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The Zeta Functions of Fermat Curves: A look at Weil's 1949 paper.

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Tags: number-theoryalgebraic-geometrycounting-good

For a long time, I’ve been very interested in the 1949 paper "Numbers of Solutions to Equations in Finite Fields" by Andre Weil, in which Weil derives explicit formulae for the number of points of a Fermat hypersurface over a finite field. It is both interesting on its own, and as history. As history, the results of this paper are what lead Weil to conjecture his "Weil conjectures". However, even in a post-Weil conjecture proven world, having a class of varieties with explicit knowledge of their zeta functions is very nice- these varieties especially give a good view into the general theory of the motives of Hecke characters, which is not something I will touch on, but the interested reader should look at the incredible book "On the Periods of Hecke Characters" by Norbert Schappacher. Primarily, these notes will only be understandable to people with a decent number theory/abstract algebra background. I tried though to make it accessible to a more general audience, but that is still the main audience.

https://github.com/daguyheis/SOME-submission-luna-dole

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The submission is comprehensible only to a small niche of mathematicians, though it does not claim otherwise. It may act as a very useful and helpful guide to mathematicians within that niche looking to understand Weil’s paper, but to borrow the phrasing within the guidelines, there is no empathy for people unfamiliar. The style of writing is informal (a definite bonus given the original paper is probably quite terse) and does not feel alienating. The motivations should be expanded upon for people not already familiar with Weil’s results and their applications

I agree with your statement “Primarily, these notes will only be understandable to people with a decent number theory/abstract algebra background.”

I think is a great piece for personal study, and I think the author did a great job processing and understanding all of this. That being said, I think this submission has a bit of an identity issue and would need to be tuned up significantly as a science communication or education piece. I think the submission really needs a lot more focus on the “why” an audience member should care and be excited about it, or the “why” learning this is useful in actual science rather than being a set of fun facts. Again though, this is a very great result from what looks like some genuine passion for the topic!