Algebraic geometry over the complex numbers, and beyond

Question

My question basically is very simple: when did mathematicians start to do algebraic geometry "outside the complex numbers" ?

In the old days, algebraic geometry was solely done over the field of complex numbers (if I am not mistaken), but at some point authors started considering other fields such as finite fields, one of the great culminating points being the Weil conjectures (and their solutions).

How did algebraic geometry evolve from "classical, over the complex numbers" to "modern, over other fields (and more general structures), and in particular finite fields"?

Did this happen before Weil? Or was he involved in this very evolution?

Answer 1

Algebraic geometry began over the field of reals. What Apollonius of Perga did would be certainly qualified today as algebraic geometry: he classified real plane curves of second order and studied their intersections, tangents, etc. in very great detail. Certainly Apollonius had neither real numbers nor the notion of degree, but nevertheless we was able to obtain many non-trivial results.

What Apollonius did over the real field Diophantus did over rationals: he essentially gave a complete study of rational points on curves of degree $2$, and much more. Diophantus' book contains no theorems, only numerical examples. But he really somehow understood some basic facts which nowadays belong to algebraic geometry (rational curves and surfaces, not only in dimension 2 and 3 but in higher dimensions, rational points on elliptic curves do really appear in his work for the first time. He probably even understood to some extent the addition law on a plane non-singular cubic; some of his examples involve it).

The general solution of $x^2+y^2=z^2$ in integers certainly belongs to algebraic geometry over rationals. It was known before Diophantus.

Until the late 19th nobody realized that the work of Apollonius and Diophantus belongs to the same area of mathematics. I think this was clearly understood for the first time in the work of Poincare Sur les proprietes arithmetiques de courbes algebriques (1901).

But one can also say that true algebraic geometry really begins with Decartes who clearly explained the relation between curves in the plane and equations $f(x,y)=0$, thus establishing the notion of algebraic curve and degree. And this was mostly real algebraic geometry. One result of Descartes was his "Rule of Signs" which has been much generalized in real algebraic geometry (fewnomials theory). An outstanding achievement of this real algebraic geometry was Newton's classification of plane real cubics. Bézout's theorem was also proved before the introduction of complex numbers to algebraic geometry.

Complex algebraic geometry begins only in 19th century, first by introduction of complex points to projective geometry.

To Dedekind and Weber belongs the discovery that the theory of algebraic functions is really similar to the theory of algebraic numbers, in the sense that both study finite field extensions, and much of this theory can be exposed in a unified way. They were able to prove many results of Riemann on algebraic curves by purely algebraic methods. From that time people started consciously "doing algebraic geometry over arbitrary field".

References:

I. G. Bashmakova, Diophantus and Diophantine equations, MAA, 1997, MR1483067

I. G. Bashmakova, Arithmetic of algebraic curves from Diophantus to Poincaré, Historia Math. 8 (1981), no. 4, 393–416.

R. Dedekind and H. Weber, Theory of algebraic functions of one variable, AMS 2010. (Several different English translations are available on Internet).

A. Weil, Number Theory. An approach through history. From Hammurapi to Legendre. Birkhauser, 1984.

Answer 2

As others have noted, algebraic geometry has always been done over fields other than the complex numbers. However, in the "old days" (early 20th century), there was a flourishing Italian school of algebraic geometry, which largely worked over the complex numbers, and you can ask when that work was extended to other fields. van der Waerden began the work of making it more general, more algebraic, and more rigorous, in the 1930s and Zariski, Weil, and others continued it. In 1940, Weil observed that if certain statements in Italian geometry made sense and were true over finite fields, then the Riemann hypothesis for curves and abelian varieties over finite fields would follow. He spent the rest of the 1940s writing three books in which he developed the foundations of algebraic geometry, especially curves and abelian varieties, over arbitrary fields, and, indeed, the statements he needed did make sense and were true (e.g., the positivity of the Rosati involution). So Weil was a major figure, but not the only one.