# Tagged Questions

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### Automorphicity of L-Factors of Zeta Functions

Associated to a variety over a number field $K$, one has a family of ``Hasse--Weil'' L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the ...

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208 views

### L-Functions of Varieties, Zeta Functions of Their Models

Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...

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295 views

### Does the Riemann hypothesis for liftable varieties over a finite field imply the Riemann hypothesis for all varieties over a finite field

The Riemann hypothesis for varieties over a finite field has been proven by Deligne. Still I would like to ask the following question.
A variety $X$ over a finite field $k$ is liftable if there ...

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133 views

### Special values of zeta functions and extensions of base fields.

Let $X$ be a scheme of finite type over a finite field $k=\mathbb{F}_{q}$ of $q$ elements.
Then, one can define the zeta function $Z_{X/k}(T)$ of $X$ ovet $k$ as $\prod_{x\in ...

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406 views

### Can we decide if an abelian variety is simple by knowing its Zeta function ?

Let $A$ be an Abelian variety defined over the finite field with $q$ elements. Let $P_i(T)$ be the characteristic polynomial of the action of the Frobenius on the $i^{th}$ étale cohomology group.
Is ...

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**1**answer

410 views

### Ihara zeta function

Is there a natural connection between the Ihara zeta function of a graph,
and (for instance) the Riemann zeta function of certain varieties over finite fields ?
Thanks.

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283 views

### Is there a notion of a zeta function of a morphism?

The Hasse-Weil zeta function is defined only for arithmetic schemes. By an arithmetic scheme I will mean a scheme $X$ together with a morphism of finite type $X\rightarrow S$, where $S$ is an affine ...

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217 views

### (why) Are the following two constructions of zeta functions equal?

Let $X$ be a variety defined over $\mathbb{Q}$. One has the usual Hasse-Weil zeta function.
Now, let's do a different construction. Base change $X$ to $\mathbb{C}$: $X_{\mathbb{C}}$. Now look at its ...

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**3**answers

1k views

### A question on K_1 of an elliptic curve

Consider an elliptic curve $E/ \mathbb{Q}$, with a regular model $\mathcal{E} / \mathbb{Z}$. We have (Beilinson regulator) maps
$$ K_1(\mathcal{E})^{(2)} \to K_1(E)^{(2)} \to H_D^3(E_{/ \mathbb{R}} , ...

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### A puzzling remark of Manin (ICM 1978)

Manin ends his 1978 ICM talk with this remark:
I would also like to mention I. M. Gel'fand's suggestion that the $\zeta$-functions of certain special differential operators should have an arithmetic ...

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239 views

### Calculate the zeta function of a scheme by from its étale covers?

The title says it all. Given a (proper smooth) scheme $X$ over Spec$\mathbb{F}_q$, is it possible to calculate the zeta function of $X$ by from its étale covers?
Like for $\mathbb{P}^n$ you can ...

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**1**answer

808 views

### Dirichlet's regulator vs Beilinson's regulator

Consider a number field $F$ with ring of integers $O_F$. The Beilinson regulator can in this particular setting be viewed as a map from $K_n(O_F)$ to a suitable real vector space. Here $n$ is any ...

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387 views

### How looks the “land of Tamagawa numbers”?

Jonah Sinick's question here, other interesting ideas he mentioned, and Franz Lemmermeyer's remark make one think at Bloch and Kato's drawing + question. What's known or guessed about that "land" by ...

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621 views

### 2d Weil conjecture

Does there exist a two variable analogue of the Weil conjecture?
What I mean is that the usual Weil involves a one-variable zeta-function which you get by using numbers $V_n = V ( GF(p^n))$ of points ...

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**3**answers

2k views

### Dwork's use of p-adic analysis in algebraic geometry

Using p-adic analysis, Dwork was the first to prove the rationality of the zeta function of a variety over a finite field. From what I have seen, in algebraic geometry, this method is not used much ...

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529 views

### How does the order of a pole of a zeta function indicate any geometric information?

Here, I'm primarily concerced about zeta functions of hypersurfaces over fields of finite characteristic.
Assume $F_q$ to be a finite field with q elements. Consider the zeta function of the ...

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714 views

### Weil Conjectures for nonprojective algebraic varieties

If we replace projective variety with algebraic variety in the statement of the Weil conjectures what happens? To me it seems the statement still makes sense. But is it still true?

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673 views

### 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?

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661 views

### Castelnuovo Positivity (Rewrite of: Weil's original proof for FP^2)

Weil's proof of the Riemann Hypothesis for projective curves relies upon the following positivity result: Let $\mathbb{F}q$ be the finite field with $q$ elements, $\overline{\mathbb{F}q}$ its closure, ...

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### Behaviour of Zeta-function under Finite Morphism

Let X ---> Y be a finite surjective morphism of smooth, projective, connected varieties over a finite field F_q. Can one describe the zeta function Z(X, t) in terms of the zeta-function Z(Y,t) of ...

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5k views

### What is the field with one element?

I've heard of this many times, but I don't know anything about it.
What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is ...