As we approach the first anniversary of Jean-Pierre Wintenberger’s death on 23 Jan 2019, Ken Ribet is giving a lecture at the JMM 2020 on 16 Jan 2020 about the possibility of simplifying the proof of Fermat’s Last Theorem. This is 25 years after it was proved as a corollary of the proof of the semistable case of the Taniyama conjecture by Andrew Wiles.

A summary of the lecture is in the January 2020 Notices of the American Mathematical Society at https://www.ams.org/journals/notices/202001/rnoti-p82.pdf

After Wiles' proof, the full Taniyama conjecture was proved in 2001 by Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor.

However as early as 1975 there has been another way to prove Fermat’s Last Theorem via Serre’s modularity conjecture. This asserts that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form.

This conjecture was proved by Chandrashekhar Khare and Jean-Pierre Wintenberger in 2009.

Unfortunately however this way isn’t much simpler than Wiles’ method.

The Khare-Wintenberger proof is at

Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (I)",

(preprint: https://www.math.ucla.edu/~shekhar/papers/results.pdf )

and

Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (II)",

(preprint: https://www.math.ucla.edu/~shekhar/papers/proofs.pdf )

A summary of the lecture is in the January 2020 Notices of the American Mathematical Society at https://www.ams.org/journals/notices/202001/rnoti-p82.pdf

After Wiles' proof, the full Taniyama conjecture was proved in 2001 by Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor.

However as early as 1975 there has been another way to prove Fermat’s Last Theorem via Serre’s modularity conjecture. This asserts that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form.

This conjecture was proved by Chandrashekhar Khare and Jean-Pierre Wintenberger in 2009.

Unfortunately however this way isn’t much simpler than Wiles’ method.

The Khare-Wintenberger proof is at

Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (I)",

*Inventiones Mathematicae*,**178**(3): 485–504(preprint: https://www.math.ucla.edu/~shekhar/papers/results.pdf )

and

Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (II)",

*Inventiones Mathematicae*,**178**(3): 505–586(preprint: https://www.math.ucla.edu/~shekhar/papers/proofs.pdf )

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