I read Edward Frenkel's Love and Math. But reanding this book made me wonder about origin of the concept of the sheaf used in algebraic geometry. I think the conclusion that I came to in the process of reading this book is that the concept of the Langlands program, which is a fusion of number theory, curves over finite fields, and Riemann surface. please see the below paragraphs.
In 1940, during the war, Andre Weil was imprisoned in France for refusing to serve in the army. As the obituary published in The Economist put it. While in prison, Weil wrote a letter to his sister Simone Weil, a famous philosopher and humanist. This letter is a remarkable document; in it, he tries to explain in fairly elementary terms the "big picture" of mathematics as he saw it.
The connection between Riemann surfaces and curves over finite fields should now be clear: both come from the same kind of equations, but we look for solutions in different domains, either finite fields or complex numbers. On the other hand, "any argument or result in number theory can be translated, word for word, "to curves over finite fields, as Weil put in his letter. Weil's idea was therfore that curves over finite fields are the objects that intermediate between number theory and Riemann surfaces.Thus, we find a bridge, between number theory and Riemann surfaces, and that is the theory of algebraic curves over finite fields. In other words, we have three parallel tracks or columns: Number Theory, Curves over Finite Fields, Riemann Surfaces
And this book says that automorphic function's geometric likeness is sheaf similar to the fundamental group of the Riemann surface acting as Galois group.
In the end, I think sheaf's motivation is Weil's thinking. This conclusion is right? If it is wrong, Could you explain background of Why Jean pierre Serre thought he should develop the sheaf theory?
