Implementing 2-descent for Jacobians of Hyperelliptic Curves

2-Descent on the Jacobians of Hyperelliptic Curves

Explicit Descent for Jacobians of Cyclic Covers of the Projective Line

We develop a general method for bounding Mordell-Weil ranks of Jacobians of arbitrary curves of the form y = f(x). As an example, we compute the Mordell-Weil ranks over Q and Q( √ −3) for a non-hyperelliptic curve of genus 8.

Decomposing Jacobians of Hyperelliptic Curves

Many interesting questions can be asked about the decomposition of Jacobians of curves. For instance, we may want to know which curves have completely decomposable Jacobians (Jacobians which are the product of g elliptic curves) [4]. We may ask about number theoretic properties of the elliptic curves that show up in the decomposition of Jacobians of curves [2]. We would also like to know how ma...

The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point

We prove that when all hyperelliptic curves of genus n ≥ 1 having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational p...

Isogenies and the Discrete Logarithm Problem on Jacobians of Genus 3 Hyperelliptic Curves

We describe the use of explicit isogenies to reduce Discrete Logarithm Problems (DLPs) on Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, which are vulnerable to faster index calculus attacks. We provide algorithms which compute an isogeny with kernel isomorphic to (Z/2Z) for any hyperelliptic genus 3 curve. These algorithms provide a rational isogeny...