STNB 2016 (30è any)

The 2004 STNB on Fontaine-Mazur and Serre's conjectures, and its influence on my proofs of cases of these two conjectures


Luis Dieulefait


we will recall how in the 2004 STNB we were among the first to emphasize the importance of "potential modularity" (a result by R. Taylor) in connection to the proof of modularity conjectures. My proof of "existence of compatible systems" and of the first cases of the Fontaine-Mazur conjecture for GL(2) was already of public domain before 2004, and right after the 2004 STNB I came with the key insight (deduced also independently by C. Khare) that potential modularity (combined with level lowering for Hilbert modular forms and other tools) could also be use to prove "existence of minimal lifts", a result that combined with my previous results on Fontaine-Mazur gave as a corollary the proof of the first cases of Serre's conjecture (small level and weight).



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