Arithmetic Geometry

In 1922 Louis Mordell proved that on an elliptic curve, a smooth cubic of genus 1, all the rational points can be generated from finitely many of them by a geometric "chord and tangent" addition. He also guessed that curves of genus 2 or more have only finitely many rational points.

That conjecture stood until 1983, when Gerd Faltings proved it, using machinery Grothendieck's school had built for entirely different reasons. The picture was now complete: genus 0, none or infinitely many rational points; genus 1, a finitely generated group; genus 2 and up, finitely many.

In 1955 a young Japanese mathematician, Yutaka Taniyama, suggested that elliptic curves might be connected to modular forms, highly symmetric functions from a quite different part of mathematics. His colleague Goro Shimura made the idea precise: every elliptic curve over the rationals should correspond to a modular form. Taniyama died in 1958. The conjecture spread through André Weil's 1967 paper, and who deserved credit for it was argued about for decades.

In the mid-1980s Gerhard Frey noticed something remarkable. A solution to Fermat's equation $a^n + b^n = c^n$ would produce an elliptic curve, $y^2 = x(x - a^n)(x + b^n)$, so strange that it could not be modular. Ken Ribet proved this in 1986. Fermat's Last Theorem would follow from the modularity conjecture.

Andrew Wiles had wanted to prove Fermat's theorem since childhood. On hearing of Ribet's result he worked in secret for seven years and announced a proof in 1993. A gap was found, and he repaired it with Richard Taylor in 1994. The proof is recorded on the algebraic geometry page, the field whose tools it used. By 2001 Christophe Breuil, Brian Conrad, Fred Diamond and Taylor had proved modularity for every elliptic curve over the rationals.

Fermat claimed that the only whole-number solutions of

are $x = 3$, $y = \pm 5$, and he was right. But in fractions there are infinitely many, and geometry produces them.

The equation defines an elliptic curve. Draw the tangent line to the curve at the point $(3, 5)$. By implicit differentiation its slope is $\frac{3x^2}{2y} = \frac{27}{10}$. A line meets a cubic curve in three points, counted with multiplicity, and the tangent touches at $(3, 5)$ twice, so it must cross the curve at exactly one more point. Because the line and the point have rational coordinates, the algebra forces that third point to be rational too. Working it out gives

and indeed $\left(\frac{383}{1000}\right)^2 = \left(\frac{129}{100}\right)^3 - 2$. Reflecting in the $x$-axis gives $\left(\frac{129}{100}, -\frac{383}{1000}\right)$. Repeating the construction with tangents and chords through known points produces ever more complicated fractions, without end. Diophantus and Bachet used tricks like this. The modern view sees an addition law: the chord-and-tangent construction makes the rational points into a group.

Mordell's 1922 theorem says that group is always generated by finitely many points. For this curve, one point, $(3, 5)$, generates all of them: the rank is 1. The Birch and Swinnerton-Dyer conjecture predicts that rank from the curve's L-function. For curves of genus 2 or more, Faltings proved that no such endless supply exists: there are only finitely many rational points.

The deepest questions remain open. The Birch and Swinnerton-Dyer conjecture, recorded with algebraic geometry, asks for the rank of an elliptic curve from its L-function. The abc conjecture, a simple-looking statement about $a + b = c$, would imply much of the subject at once. In 2012 Shinichi Mochizuki released a claimed proof of more than 500 pages in a framework almost no one else uses. Six years later Peter Scholze and Jakob Stix identified a step they believe fails. The proof has been published, but it has not persuaded most of the field. Whether the fog has lifted there is itself disputed.