In what you have written the base space is the coset space, however.

I spend much of my time on historical biography. You might want more on Künneth for reasons of family history, local history, institutional history and so on. There may not be so many salient facts, though.

If it ain't necessarily so, it's possibly not.

Well, OK, you don't say why you are interested in the question. For me it is probably related to 2x2 matrices and the geometry of the singular set.

Try Serre's article in Arithmetical Algebraic Geometry. Proceedings of a Conference held at Purdue University, December 5-7, 1963. Edited by O. F. G. Schilling, Harper & Row, Publishers, New York 1965. And en.wikipedia.org/wiki/Local_zeta-function which you are supposed to read before posting here, in fact. Sandwiches which are too think can dislocate your jaw. Split multiplicative has the local zeta of the projective line minus two points: think about that for intuition.

If you are trying to understand the Hasse-Weil functions before the local zeta function of an elliptic curve, that is pretty much trying to run before you can walk.

Usually a change of basis is applied, so you count ordinary lattice points in an certain type of ellipse. The general "lattice points in an ellipse" problem is well known, and you'll have more luck with it on Google. The asymptotic formula is the area (could hardly be anything else).

You might get some insight by considering the whole set or subsets as convex sets. The analysis is hardly mysterious, so geometry could help.

Good luck finding more by "pure thought" than is in Kaplansky's treatment.

It's actually not hard to find the general solution for the $$z_n$$: there is a particular solution, and then you add the general solution of the homogeneous recurrence. So you should try the same strategy for the $$y_n$$.