Implementing 2-descent for Jacobians of hyperelliptic curves

By Michael Stoll. 2001

172 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/39cf5f1d4d0a7c1c7ea6f7e195d232063241a6e4

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Citations in the corpus

Explicit Chabauty over number fields (2010, methodology): “culations. In particular, MAGMAincludes implementations by Nils Bruin and Michael Stoll of 2-descent on Jacobians of hyperelliptic curves over number ﬁelds; the algorithm is detailed in Stoll’s paper [53]. MAGMAalso includes an implementation of Chabauty for genus 2 curves over Q. 6.1. Case I:y is o”
Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation (2004, background): “We do not assume in this section that D is necessarily in the range (2), merely that −D is not a square.”
Chabauty for symmetric powers of curves (2009, background): “Obtaining a basis for the full Mordell–Weil group is often time consuming for curves of genus 2 and simply not feasible in the present state of knowledge for curves of higher genus.”
Powers from Products of Consecutive Terms in Arithmetic Progression (2006, methodology): “A straightforward 2-descent on JacC (Q) (see for example [7, 40]) shows that the {δi} lie in a group isomorphic to (Z/2Z)(6), generated by (−3,−5, 5, 15, 5), (3, 1,−1,−15, 5), (2, 5, 1, 5, 2), (3, 6, 1, 15, 30), (15, 15, 10, 3, 30), (3, 1, 5, 30, 2).”
How to do a p-descent on an elliptic curve (2003, background): “we now simply find points in E(Qq) (in a random or systematic way, compare [33] for a description in the case of a 2-descent) until their images under Fq generate a space of the correct size.”
Most odd degree hyperelliptic curves have only one rational point (2013, methodology): “eck each of its points for membership in C(Q 2) to compute T:= C(Q 2)∩J(Q 2) tors. Since J(Q 2) tors ⊆J(Q′ 2), we have T⊆C(Q′ 2). If any T∈Tis of odd order greater than 1, then return “no”. Next, use [Sto01] to compute the group Sel 2J, the group J(Q 2)/2J(Q 2), and the map between them. Compute J(Q”
Elliptic curves over ℚ $\mathbb {Q}$ and 2-adic images of Galois (2014, methodology): “via RankBound, an implementation of [46]); computation of the rank of the Jacobian of a genus 3 plane curve has recently been worked out [7], but is often impractical [7, Remark 1.”
Elliptic curves over real quadratic fields are modular (2013, methodology): “Using 2-descent [75] we find that J(Q) has Mordell-Weil rank 0.”
Integral points on hyperelliptic curves (2008, methodology): “The MAGMA programs used for this step are based on Stoll’s papers [44], [45], [46].”