The great French mathematician André Weil spent the months of February-May 1940 in a prison in Rouen, as a result of what he referred to as “a disagreement with the French authorities on the subject of my military obligations”. Others might have called this “draft evasion”, and the story has something to do with why one of the most famous French mathematicians spent his post-war career not in France, but in Chicago and Princeton.
Weil’s sister was Simone Weil, who could variously be described as a moral, political and religious philosopher, an activist and mystic. She died in 1943 in England, from some combination of tuberculosis and starving herself out of sympathy with her compatriots in occupied France. During her brother’s prison stay they exchanged letters which were later published. One of these letters is a remarkable mathematical document that André Weil wrote to his sister, although she would have had little chance of understanding what he was talking about. It is reproduced in his collected works, and an English translation has just appeared in the latest Notices of the AMS.
The focus of Weil’s letter is the analogy between number fields and the field of algebraic functions of a complex variable. He describes his ideas about studying this analogy using a third, intermediate subject, that of function fields over a finite field, which he thinks of as a “bridge” or “Rosetta stone”. For function fields over a finite field, the analogies with number fields are quite close and many facts one knows about one subject can be used to make conjectures about what is true for the other. Some examples include the Riemann-Roch theorem and the Riemann hypothesis. After getting out of prison and leaving for the U.S., in 1941 Weil was able to prove the Riemann hypothesis for the function field case; of course for the number field case it remains an open problem.
For much more detail about this analogy, there’s an interesting textbook by Dino Lorenzini called An Invitation to Arithmetic Geometry.
Last Updated on