Do you, by chance, have the Manders and Adleman paper as a pdf?

No, sorry, complexity theory is not something I am particularly interested in, though I have been following discussion of Vinay Deolalikar’s “Proof” at this blog

It's indeed clear to me that SAT instances can be encoded as diophantine equations, but the intuitively obvious encoding doesn't give equations with such simple structure, does it?

If that is clear to you, then you are way ahead of me here. Perhaps it has something to do with coding arbitrary diophantine problems into that simple three-parameter two-variable quadratic. But I don't know enough number theory to suggest how that might be possible.