Number theoretic problems leading to elliptic curves

Question

I am looking for basic number theoretic problems that can be addressed by the use of elliptic curves. I know few of problems such as congruent number problem, or the problem of extending a Diophantine triple $\{a,b,c\}$ to a quadruple $\{a,b,c,x\} $satisfying $ax+1 = \square, ~bx+1 = \square, ~cx+1 = \square,$ where a Diophantine triple is set consisting three integers with the property that product of any two increased by one is always a perfect square e.g. $\{1,3,8\}$ and $\square$ denotes perfect square. This problem can be addressed by the elliptic curve $E: y^2 = (ax+1)(bx+1)(cx+1).$ I am looking for some more problems that can have relations with elliptic curve. What are those problems and what can be a source to understand them?

Answer 1

Um, all of them? (That's an exaggeration, but only slightly.)

For example consider the $n=3$ case of Fermat's Last Theorem. You're looking for nontrivial integer solutions to $a^3 + b^3 = c^3$. Dividing by $c^3$ and setting $x = a/c$ and $y = b/c$, we get the new equation $x^3 + y^3 = 1$, where now we are looking for rational solutions instead of integer solutions (since if $a,b,c$ are integers then $x,y$ are rational numbers, and vice-versa). If you know enough general theory, you may recognize that $x^3+y^3 = 1$ is an elliptic curve. Explicitly, if we set $C : x^3 + y^3 = 1$ and $E : Y^2 - 9Y = X^3 - 27$, then the transformation \begin{align*} X &= \frac{3 y}{1-x} \\ Y &= \frac{9 x}{x-1} \end{align*} maps $C$ to $E$, and the inverse transformation is \begin{align*} x &= \frac{Y}{Y - 9} \\ y &= \frac{3 X}{9 - Y} \end{align*}

mapping $E$ to $C$. Hence by finding rational points on the elliptic curve $E$, we can find rational points on $C$, and vice-versa. To find rational points on $E$, we can use the general theory of elliptic curves. For example, as told by Sagemath:

sage: E = EllipticCurve([0,0,-9,0,-27])
sage: E.rank()
0
sage: E.torsion_points()
[(0 : 1 : 0), (3 : 0 : 1), (3 : 9 : 1)]

This computation indicates that $\{\mathcal{O}, (3,0), (3,9)\}$ are the only rational points on $E$. Translating back to $C$ using the above equations, we find that $(x,y) = (1,0)$ and $(x,y) = (0,1)$ are the only rational points on $C$. So that's how you prove one of the cases of Fermat's Last Theorem using the general theory of elliptic curves.

A similar trick works with the $n=4$ case. If $a^4 + b^4 = c^2$, then divide by $b^4$ and substitute $x = a/b$ and $y = c/b^2$ to obtain $C : x^4 + 1 = y^2$. Then $C$ is an elliptic curve, birational to $E : Y^2 = X^3 - 4X$, with equations \begin{align*} X &= \frac{2x(y+1)}{x^3} \\ Y &= \frac{4(y+1)}{x^3} \\ x &= \frac{2X}{Y} \\ y &= \frac{2X^3-Y^2}{Y^2} \end{align*} mapping $C$ to $E$ and vice-versa. The only rational points on $E$ are $\{\mathcal{O}, (0,0), (\pm 2,0)\}$, from which one can deduce that $a^4+b^4=c^2$ has no nontrivial integer solutions.

Unfortunately, the elliptic curve train ends here. The $n=5$ and higher cases of Fermat's Last Theorem are not amenable to elementary elliptic curve manipulations (not counting Wiles' proof here, which of course is based on elliptic curves, but uses very advanced theory).