108 research outputs found
Second p descents on elliptic curves
Let p be a prime and let C be a genus one curve over a number field k
representing an element of order dividing p in the Shafarevich-Tate group of
its Jacobian. We describe an algorithm which computes the set of D in the
Shafarevich-Tate group such that pD = C and obtains explicit models for these D
as curves in projective space. This leads to a practical algorithm for
performing 9-descents on elliptic curves over the rationals.Comment: 45 page
Second Isogeny Descents and the Birch and Swinnerton-Dyer Conjectural Formula
Let h be a p-isogeny of elliptic curves. We describe how to perform
h-descents on the nontrivial elements in the Shafarevich-Tate group which are
killed by the dual isogeny h'. This makes computation of p-Selmer groups of
elliptic curves admitting a p-isogeny over Q feasible for p = 5,7 in cases
where an isogeny descent is insufficient and a full p-descent would be
infeasible. As an application we complete the verification of the full Birch
and Swinnerton-Dyer conjectural formula for all elliptic curves over Q of rank
zero or one and conductor less than 5000.Comment: version 2: Definition 2.1 has been corrected; other minor edits and
correction
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective
line, also known as Belyi maps. These methods include a direct approach,
involving the solution of a system of polynomial equations, as well as complex
analytic methods, modular forms methods, and p-adic methods. Along the way, we
pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French
abstract; revised according to referee's suggestion
Explicit Methods in Number Theory
These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes asymptotics for field extensions and class numbers, random matrices and L-functions, rational points on curves and higher-dimensional varieties, and aspects of lattice basis reduction
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