# Questions tagged [weil-conjectures]

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52
questions

**22**

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**0**answers

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### Weil's reaction to Deligne's proof [migrated]

How did André Weil react when Pierre Deligne finally solved the most important and hardest of the Weil conjectures ?
Is there any written account on this ?
I guess Serre's and Grothendieck's (...

**2**

votes

**1**answer

147 views

### Cancellation in a particular sum

In an attempt to compute cycle counts in an of a certain number theoretic graph, the following estimate was needed.
It is true that
$$\bigg|\sum_{a,b,c\in \mathbb{Z}/p\mathbb{Z}}\bigg(\sum_{d=1}^{p-1}\...

**4**

votes

**1**answer

165 views

### Which $p$-adic valuations of Weil numbers (that is, eigenvalues of Frobenius) are possible?

Let $C$ be a smooth projective curve over a finite field $\mathbb F_q$, $q$ is a power of the characteristic $p$. It is well-known that if $\alpha$ is an eigenvalue of Frobenius acting on $H^1_{et}(C,\...

**1**

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**1**answer

189 views

### Computing weights of $\bar{\mathbb{Q}}_l(1)$ from the definition

This seems to be a trivial question, but I am genuinely confused about it.
The notion of weights as in Deligne's Weil II are defined in terms of eigenvalues of automorphisms that Frobenius morphisms ...

**1**

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**0**answers

94 views

### What is behind the constant in the functional equation for the Hasse-Weil zeta function?

Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ of dimension $n$. The Weil conjectures assert that the zeta function $Z(X_0,t)$ satisfies the functional equation
$$Z(X_0,t) = \pm q^{\...

**20**

votes

**1**answer

993 views

### Is the Hilbert–Pólya intuition vindicated in the function field case?

The Hilbert–Pólya conjecture is the name given to the idea that the "reason" or "explanation" for the collinearity of the non-trivial zeros of the Riemann zeta function $\zeta(s)$ is that they are the ...

**3**

votes

**1**answer

190 views

### Purity of vanishing cycle for proper scheme over DVR with smooth generic fiber

Let $X$ be a scheme proper and flat over a complete discrete valuation ring $O$ with finite residue field $k$, and choose a prime $l$ not equal to characteristic of $k$. Consider the Galois ...

**15**

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**1**answer

445 views

### What is the automorphic interpretation of the Weil conjectures over finite fields

I am very much a beginner in the theory of automorphic forms and I might (will?) make mistakes in what follows. Please correct me.
A loose interpretation of the Langland's philosophy is that to any ...

**2**

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**0**answers

162 views

### Is there any generalization of Weil conjecture for non-smooth variety?

Is there any generalization of Weil conjecture for any non-smooth geometric-connected variety? For example, for more general curve, or at least some numerical evindence (example)?

**7**

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214 views

### The geometric meaning of the sign in the functional equation

Let $X$ be a smooth projective variety of dimension $n=\dim X$ over a finite field $\mathbb{F}_q$. As is well known, its zeta function satisfies a functional equation of the form
$$Z(X,q^{-n}T^{-1})=\...

**4**

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155 views

### Original motivation for pairing definitions

Today the Weil and Tate pairings are used a lot in cryptography. I'm curious, what was the original motivation of Weil and Tate for defining them? (Especially curious about Weil.)
I've understood Weil ...

**4**

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**1**answer

356 views

### Are degrees of polynomials in Weil's zeta function equal/bounded to/by dimensions of SOME cohomologies in non-smooth or non-projective case?

[Edit] Let me make question more focused. It is about details of Weil conjectures.
Rationality of zeta function does NOT require the manifold to be smooth & projective, so zeta function is a ...

**11**

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**0**answers

822 views

### Nick Katz observation: “the rationality of the zeta function!”

In the proceedings "Algebraic Geometry - Arcata 1974" edited by R. Hartshorne there is an article by Nick Katz called "$p$-adic $L$-functions via moduli of elliptic curves". He starts by recalling $p$-...

**6**

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**1**answer

896 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 ...

