On principal congruences and the number of congruences of a lattice with more ideals than filters

Let lambda and kappa be cardinal numbers such that kappa is infinite and either 2 <= lambda <= kappa, or lambda = 2(kappa). We prove that there exists a lattice L with exactly lambda many congruences, 2(kappa) many ideals, but only kappa many filters. Furthermore, if lambda >= 2 is an integer of the form 2(m) center dot 3(n), then we can choose L to be a modular lattice generating one of the minimal modular nondistributive congruence varieties described by Ralph Freese in 1976, and this L is even relatively complemented for lambda = 2. Related to some earlier results of George Gratzer and the first author, we also prove that if P is a bounded ordered set (in other words, a bounded poset) with at least two elements, G is a group, and kappa is an infinite cardinal such that kappa >= vertical bar P vertical bar and kappa >= vertical bar G vertical bar, then there exists a lattice L of cardinality kappa such that (i) the principal congruences of L form an ordered set isomorphic to P, (ii) the automorphism group of L is isomorphic to G, (iii) L has 2(kappa) many ideals, but (iv) L has only kappa many filters.