Elliptic fibrations on quartic K3 surfaces with large Picard numbers

Masato Kuwata

Vol. 171 (1995), No. 1, 231–243

DOI: 10.2140/pjm.1995.171.231

Abstract

Let q₁ and q₂ be two binary quartic forms. We consider the diophantine equation q₁(x,y) = q₂(z,w) from the geometric view point. Under a mild condition we prove that the K3 surface defined by the above equation admits an elliptic fibration whose Mordell-Weil group over C(t) has rank at least 12. Next, we choose suitable q₁ and q₂ such that the Mordell-Weil group contains a subgroup of rank 12 defined over ℚ(i) and a subgroup of rank 8 defined over ℚ.

Mathematical Subject Classification 2000

Primary: 14J28

Secondary: 14J27, 14J05, 11G05

Milestones

Received: 6 January 1993

Revised: 22 October 1993

Published: 1 November 1995