On the height constant for curves of genus two
Implementing 2-descent for Jacobians of hyperelliptic curves
On the height constant for curves of genus two, II
On the number of rational squares at fixed distance from a fifth power
Rational points on curves
This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve over . The focus is on practical aspects of this problem in the case that the genus of is at least , and therefore the set of rational points is finite.
Explicit Selmer groups for cyclic covers of ℙ¹
For any abelian variety J over a global field k and an isogeny ϕ: J → J, the Selmer group is a subgroup of the Galois cohomology group , defined in terms of local data. When J is the Jacobian of a cyclic cover of ℙ¹ of prime degree p, the Selmer group has a quotient by a subgroup of order at most p that is isomorphic to the ‘fake Selmer group’, whose definition is more amenable to explicit computations. In this paper we define in the same setting the ‘explicit Selmer group’, which is isomorphic...
On the Diophantine equation
The Cassels-Tate pairing on polarized Abelian varieties.
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