serre's uniformity conjecture for semistable elliptic curves over totally real fields
Let be a totally real field, and let be a finite set of
non-archimedean places of . It follows from the work of Merel, Momose and
David that there is a constant so that if is an elliptic curve
defined over , semistable outside , then for all , the
representation is irreducible. We combine this with
modularity and level lowering to show the existence of an effectively
computable constant , and an effectively computable set of elliptic
curves over with CM such that the following holds. If
is an elliptic curve over semistable outside , and is prime,
then either is surjective, or for some .Comment: 7 pages. Improved version incorporating referee's comment
A table of elliptic curves over the cubic field of discriminant -23
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
Finding ECM-friendly curves through a study of Galois properties
In this paper we prove some divisibility properties of the cardinality of
elliptic curves modulo primes. These proofs explain the good behavior of
certain parameters when using Montgomery or Edwards curves in the setting of
the elliptic curve method (ECM) for integer factorization. The ideas of the
proofs help us to find new families of elliptic curves with good division
properties which increase the success probability of ECM
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