Euler and algebraic geometry

Euler’s work on elliptic integrals is a milestone in the history of algebraic geometry. The founders of calculus understood that some algebraic functions could be integrated using elementary functions (logarithms and inverse trigonometric functions). Euler realized that integrating other algebraic functions leads to genuinely different functions, elliptic integrals. These functions are not something ugly. They have a richer symmetry than the elementary functions. As Abel discovered, their inverses are doubly periodic functions on the complex plane. Although elliptic curves had been studied earlier, indeed in great depth by Fermat, Euler’s analysis clarifies the key points: elliptic curves (algebraic curves of genus 1) are fundamentally different from rational curves, and not only in a negative way. They have a different kind of symmetry, the famous group structure possessed by an elliptic curve. This paper considers two main themes in algebraic geometry descended from Euler’s work: integrals of algebraic functions (in fancier terms, Hodge theory) and birational geometry. In section 1, we reach a major open problem of algebraic geometry: which representations of the fundamental group are summands of the cohomology of some family of algebraic varieties? Or, equivalently: which linear differential equations can be solved by integrals of algebraic functions? One might not expect any good answer to these questions, but in fact there are two promising approaches (the Simpson and Bombieri-Dwork conjectures). Section 2, more elementary, gives an introduction to birational geometry. I hope to explain the significance of the problem of finite generation of the canonical ring, which has just been solved. Thanks to Carlos Simpson for his comments on an earlier version.