remember that millenium problem solved by perelman? its a problem in topology, but the proof is expressed in terms of elliptic curves which lives in the number theory space, and so far, we havent found a way to express it in terms of topology.
this is pissing topologists off a good bit. that a number theory guy solved their problem.
similar things could happen to the riemann conjecture. proven in unforeseen ways without revealing prime number stuffs. betting on how a particular conjecture is GOING TO be proven sounds far fetched at best.
"an eigenvalue of a certain elliptic equation" is probably an elliptic curve. I know very little about number theory.
There are many techniques involved. What I know about the Poincare Conjecture comes from a story a professor told during one of his classes.
As far as I know, there is no direct link between elliptic PDEs and elliptic curves.
Elliptic PDEs are called like that because the coefficient in the second order term look like those of the equation of an ellipse (as opposed to parabolic or hyperbolic PDEs)
Elliptic curves take their name because they were first studied in the context of elliptic functions, which are certain integrals that arise in computing the arc length of an ellipse. Even then, the kind of elliptic curves used by number theorists are generalizations where the algebraic form of the curve is the same, but the field of coefficients is a finite extension of the rationals (while the first appearance was with complex numbers). There is a relation between the two, but in general the study of the curve over the complex field only tells part of the story.
I am not aware of any more direct relations, but it would be fascinating to learn that there is something that relates them!