# Questions tagged [weil-conjectures]

The weil-conjectures tag has no usage guidance.

69
questions

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### Pure varieties which are neither smooth nor projective

Recall that a variety $X$ over a finite field $k$ is said to be pure if the eigenvalues of the Frobenius on $i^{\mathrm{th}}$ etale cohomology of $\overline{X}:=X\otimes_k \overline{k}$ have ...

3
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0
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### Is the weight-monodromy conjecture known for unramified representations?

Let $X$ be a smooth proper variety over a number field $K$, $v$ a place of $K$ lying over a prime number $p \neq \ell$, and $V := H^n(X_{\overline{K}};\mathbb{Q}_{\ell})$. Suppose $V$ is unramified at ...

3
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1
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### Purity of Frobenius on cohomology of a projective variety over $\mathbb F_q$ with isolated singularities

Let $X_0$ be a projective variety of dimension $n>0$ over a finite field $\mathbb F_q$ of characteristic $p$. Let $X$ denote its base change to an algebraic closure. Let $\ell$ be a prime number ...

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0
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### Deformation of complex manifolds that admit reduction modulo $p$

Let $(M,B,\omega)$ be a complex analytic family of compact (projective non singular) complex manifolds, where $B \subset \mathbb{C}^{m}$ is some domain. Lets consider a subclass of such manifolds $\{...

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### History of algebraic geometry over finite fields

My question is of historical nature: when did mathematicians start studying algebraic geometry over finite fields in a systematic way, and who were the main driving forces ?
Did it start with Weil (...

2
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0
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### $L$-series and Riemann zeta function

I am currently reading SGA 4$\frac{1}{2}$, exposé 2: Rapport sur la formule des traces.
The $L$-series associated to a scheme $X$ of finite type over $\mathbb{F}_{p}$ is defined as
$$L(X,s):=\prod_{x\...

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0
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### Do Weil cohomology theories for schemes over arbitrary rings exist, and do the standard theorems (Lefschetz fixed point, Tr. Formula etc.) still hold?

A Weil cohomology theory is a functor that assigns to a smooth projective variety $X$ of dimension $d$ over a field $k$ a graded ring of cohomology groups with values in a field $K$ of characteristic $...

6
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1
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### Zeta function of $X = \mathbb{F}_p \mathbb{P}^1$

I'm trying to produce a toy version of the RH Weil conjecture. Solving this could help me to get a good start at understanding where the $1/2$'s come in here, ideally without having to prove the Hard ...

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2
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### Algebraic geometry over the complex numbers, and beyond

My question basically is very simple: when did mathematicians start to do algebraic geometry "outside the complex numbers" ?
In the old days, algebraic geometry was solely done over the ...

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0
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### Are the zeroes of the finite characteristic zeta functions dense in $\left\{s\in\mathbb{C}\mid\mathfrak{Re}(s)=\frac{1}{2}\right\}$?

If $p$ is a prime, $n\in\mathbb{N}$ is a natural number and $C$ is a nonsingular curve over $\mathbb{F}_{p^{n}}$, the $\zeta$ function associated to $C\mid_{\mathbb{F}_{p^{n}}}$ is defined as
\begin{...

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2
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### Could the Weil zeroes of curves be evenly distributed?

If $X$ is a smooth, geometrically connected, projective curve of genus $g$
over $\mathbb{F}_q$, then the zeta function of $X$ is of the form $P(s)/(1 - s)(1 - qs)$, where $P(s)$ is a polynomial of ...

2
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1
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394
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### Why geometric generic point (in abstract algebraic geometry) replace general points in the unit disk?

In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:
On $\mathbb{C}$ Lefshietz local results are as follows. Let $X$ be a ...

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0
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### Meaning of "the" general fiber in the paper "La conjecture de Weil : I"

In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:
Let $X$ be a non singular analytic space and purely of dimension $n+1$....

4
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1
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322
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### The numbers of isomorphism classes of abelian variety over finite fields

It is known that there are only finitely many isomorphism classes of abelian variety over a finite field. I am curious about the exact number of these isomorphism classes.
Explicitly, fix $g$, let $\...

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### Generalization of Weil Conjectures

is there a reference in English, besides Deligne's original publication: "La conjecture de Weil: II", not synthetic but complete that deals with the original argument of the generalization ...

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### Deligne's theorem on exponential sums

I'm an analyst who needs to use Deligne's Theorem 8.4 in 1, but I feel lost in the maze of definitions, and I don't trust my geometric intuition here.
Theorem 8.4: Let $Q$ be a polynomial in $n$ ...

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0
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### Eilenberg-Steenrod cohomological theory versus Weil cohomological theory [closed]

Can someone enlighten me what is the difference between an Eilenberg-Steenrod cohomological theory ( See here, https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms ), and a Weil ...

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### Effective weight-monodromy conjecture

$\DeclareMathOperator\Gr{Gr}$Let $G$ be the absolute Galois group of a finite extension of $\mathbb{Q}_p$ with inertia subgroup $I$, and let $V$ be an $\ell$-adic representation of $G$. Grothendieck's ...

3
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1
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### Asymptotic estimate of the number of points of variety over finite field

EDIT: Let $X$ be a geometrically irreducible $n$-dimensional variety over finite field $\mathbb{F}_{q_0}$. Let $\mathbb{F}_q$ denote any finite extension of $\mathbb{F}_{q_0}$.
It is known (e.g. ...

2
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1
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238
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### 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}\...

5
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1
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369
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### 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,\...

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1
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### 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 ...

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0
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### 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^{\...

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1
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### 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 ...

4
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1
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### 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 ...

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1
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### 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 ...

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0
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### 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)?

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### 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})=\...

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### 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 ...

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### 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 ...

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### 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 $p$-...

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### 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 ...

3
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1
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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### 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 ...

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### 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 ...

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3
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### 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, ...

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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 ...

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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 ...

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### 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
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1
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### "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)(...

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1
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### 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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2
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### 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
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### 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 ...

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### 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 $B$, ...

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1
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### 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
$$ |\...

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2
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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 ...