The minimal height of Jacobian fibrations on \(K3\) surfaces (Q1362568)

A Jacobian fibration \(\Phi:X\to C\) on a surface \(X\) is an elliptic fibration on \(X\) with a global section. Its Mordell-Weil group \(\Lambda\) (automorphisms which act as translations on fibers) is equipped with a bilinear form \(\phi\), called ``height pairing''. The minimum for the height is known for rational surfaces, since the Jacobian fibrations on them are classified. The author proves here that on a K3 surface, the minimum for the height is \(11/420\) and this value is attained by some Jacobian fibrations. The proof is obtained by studying, with the aid of a computer, all the possibilities for the root lattice of \((\Lambda,\phi)\).