Questions tagged [weil-conjectures]

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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 ...
Dr. Evil's user avatar
  • 2,641
3 votes
0 answers
227 views

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 ...
David Corwin's user avatar
3 votes
1 answer
284 views

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 ...
Suzet's user avatar
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2 votes
0 answers
102 views

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 $\{...
Mishkaat's user avatar
  • 321
4 votes
0 answers
205 views

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 (...
THC's user avatar
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2 votes
0 answers
148 views

$L$-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\...
The Thin Whistler's user avatar
3 votes
0 answers
226 views

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 $...
The Thin Whistler's user avatar
6 votes
1 answer
389 views

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 ...
Ronald J. Zallman's user avatar
11 votes
2 answers
2k views

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 ...
THC's user avatar
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0 votes
0 answers
93 views

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{...
The Thin Whistler's user avatar
6 votes
2 answers
664 views

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 ...
LeechLattice's user avatar
  • 9,411
2 votes
1 answer
394 views

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 ...
Roxana's user avatar
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0 answers
340 views

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 $n+1$....
Roxana's user avatar
  • 519
4 votes
1 answer
322 views

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 $\...
Yuan Yang's user avatar
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0 answers
500 views

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 ...
Alessandro's user avatar
12 votes
1 answer
1k views

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 $n$ ...
user90189's user avatar
  • 398
1 vote
0 answers
260 views

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 ...
Bradley04's user avatar
  • 487
11 votes
1 answer
2k views

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 ...
David Corwin's user avatar
3 votes
1 answer
237 views

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. ...
asv's user avatar
  • 20.9k
2 votes
1 answer
238 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}\...
Mehtaab Sawhney's user avatar
5 votes
1 answer
369 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,\...
Mikhail Bondarko's user avatar
1 vote
1 answer
221 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 ...
user545's user avatar
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1 vote
0 answers
138 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^{\...
Kim's user avatar
  • 4,024
21 votes
1 answer
2k 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 ...
Timothy Chow's user avatar
  • 77.9k
4 votes
1 answer
307 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 ...
sawdada's user avatar
  • 6,128
16 votes
1 answer
599 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 ...
Asvin's user avatar
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2 votes
0 answers
223 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)?
Bonbon's user avatar
  • 806
7 votes
0 answers
243 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})=\...
Alex Gavrilov's user avatar
4 votes
0 answers
169 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 ...
relG's user avatar
  • 141
5 votes
1 answer
442 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 ...
Alexander Chervov's user avatar
11 votes
0 answers
1k 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 $p$-...
efs's user avatar
  • 3,089
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
3 votes
1 answer
229 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 ...
peterx's user avatar
  • 693
25 votes
8 answers
3k 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 ...
user avatar
0 votes
0 answers
225 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 ...
THC's user avatar
  • 4,253
18 votes
1 answer
560 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 ...
Tyler Foster's user avatar
3 votes
1 answer
360 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?
Turbo's user avatar
  • 13.6k
8 votes
0 answers
354 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 ...
Vesselin Dimitrov's user avatar
15 votes
1 answer
2k 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 ...
Geordie Williamson's user avatar
3 votes
3 answers
551 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, ...
David Loeffler's user avatar
16 votes
1 answer
2k views

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 ...
YUAN Zhiri's user avatar
14 votes
2 answers
1k views

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 ...
wkf's user avatar
  • 637
24 votes
1 answer
2k 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-...
Tim Dokchitser's user avatar
4 votes
1 answer
177 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)(...
inv's user avatar
  • 41
10 votes
1 answer
635 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 ...
M. Carmona's user avatar
2 votes
2 answers
1k 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 ...
user avatar
1 vote
0 answers
139 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 ...
js21's user avatar
  • 7,189
5 votes
3 answers
804 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 $B$, ...
Martin Bright'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
  • 107k
6 votes
2 answers
2k views

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 ...
user37849's user avatar