Nontrivial elements of Sha explained through K3 surfaces

Abstract

Abstract. We present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of this principal homogeneous space. In an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose Jacobians have nontrivial Tate-Shafarevich group.

Homogeneous spaces and degree 4 del Pezzo surfaces

“…Our main aim will be to show how, given any degree 4 del Pezzo surface V , to find C, δ such that V is the same as V δ (up to linear change in variable); the algorithm in the next section will not require anything of the geometry of H δ or J. For a description of the underlying geometry, see [12], [20] and Lemma 6.1 of [8].…”

“…The following example is the first using this method to have only two Weierstrass points (whereas the examples in [3], [4] have three Weierstrass points; see also the examples in [6], [12], [14]). …”

Rational points on curves

“…In some cases, it is possible to show that there are non-trivial such elements, thereby improving the upper bound on the rank. Two techniques that have been suggested and also used are visualization [6] and the Brauer-Manin obstruction on certain related varieties [1,25,32].…”

Explicit computations on the desingularized Kummer surface

“…Cassels and Flynn already suggested that the 2-Selmer group could be investigated by using twists of S. In 2007 A. Logan and R. van Luijk ( [7]) and P. Corn ([2]) made use of twists of S to find specific curves with nontrivial 2-torsion elements in the Tate-Shafarevich groups of their Jacobians.…”