Questions tagged [weil-conjectures]
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70 questions
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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 ...
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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......
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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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8
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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 ...
user97565
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3
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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 ...
- 7,442
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1
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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-...
- 5,363
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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 ...
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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 ...
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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 ...
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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 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 ...
- 1,525
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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 ...
- 7,609
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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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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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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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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 $...
- 5,052
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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 ...
- 6,348
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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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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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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 ...
- 4,575
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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 ...
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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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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 ...
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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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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?
- 13.2k
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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 ...
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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 ...
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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 ...
- 13.8k
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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 ...
- 71
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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})=\...
- 6,891
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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 $p$) $\...
- 3,555
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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 ...
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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 ...
- 1,810
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1
answer
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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 ...
user30211
6
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1
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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 ...
user108974
6
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1
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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!
- 2,842
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2
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Weil Conjectures for Grassmannians
To establish the Weil conjectures for $n$-dimensional projective space over a finite field is elementary. Does there exist a simple direct proof of the conjectures for finite field Grassmannians?
- 1,462
5
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3
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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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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{...
- 6,348
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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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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
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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 ...
- 51
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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. ...
- 21.8k
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 ...
- 6,622
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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 ...
- 2,751
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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)(...
- 41
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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 $\...
- 547
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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 (...
- 4,575
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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 ...
- 141
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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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