Narrow Results

Reflection walls of certain primitive vectors in the anti-invariant sublattice of the K3 lattice define Heegner divisors in the period space of Enriques surfaces. We show that depending on the norm of these primitive vectors, these Heegner divisors are either irreducible or have two irreducible components. The two components are obtained as the walls orthogonal to primitive vectors of the same norm but of different type as ordinary or characteristic.

We study well-rounded lattices which come from ideals in quadratic number fields, generalizing some recent results of the first author with Petersen [On ideal well-rounded lattices, Int. J. Number Theory8(1) (2002) 189–206]. In particular, we give a characterization of ideal well-rounded lattices in the plane and show that a positive proportion of real and imaginary quadratic number fields contains ideals giving rise to well-rounded lattices.