May 13, 2024: Congruent numbers, elliptic curves, the Birch–Swinnerton-Dyer rank conjecture, and applications (for a general audience)

Congruent numbers. We say that a number $n \in \mathbf{Z}_{\geq 1}$ is a congruent number if there are $x,y,z \in \mathbf{Q}$ such that $y^2 = x^2 + n$ and $z^2 = y^2 + n$ , i.e., if the distance of successive numbers $x^2,y^2,z^2$ is $n$ . (One can prove that $n$ is congruent if and only if there is a right triangle with rational side lengths and area $n$ .)

Geometrically, we get an intersection of two surfaces of degree $2$ , and one can prove that the intersection is a plane cubic, e.g., the elliptic curve $E\colon y^2 = x^3 - x$ from the blog entry from December 24, 2023.

The elliptic curve $E\colon y^2 = x^3 - x$ .

The equation of the cubic can be computed using a computer algebra system like Magma.

Elliptic curves. What have we gained? We are in a situation where we have a lot of theoretical results and algorithms. Solutions to plane cubics $E$ carry the structure of an abelian group $E(\mathbf{Q})$ , i.e., one can add points to get new ones. Furthermore, the theorem of Mordell (1922) says that the Mordell–Weil group $E(\mathbf{Q})$ is finitely generated, i.e., that any solution can be obtained by adding finitely many points. Only because of this result, we can represent $E(\mathbf{Q})$ in a computer!

Finite generation means that $E(\mathbf{Q}) \cong E(\mathbf{Q})_\mathrm{tors}\times \mathbf{Z}^r$ for some (algebraic) rank $r \geq 0$ . It turns out that the finite torsion subgroup $E(\mathbf{Q})_\mathrm{tors}$ is easily computed (there are only finitely many possibilities, see e.g. the blog entry from May 14, 2023), but it is a big unsolved open problem to compute $r$ . If one knows $r$ , one can compute a generating set of $E(\mathbf{Q})$ .

The BSD rank conjecture. Conjecturally, one can compute $r$ using the Birch–Swinnerton-Dyer conjecture: $r = r_\mathrm{an}:= \operatorname{ord}_{s=1}L(E,s)$ Note that the analytic rank $r_\mathrm{an}$ is an analytic and local invariant (more about this in a later blog entry), while the rank is algebraic and global in nature.

In algebra, one often has an equality (number we want to compute) $=$ (number we can compute). It is more difficult here: We cannot compute both numbers individually, but assuming we know they are equal, we can compute both of them at the same time.

Applications. For example, this conjecture says that $E(\mathbf{Q})$ is finite iff $r = 0$ iff $L(E,1) \neq 0$ , and this is something computable (Cremona, Dokchitser). For example, the elliptic curve above has $L(E,1) \approx 0.655514388573029952616209897473$ , which proves that it is non-zero (note that a computation can show that an approximate complex number is non-zero, but proving equality to $0$ is much harder!), hence $E(\mathbf{Q}) = E(\mathbf{Q})_\mathrm{tors}$ is finite. One can easily prove that $E(\mathbf{Q})_\mathrm{tors}\cong(\mathbf{Z}/2)^2$ , and this implies that $1$ is not a congruent number!

Another application is Fermat’s Last Theorem for the exponent $n = 3$ . One can compute that the equation $x^3 + y^3 = z^3$ is equivalent to the elliptic curve $E\colon y^2 = x^3 - 432$ and that $r = r_\mathrm{an}= 0$ . One has $E(\mathbf{Q}) \cong\mathbf{Z}/3$ , and this shows that the original equation only has trivial solutions $xyz = 0$ over $\mathbf{Q}$ .