2001/01/01 by Michael Stoll · 1 citation
Computer Science · Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Computer science #Cryptography and Residue Arithmetic #Descent (aeronautics) #Geography #Hyperelliptic curve #Hyperelliptic curve cryptography #Mathematics #Operating system #Polynomial and algebraic computation #Pure mathematics
paper · pdf · doi:10.4064/aa98-3-4
crossref issued 2001/01/01 · crossref published 2001/01/01 · crossref published-online 2001/01/01 · openalex publication_date 2001/01/01 · crossref created 2008/11/27 · crossref deposited 2011/04/20 · openalex created_date 2025/10/10 · crossref indexed 2026/07/29 · openalex updated_date 2026/07/29
. This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2-Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Q-rational Weierstra point. 1. Introduction Given some curve C over Q , one would like to determine as much as possible of its arithmetical properties. One of the more important invariants is the MordellWeil rank of its Jacobian J , i.e., the free abelian rank of J(Q ) (finite by the Mordell-Weil Theorem). There is no algorithm so far that provably determines this rank, but it is possible (at least in theory) to bound it from above by computing the size of a suitable Selmer group. It is also fairly easy to find lower bounds by looking for independent rational points on the Jacobian. (It can be difficult, however, to find the right number of independent points, when some of the generators are large.) With some luck, both bounds coincide, and the rank is determined. In...