Questions tagged [weil-conjectures]

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24 votes
3 answers
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Are there "motivic" proofs of Weil conjectures in special cases?

This is a question meant as a first step to get into reading more on Weil conjectures and standard conjectures. It is known that the standard conjectures on vanishing of cycles would imply the Weil ...
Anweshi's user avatar
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35 votes
4 answers
8k views

What would a "moral" proof of the Weil Conjectures require?

At the very end of this 2006 interview (rm), Kontsevich says "...many great theorems are originally proven but I think the proofs are not, kind of, "morally right." There should be better proofs......
bhwang's user avatar
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39 votes
6 answers
4k views

Elementary examples of the Weil conjectures

I'm looking for examples of the Weil conjectures---specifically rationality of the zeta function---that can be appreciated with minimal background in algebraic geometry. Are there varieties for which ...
Jonathan Wise's user avatar
31 votes
1 answer
3k views

Is there a cheap proof of power savings for exponential sums over finite fields?

Let $p$ be a large prime, and let $f(x) = P(x)/Q(x)$ be a non-constant rational function over ${\Bbb F}_p$ of bounded degree. From the Weil conjectures for curves, we have a bound of the form $$ |\...
Terry Tao's user avatar
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19 votes
5 answers
4k views

Equivalent statements of the Riemann hypothesis in the Weil conjectures

In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
Brandon Levin's user avatar
17 votes
4 answers
2k views

Weil conjecture for algebraic surfaces

Deligne's proof of the Weil conjecture is difficult. On the other hand, there are some "simpler" proofs of the Weil conjecture in the case of algebraic curves. For instance, in GTM52, one see it ...
Bo Peng's user avatar
  • 1,495
11 votes
0 answers
1k views

Do the Standard Conjectures imply parts of the "Weil II" Riemann Hypothesis?

It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...
bhwang's user avatar
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10 votes
1 answer
1k views

Motivic proof of Weil-conjectures?

Assuming the standard conjectures (and whatever is needed in addition), is there a nice proof of the Weil-conjectures written completely in the language of motives?
Jan Weidner's user avatar
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6 votes
1 answer
2k views

The connection between the Weil conjectures and Ramanujan's conjecture

I'm writing an essay about Ramanujan's conjecture and have some questions: 1 How is Ramanujan's conjecture connected with the Weil conjectures? 2 How could Ramanujan's conjecture be assumed true or ...
user avatar
5 votes
1 answer
1k views

In what way do the Weil Conjectures pertain to Langlands?

For a relative variety $X$ over a ring of integers $O_K$, we can define a zeta function. This zeta function is defined as the product of the zeta functions of the variety specialized to $O_K/\mathfrak{...
James D. Taylor's user avatar