Questions tagged [weil-conjectures]

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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 $p$-...
efs's user avatar
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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 ...
bhwang's user avatar
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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 ...
Vesselin Dimitrov's user avatar
7 votes
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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})=\...
Alex Gavrilov's user avatar
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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 ...
Alessandro's user avatar
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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 (...
THC's user avatar
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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 ...
relG's user avatar
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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 ...
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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 $...
The Thin Whistler's user avatar
3 votes
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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 $n+1$....
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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 $\{...
Mishkaat's user avatar
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$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\...
The Thin Whistler's user avatar
2 votes
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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)?
Bonbon's user avatar
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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^{\...
Kim's user avatar
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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 ...
js21's user avatar
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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{...
The Thin Whistler's user avatar
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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 ...
THC's user avatar
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