Langrands Program & Weil’s Rosetta stone

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)
to:
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…