Why are mathematicians more interested in elliptic curves than other algebraic curves?

Question

Why are mathematicians more interested in elliptic curves than other algebraic curves?

There must be some reason that motivates mathematicians to research elliptic curves specifically.

Answer 1

Number theorists (arithmetic geometers) are interested in finding rational points on all diophantine equations. The case of elliptic curves is the first one that we do not know how to solve:

• Diophantine equations in one variable, i.e., polynomials $p(x)$ in one variable with integer coefficients: it is easy to find all the rational roots of a given polynomial $p(x)$ with integer coefficients, using the so-called rational root theorem.

• Diophantine equations in two variables, i.e., planar curves given by $f(x,y)=0$, where $f$ is a polynomial in two variables with integer coefficients. We classify curves according to their genus.

  • Genus 0: the (non-singular) curves of genus $0$ are lines or conic sections. They either have none or infinitely many rational points, and it is well-known how to determine all points (for instance, the Hasse-Minkowski theorem tells us when there is at least one rational point).

  • Genus 1: the (non-singular) curves of genus $1$ with at least one rational points are elliptic curves, which can have exactly $1$ point, finitely many points, or infinitely many points. A lot is known about elliptic curves, but we have not been able to prove that there exists an algorithm that will compute all the rational points on an elliptic curve in a finite amount of time. There is a candidate for an algorithm, known as descent, but this method may not terminate if the Tate-Shafarevich group is infinite.

  • Genus $\geq 2$ ("higher genus"): Faltings' theorem says that a (non-singular) curve of genus $\geq 2$ has only finitely many rational points, but finding all the rational points on a given curve of higher genus can be extremely difficult, and almost nothing is known if the genus is $\geq 4$.

• Diophantine equations in three or more variables... we don't even know how to find the rational points on curves! And yet there are some equations that have mesmerized mathematicians for centuries...