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Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of
-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over
, ordered by height. We describe databases of elliptic curves over
, ordered by height, in which we compute ranks and
-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.
We study elliptic curves over quadratic fields with isogenies of certain degrees. Let
be a positive integer such that the modular curve
is hyperelliptic of genus
and such that its Jacobian has rank
. We determine all points of
defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to
-isomorphism, every elliptic curve over a quadratic field
-isogenous, for some
, to the twist of its Galois conjugate by a quadratic extension
. We determine
explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to
-isomorphism, all elliptic curves with
-isogenies over quadratic fields are in fact
We develop algorithms to turn quotients of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and their behavior under certain quotients. We illustrate the power of our ideas in a new modular normal form algorithm for modules over rings of integers, vastly outperforming classical algorithms.
We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we rigorously prove the formula for 16714 of the 16725 such curves of conductor less than 5000.
We present p-adic algorithms for computing Hecke polynomials and Hecke eigenforms associated to spaces of classical modular forms, using the theory of overconvergent modular forms. The algorithms have a running time which grows linearly with the logarithm of the weight and are well suited to investigating the dimension variation of certain p-adically defined spaces of classical modular forms.
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