Questions tagged [weil-conjectures]
The weil-conjectures tag has no usage guidance.
70 questions
0
votes
0
answers
123
views
Roots of weight of a characteristic polynomial of Frobenius
We are expected to solve a conjecture of the title. Reference is Jean-Pierre Serre — Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques.
Precisely;
Conjecture A:...
- 153
4
votes
1
answer
264
views
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 ...
- 2,751
3
votes
0
answers
288
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 ...
- 15.4k
3
votes
1
answer
331
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 ...
- 769
2
votes
0
answers
107
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 $\{...
- 331
4
votes
0
answers
270
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 (...
- 4,575
2
votes
0
answers
154
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\...
- 1,805
3
votes
0
answers
248
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 $...
- 1,805
6
votes
1
answer
402
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 ...
user30211
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 ...
- 4,575
0
votes
0
answers
95
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{...
- 1,805
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 ...
- 9,501
2
votes
1
answer
432
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 ...
- 519
3
votes
0
answers
377
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$....
- 519
4
votes
1
answer
364
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 $\...
- 547
5
votes
0
answers
524
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 ...
- 51
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$ ...
- 408
1
vote
0
answers
271
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 ...
- 595
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 ...
- 15.4k
4
votes
1
answer
266
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. ...
- 21.8k
2
votes
1
answer
264
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}\...
5
votes
1
answer
406
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,\...
- 16.9k
1
vote
1
answer
224
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 ...
- 83
1
vote
0
answers
153
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
$$Z(X_0,t) = \pm q^{\...
- 4,164
21
votes
1
answer
2k
views
Is the Hilbert–Pólya intuition vindicated in the function field case?
The Hilbert–Pólya conjecture is the name given to the idea that the "reason" or "explanation" for the collinearity of the non-trivial zeros of the Riemann zeta function $\zeta(s)$ is that they are the ...
- 82.7k
4
votes
1
answer
336
views
Purity of vanishing cycle for proper scheme over DVR with smooth generic fiber
Let $X$ be a scheme proper and flat over a complete discrete valuation ring $O$ with finite residue field $k$, and choose a prime $l$ not equal to characteristic of $k$. Consider the Galois ...
- 6,622
16
votes
1
answer
641
views
What is the automorphic interpretation of the Weil conjectures over finite fields
I am very much a beginner in the theory of automorphic forms and I might (will?) make mistakes in what follows. Please correct me.
A loose interpretation of the Langland's philosophy is that to any ...
- 7,746
2
votes
0
answers
242
views
Is there any generalization of Weil conjecture for non-smooth variety?
Is there any generalization of Weil conjecture for any non-smooth geometric-connected variety? For example, for more general curve, or at least some numerical evindence (example)?
- 806
7
votes
0
answers
248
views
The geometric meaning of the sign in the functional equation
Let $X$ be a smooth projective variety of dimension $n=\dim X$ over a finite field $\mathbb{F}_q$. As is well known, its zeta function satisfies a functional equation of the form
$$Z(X,q^{-n}T^{-1})=\...
- 6,891
4
votes
0
answers
186
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 ...
- 141
5
votes
1
answer
464
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 ...
- 24.9k
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 $p$-...
- 3,107
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 ...
user108974
3
votes
1
answer
265
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 ...
- 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 ...
user97565
0
votes
0
answers
231
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 ...
- 4,575
18
votes
1
answer
571
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 ...
- 954
3
votes
1
answer
367
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?
- 13.9k
8
votes
0
answers
357
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 ...
- 13.8k
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 ...
- 6,162
3
votes
3
answers
578
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, ...
- 37k
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 ...
- 363
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 ...
- 647
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-...
- 5,363
4
votes
1
answer
181
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
$$
Z_C(t)=\frac{P(t)}{(1-t)(...
- 41
10
votes
1
answer
669
views
What is the current state of the crystalline analogue of the Weil conjectures?
In "F-isocrystals on open varieties results and conjectures" Faltings says:
"Finally, we extend the theory of weights and show as much as possible of the crystalline analogue of the Weil ...
- 545
2
votes
2
answers
1k
views
About Weil's proof of "Weil conjectures for curves and abelian varieties"
I know that the Weil's proof of the Weil conjectures for curves and abelian varieties is made under the lenguage of his "Foundation of algebraic geometry", however in "Polarizations and Grothendieck's ...
user56715
1
vote
0
answers
142
views
Is semistability of smooth Weil sheaf preserved under tensor product?
Let $X_0$ be a smooth, geometrically connected scheme over $\mathbb{F}_q$. As usual, let $\tau : \bar{\mathbb{Q}}_{\ell} \simeq \mathbb{C}$ be a fixed isomorphism. Let $\mathcal{C}$ be the category of ...
- 7,249
5
votes
3
answers
835
views
Reference for counting points over finite fields
The following fact is extremely well known:
Fact. Let $Y$ be a geometrically irreducible variety (not necessarily smooth or proper) over a finite field $k$.
Then there is a constant $B$, ...
- 4,185
32
votes
1
answer
3k
views
Is there a cheap proof of power savings for exponential sums over finite fields?
Let $p$ be a large prime, and let $f(x) = P(x)/Q(x)$ be a non-constant rational function over ${\Bbb F}_p$ of bounded degree. From the Weil conjectures for curves, we have a bound of the form
$$ |\...
- 114k