**2**

votes

**1**answer

165 views

### Geometric (or at least non-cohomological) proof of Lefschetz trace formula for curves

There is an isomorphism between (rational) correspondences on a curve $C/\mathbb{F}_p$ orthogonal to the "valence zero" ones (i.e. orthogonal under intersection pairing to $\{*\}\times C$ and $C\times ...

**24**

votes

**8**answers

2k views

### Relatively concise English expositions of the proofs of the various Weil conjectures

Where can I find relatively concise (i.e. not excessively wordy and waxing poetic about history and intuitions and such, doesn't spend an eternity carefully developing various parts of the theory of ...

**0**

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**0**answers

178 views

### Unexpected isomorphisms between “unrelated fields”

I read in the post Why worry about the Axiom of Choice ? that the existence of isomorphisms between $\overline{\mathbb{Q}_p}$, $p$ any prime, and $\mathbb{C}$, makes some worry about the Axiom of ...

**18**

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**1**answer

503 views

### Weil conjectures for higher dimensional cycles?

Let $X$ be a smooth projective variety over $\mathbb{F}_{q}$. For each pair of positive integers $n$ and $d$, let $\text{Chow}_{n,d}(X)$ denote the (coarse) moduli space of $n$-cycles of degree $d$ on ...

**3**

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**1**answer

310 views

### Weil Conjectures Analog for Multivariate Zeta Functions

We know that the Riemann zeta function can be generalized to multivariate zeta functions.
Is there a multivariate analog of the Weil conjectures?

**8**

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**0**answers

303 views

### Does Stepanov's method extend to complete intersections?

Stepanov (circa 1970) created the polynomial method to limit the rational points of an algebraic curve over $\mathbb{F}_q$, leading to one of several alternative proofs of Weil's Riemann hypothesis ...

**14**

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**1**answer

1k views

### Gabber's original proof of his purity theorem

Gabber's purity theorem is the statement that if $\mathscr{F}$ is a pure perverse sheaf on an open subvariety $j : U \hookrightarrow X$ then so is $j_{!*} \mathscr{F}$.
It is remarkable because it ...

**3**

votes

**3**answers

425 views

### Curve with given Frobenius polynomial

Does there exist a prime $p$ and a smooth genus 2 curve $C / \mathbf{F}_p$ such that the characteristic polynomial of Frobenius on the Tate module of $J(C)$ is given by $(T^2 - p)^2$?
More generally, ...

**13**

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**1**answer

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### Relation between Weil Conjecture and Langlands Program

Recently I read Gelbart's An Elementary Introduction To The Langlands Program, which explained the origin of the program, and this question came to me. For an elliptic curve over finite field, the ...

**12**

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**2**answers

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### Idea of using etale site

I have just read an article which mentions that, when Grothendieck considered using etale morphism, he did borrow the idea from Riemann that multivalued function on an open subset of complex plane ...

**22**

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**1**answer

1k views

### When is “independence of l” known?

My question is for which varieties over local fields is "independence of l" known for
etale cohomology. Say $X/{\mathbb Q}_p$ is a complete non-singular variety and $W_l$ is the (complex) Weil-...

**4**

votes

**1**answer

161 views

### “Inverse problem” for the zeta function [duplicate]

Let $C$ be a smooth, projective, geometrically irreducible curve, of genus $g$, over a finite field $\mathbb{F}_q$. By the Weil conjectures, the zeta function has the shape
$$
Z_C(t)=\frac{P(t)}{(1-t)(...

**8**

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**1**answer

543 views

### What is the current state of the crystalline analogue of the Weil conjectures?

In "F-isocrystals on open varieties results and conjectures" Faltings says:
"Finally, we extend the theory of weights and show as much as possible of the crystalline analogue of the Weil ...

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vote

**2**answers

712 views

### About Weil's proof of “Weil conjectures for curves and abelian varieties”

I know that the Weil's proof of the Weil conjectures for curves and abelian varieties is made under the lenguage of his "Foundation of algebraic geometry", however in "Polarizations and Grothendieck's ...

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**0**answers

129 views

### Is semistability of smooth Weil sheaf preserved under tensor product?

Let $X_0$ be a smooth, geometrically connected scheme over $\mathbb{F}_q$. As usual, let $\tau : \bar{\mathbb{Q}}_{\ell} \simeq \mathbb{C}$ be a fixed isomorphism. Let $\mathcal{C}$ be the category of ...

