I think it is an elliptic equation as in elliptic partial differential equation https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation https://en.

I think it is an elliptic equation as in elliptic partial differential equation https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation https://en.wikipedia.org/wiki/Elliptic_partial_differential_... (the simplest one being the Laplace equation, which is satisfied by harmonic functions). 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!