To Shenghao: About papers on L-functions

[Yinbang Lin]

Dear Shenghao,

Previously, I asked you about my thesis on L-functions. You've recommended several books. But it's too huge a topic, it's impossible to read the books in such a short time. Maybe following some papers not too difficult would be an effective way. So I would like to know about your experience of studying Weil conjeture.

During your studying Weil conjecture, are there any interesting and important problems relating to L-functions? Solved or unsolved, both are all right. Could you recommend some papers and paper sources? If it's possible, please order the papers according to their difficulty.

By the way, I am working on this thesis with another student, 叶之林.

Thank you for your reply.

Best wishes,

Yinbang

[tt]

Hartshorne's Algebraic Geometry, Appendix C is a good introduction to
the Weil Conjectures, and it is less technical (compared to Milne's
Etale Cohomology book / lecture note, for instance). You may try to
solve some exercises in Hartshorne's book. My course paper <Weil Conj
of Elliptic curves> (available on my webpage) is to solve some
exercises in Hart 4.4, following Hasse's idea of proof. One can also
find the proof of the Weil conj for elliptic curves in Silverman I.
There are some exercises in Hart 5.1 and App. C on Weil's proof of his
conjecture for general curves, which you may try (after learning a bit
intersection theory on surfaces in 5.1).

The rationality part of the Weil conjecture is not easy to prove
(originally due to Dwork), unless one uses etale cohomology and in
particular, the Lefschetz-Grothendieck trace formula. So maybe you can
reserve this for graduate study. For general varieties, the Riemann
Hyp part of the Weil conj is much more difficult than the rationality
part, but for curves, it's possible to give an elementary proof using
only geometry. The reason Weil could prove his conjecture for curves,
even before the invention of etale cohomology, I guess, is that for
curves, the only interesting cohomology group is H^1, and the Jacobian
of a curve tells us all info of its H^1. More precisely, the Tate
module of the Jacobian (viewed as the "Lie algebra" of this group
variety) can be identified with H_1, the dual of H^1. You may remember
that, in my talks in Zheda last summer, we discussed Tate modules of
elliptic curves but not etale cohomology, and that was enough to prove
the Weil conj in this case.

good luck for both of you!

[Yi Zhu]

Hi, Yingbang,

Your proposal for the thesis sounds very fun. I dont know much on the L-functions but if you want to get the hands on the Weil conjecture. I have some advice.

I think first you should know how to prove weil conjecture for projective spaces. That is a direct computation and trivial. The next nontrivial and fundamental example is elliptic curves. I recommend Silverman's book on elliptic curve, the chapter on elliptic curves over finite fields. To prove Weil on elliptic curves, we dont need etale cohomology. Shenghao also has notes on elliptic curves on his webpage. After elliptic curves, one should try to prove for general curves and abelian varieties. Maybe it is too hard if you want to do it in Grothendieck's way. However, for Weil conjecture of Curves, except the Riemann Hypothesis, classically it can be proved by Riemann-Roch computation. That is worth to do it. The proof appears in Eichler -- page 103 notes of "Modular Functions and Modular Forms" - J.S. Milne.

Best,

Yi

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