Algebraic Geometry

The second reason came from topology. In 1857 Bernhard Riemann looked at the complex solutions of a polynomial equation in two variables and saw a surface, a real two-dimensional object. For a smooth cubic curve, that surface is a torus. The number of holes, the genus, turned out to govern the curve's algebra: which functions live on it, and how it can be mapped to other curves.

From then on, the topology of complex varieties and the algebra of their equations were studied together, which is why this field's second parent is algebraic topology. Lefschetz and Hodge carried the connection into higher dimensions in the first half of the twentieth century. The link became decisive when the topology of equations over finite fields was needed.

Around 1890–1930, the Italian school of Guido Castelnuovo, Federigo Enriques and Francesco Severi achieved something extraordinary: a classification of algebraic surfaces, driven by brilliant geometric intuition. But intuition ran ahead of proof. Some arguments turned out to have gaps, some claimed results could not be checked, and by the 1930s no one could say with confidence which theorems were solid.

Oscar Zariski and André Weil responded by rebuilding the field on commutative algebra, the theory of polynomial rings and their ideals. It was a second foundational repair after Hilbert's rebuilding of Euclid, and it made the subject both more rigorous and far more abstract.

In 1949 Weil made a conjecture that shaped the next quarter-century. Take a polynomial equation and count its solutions in finite number systems, the fields with $p$, $p^2$, $p^3, \ldots$ elements. He predicted that these counts are controlled by a zeta function with a rigid structure, and that the structure is dictated by the topology of the same equation's complex solutions. Somehow, counting over finite fields should see holes in a surface.

Proving it required a topology for varieties over finite fields, where there is no ordinary notion of nearness. Alexander Grothendieck built one. Through the 1960s he rewrote the entire field in the language of schemes, which let geometry happen over any commutative ring, including the integers. With collaborators in his Paris seminar he constructed étale cohomology. In 1974 his student Pierre Deligne used it to prove the deepest of Weil's conjectures.

Weil's conjectures begin with a simple act: count the solutions of an equation in a finite number system. Take the elliptic curve

$$y^2 = x^3 - x$$

and work modulo 5, where the only numbers are $0, 1, 2, 3, 4$ and arithmetic wraps around. The squares modulo 5 are $0, 1, 4$ ($2^2 = 4$, $3^2 = 9 \equiv 4$, $4^2 = 16 \equiv 1$). Try each $x$:

$x$	$x^3 - x \pmod 5$	solutions $y$
0	0	$y = 0$
1	0	$y = 0$
2	$6 \equiv 1$	$y = 1, 4$
3	$24 \equiv 4$	$y = 2, 3$
4	$60 \equiv 0$	$y = 0$

That is 7 solutions, plus one "point at infinity" in the projective plane, 8 in all. A naive guess, one point per value of $x$ plus the point at infinity, would give $p + 1 = 6$. In 1933 Helmut Hasse proved that for any elliptic curve and prime $p$ the count $N$ satisfies

$$|N - (p + 1)| \le 2\sqrt{p} .$$

Here $|8 - 6| = 2 \le 2\sqrt 5 \approx 4.47$. Hasse's bound is the Riemann hypothesis for elliptic curves: it says the zeros of the curve's zeta function lie on a critical line. Weil proved the analogue for all curves in the 1940s and conjectured it for all varieties. The error term $2\sqrt p$ is controlled by the curve's genus, its topology over the complex numbers. That link between counting and topology is what Grothendieck's cohomology and Deligne's 1974 proof finally explained.

By then algebraic geometry had become the working language of number theory. Its most famous result is Fermat's Last Theorem: $x^n + y^n = z^n$ has no solutions in positive whole numbers for $n > 2$. It was proved by showing that a hypothetical solution would produce an elliptic curve, a cubic of genus 1, too strange to exist. Andrew Wiles announced the proof in 1993. After a gap was found, he repaired it with Richard Taylor in 1994. The number-theoretic side of that story, from Mordell and Taniyama to Frey and Ribet, is told in arithmetic geometry, where this field and the Number Theory Thread meet.

The field's frontier now runs through two Millennium Prize Problems. The Hodge conjecture asks whether topology is always witnessed by algebra. The Birch and Swinnerton-Dyer conjecture asks whether an elliptic curve's rational points can be read from an analytic function. Beneath both lies an older foundational question, settled by Hironaka in 1964 for characteristic zero but still open over fields of prime characteristic: whether every singular variety can be smoothed out.