Weil’s Rosetta stone (or Conjecture):
Number Theory (1) | Curves over Finite Fields (2) | Riemann Surfaces (3)
Weil wanted to link up these 3 distinct Maths, as in the Langands Program.
Langrands’ original idea on the Left Column (1) Number Theory & the Middle Column (2):
1. He related :
representations of the Galois groups of number fields (objects studied in number theory)
automorphic functions (objects in harmonic analysis).
2. The middle column (2):
Galois group relevant to curves over finite fields.
Also there exists a branch of harmonic analysis for automorphic functions.
3. How to translate column (3) Riemann Surfaces ?
We have to find geometric analogues of the Galois groups and automorphic functions in the theory of Riemann surfaces.
Next we have to find suitable analogues of the automorphic functions ?
It was a mystery until 1980 solved by the Russian Vladimir Drinfeld (Fields medalist for inventing Quantum Group). He replaced the automorphic functions by Sheaves. His formulation is known as the Geometric Langrands Program.
What is the geometric analogue of the Galois Group in Riemann surfaces ?
It is called the Fundamental Group of a Riemann surface, which focuses on the most salient features of geometric shapes (such as the number of “holes” in a Riemann surface).