30 years of modularity: number theory since the proof of Fermat

13 Responses to 30 years of modularity: number theory since the proof of Fermat

1. Z Y says:

   July 8, 2022 at 5:23 pm

   I’m in that orthogonal space and sincerely appreciate your talk; you touched and linked many of the ideas I wanted to understand and illustrated them with clear examples. I wanted to ask you if you know who were the discoverers of the fact that the trace of the Frobenius gives the point counts? and if you know a paper about how this discovery came to be, from the outside, it seems a fantastic connection

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2. Kevin Ventullo says:

   July 9, 2022 at 2:22 am

   As someone who can only casually keep up with number theory these days, I’d say I am in the intersection of the two audiences. Thank you for sharing!

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3. Bill says:

   July 9, 2022 at 12:55 pm

   Thank you so much, this is so much better than the low quality zoom stream.

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4. VP says:

   July 10, 2022 at 8:53 pm

   @ZY Maybe Lefschetz https://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem

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5. Will Sawin says:

   July 11, 2022 at 2:24 am

   @Z Y The statement that “the trace of Frobenius on étale cohomology is equal to the count of points” was proven by Grothendieck, but, like all early foundational results in étale cohomology, can also be attributed to an intensive collaborative effort of Grothendieck’s school.

   But if you’re interested in who first figured out it should be true, rather than who proved it, it becomes a more complicated question. Well before étale cohomology was conceived, the experts had good reason to believe there would exist some cohomology theory for varieties over finite fields such that the trace of Frobenius on it gives the count of points. Grothendieck was confident that he had found this theory before he had even settled on the best definition for étale cohomology.

   So the idea that the trace of Frobenius on étale cohomology should give the points was due to Grothendieck. The philosophy that there should exist some cohomology theory with this property was, I think, due to Weil more than anyone else. Weil found formulas for the number of points on varieties defined by certain particularly simple equations over finite fields that looked like the trace of an operator acting on the cohomology of the solution set of similar equations over the complex numbers. He wrote down precise conjectures explaining how this should generalize to more complicated equations. His conjectures didn’t mention cohomology, but it was clear that one could prove them if one had a cohomology theory for varieties in finite fields satisfying the property that the trace of Frobenius is the number of points as well as other nice properties.

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   • Persiflage says:

     July 11, 2022 at 3:05 am

     Weil was the name that came to my mind as well.

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6. tk says:

   July 11, 2022 at 7:49 pm

   Weil’s survey from 1949 (“Numbers of solutions of equations in finite fields”) does not seem to mention the Frobenius yet. I remember having read somewhere that Weil were aware that his conjectures would follow from an appropriate cohomology theory with Frobenius action, but that it was Serre, who believed more than Weil in its actual existence and made Grothendieck work on what became the development of etale cohomology.

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8. Ian Agol says:

   July 18, 2022 at 1:13 am

   Great talk! Did you film and edit it yourself, or get some help?

   One very minor point, it was Riley and Jorgensen that independently discovered the hyperbolic metric on the figure 8 knot complement (although I’m not sure if Jorgensen published his results, it appeared in a conference proceedings much later). Thurston found a much more direct way to describe the hyperbolic metric via a polyhedral decomposition after he learned of their result. Thurston credited Jorgensen’s work (which also applied to any torus bundle) partly for inspiring him to formulate the geometrization conjecture and prove it for Haken manifolds.

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   • Persiflage says:

     July 18, 2022 at 10:30 am

     If you think I edited that myself then I’m guessing you haven’t tried editing videos before (or you have too high an opinion of my skills). The University of Chicago has an in-house team who helped out with professional lighting/video/editing, and I paid for it with funds that would otherwise have paid for my travel.

     Concerning the claim about the Borromean rings, I was quite careful to phrase things (“Theorems of Riley and Thurston prove that…”) in a way that didn’t claim that this was *due* to Thurston originally. The implication in this case that this is the world in which Thurston was thinking about and proving theorems about I think is still accurate. My fact checker on this point said my phrasing was admissible!

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     • Ian agol says:

       July 18, 2022 at 1:53 pm

       Ah, of course. Anyway, it was enlightening to learn why you number theorists are interested in 3-manifolds, somehow I hadn’t heard that story before.

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10. a reader says:

    August 4, 2022 at 8:12 pm

    I thought this was very well done. Thanks for sharing!

    Reply