**5**

votes

**3**answers

548 views

### Reference for counting points over finite fields

The following fact is extremely well known:
Fact. Let $Y$ be a geometrically irreducible variety (not necessarily smooth or proper) over a finite field $k$.
Then there is a constant $B$, ...

**27**

votes

**1**answer

2k 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
$$ |\...

**6**

votes

**2**answers

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### How did Weil prove the Weil conjectures for curves?

I understand that Weil proved the Weil conjectures for curves. I have seen his proof of the third and trickiest part, the "Riemann Hypothesis for curves," but I am curious about how he showed ...

**18**

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**1**answer

2k views

### How many proofs of the Weil conjectures are there?

I hope this this is not seen as too much as jumping on the band-wagon, but here goes.
Deligne's proof of the last of the Weil conjectures is well-known and just part of a huge body of work that has ...

**6**

votes

**1**answer

949 views

### weight monodromy conjecture for curves?

Hi,
Is there a simple proof of the weight monodromy conjecture in the case of a curve over a mixed characteristic local field?
Thanks!

**36**

votes

**6**answers

3k 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 ...

**4**

votes

**4**answers

978 views

### Hodge numbers of reduction mod $p$

Let $X$ be a projective variety defined over a number field $K$, and $p \in \textrm{Spec }\mathcal{O}_K$ a maximal ideal, so that reduction mod $p$ makes sense, and the resulting scheme (mod $p$) $\...

**9**

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**1**answer

915 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?

**17**

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**1**answer

932 views

### On the Hasse-Weil L-function of $P^n$

So let us start with the "simplest" scheme over $Spec(\mathbf{Z})$ namely $X_0=Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_0$ at a prime $p$ is given by $Z_p(T)=1-T$ (a ...

**14**

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**2**answers

1k views

### Can one find the hodge number by counting points over finite fields?

Given a proper smooth variety $X$ of dimension $n$ over $\mathbb{C}$, assume it has a model over a DVR of mixed characteristic $(0,p)$ with residue field $\mathbb{F}_q$, and assume the closed fiber $...

**2**

votes

**2**answers

650 views

### Is the integrality of the zeta function easy?

I'm trying to get the gist of the proof of the Weil conjectures. Let $X$ be a variety over $\mathbb{F}_{p^n}$. A priori $Z(X,t)\in \mathbb{Q}[[t]]$. Due to the Grothendieck-Lefschetz fixed point ...

**5**

votes

**1**answer

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### 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{...

**6**

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**1**answer

932 views

### Serre's Analogue of the Weil Conjectures for Non-Compact Kahler Manifolds

The classical Riemann Hypothesis concerns the locations of zeroes of the Riemann zeta-function, or more generally the Dedekind zeta-functions of number fields. Its analogue for varieties defined over ...

**11**

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**2**answers

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### How would a motivic proof of the Riemann hypothesis over finite fields go?

It is well known that Grothendieck had a different idea than Deligne about how one should go about proving the Riemann hypothesis for finite fields. However, since Grothendieck's desired proof never ...

**8**

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**3**answers

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### Why is the zeta function of a variety over a finite field not a polynomial? (question about motives)

I've been doing some light(?) reading on motives and the standard conjectures in an attempt to put various things that I tangentially know in perspective.
The question is this: the Weil conjectures ...

**5**

votes

**2**answers

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### Direct proof of special case of Hasse's theorem for elliptic curves

Consider the elliptic curve $y^2 = x^3 + x$ over $\mathbb{F}_p$, where $p \equiv 1 \pmod 4$.
If memory serves correctly, the number of points (excluding the point at infinity) is $p - a$ where $a$ is ...

**26**

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**0**answers

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### Is there a Grothendieck-Riemann-Roch type of theorem generalizing Grothendieck's Lefschetz trace formula

Grothendieck deduced that the L-function of a (constructible) $\ell$-adic sheaf on a variety over $\mathbf{F}_p$ is rational from the generalized trace formula.
My first question is based on the ...

**11**

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**0**answers

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### 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 ...

**19**

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**3**answers

3k views

### 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 ...

**17**

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**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 ...

**32**

votes

**4**answers

6k 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......