General Langlands conjectures

Establish the general conjectures of the Langlands program, including proving that every motivic L-function equals an automorphic L-function and clarifying the predicted connections between Galois representations and automorphic representations.

Background

The paper summarizes the Langlands program as a vast web of conjectures connecting Galois representations and automorphic representations. A central prediction is that motivic L-functions are automorphic.

Despite major successes (e.g., modularity results leading to Fermat’s Last Theorem), the broad conjectural framework remains unproven in general.

References

This extension of class field theory to the non-abelian case is a vast web of conjectures, initiated by Langlands in the 1960s, it predicts deep connections between Galois representations and automorphic representations. The program has achieved spectacular successes (Wiles' proof of Fermat's Last Theorem used the Taniyama-Shimura conjecture, now a theorem) but the general conjectures remain open.

— The Riemann Hypothesis: Past, Present and a Letter Through Time  (2602.04022 - Connes, 3 Feb 2026) in Subsubsection Langlands Program