The Cassels–Tate pairing and the Platonic solids

Abstract

We perform descent calculations for the families of elliptic curves whose m-torsion splits as μ m × Z/ m Z for m=3,4 or 5. These curves are parametrised by the modular curve X( m)≃ P 1, 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)).