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…
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