The Cassels–Tate pairing and the Platonic solids Original Research Article

We perform descent calculations for the families of elliptic curves whose m-torsion splits as μm×Z/mZ for m=3,4 or 5. These curves are parametrised by the modular curve X(m)similar, equalsP1, whose cusps are arranged as the vertices of one of the Platonic solids. Following McCallum (Invent. Math. 93 (1988)) we write the Cassels–Tate pairing as a sum of local pairings. In the case m=5 our results extend the work of Beaver (J. Number Theory 82 (2000)).