All Questions
20 questions
6
votes
0
answers
338
views
Deeper meaning behind the occurrence of the factor $\frac{\log q}{i}$ in Deninger's results
In two papers Deninger proved the following:
If $q=p^{n}$ and $p$ is a finite prime of $\mathbb{Z}$, $B=\mathbb{C}[\mathbb{C}]$ is generated by symbols of the form $e^{\alpha}$, $\alpha\in\mathbb{C}$,...
- 1,805
27
votes
4
answers
3k
views
Why do we care about the eigenvalues of the Frobenius map?
The Riemann hypothesis for finite fields can be stated as follows: take a smooth projective variety X of finite type over the finite field $\mathbb{F}_q$ for some $q=p^n$. Then the eigenvalues $\...
- 1,969
6
votes
0
answers
243
views
Computing Hodge numbers by point counting
In the lecture note of Bhatt from Arizona winter school 2017, there is an exercise which claims if X is a proper smooth scheme defined over $\mathbb{Z}[1/N]$ and if there is a polynomial $P$ such that ...
- 1,093
3
votes
1
answer
179
views
The local zeta-functions of some cubic plane curves
This question is motivated by the work presented in article 358 of Gauss' Disquisitiones Arithmeticae. For the sake of completeness, let me say something about the background and present the question ...
- 935
4
votes
1
answer
250
views
How to measure how much a rational function/a singularity of variety is complicated?
There are some theorems about various zeta functions which states the rationality of those.
For example, when you consider Igusa's zeta function, roughly the generating series of solutions mod $p^n$ ...
- 485
3
votes
1
answer
190
views
Igusa zeta functions of univariate polynomials: $\mathbb{Z}_p$ or $\mathbb{Q}_p$ in this statement
Let $f\in\mathbb{Z}_p[X]$ and let $Z_{f,p}(T)\in\mathbb{Z}_{(p)}(T)$ be the $p$-adic Igusa zeta polynomial (i.e. $Z_{f,p}(p^{-s})$ is the $p$-adic Igusa zeta function in the complex variable $s$, with ...
30
votes
4
answers
3k
views
Motivation for zeta function of an algebraic variety
If $p$ is a prime then the zeta function for an algebraic curve $V$ over $\mathbb{F}_p$ is defined to be
$$\zeta_{V,p}(s) := \exp\left(\sum_{m\geq 1} \frac{N_m}{m}(p^{-s})^m\right). $$
where $N_m$ is ...
- 901
4
votes
2
answers
571
views
What are positive divisors of degree 2 on Elliptic Curve $y^2=x^3-x-1$ over $\mathbb{F}_3$?
Let (E) be an elliptic curve $y^2=x^3-x-1$ over $\mathbb{F}_3$, $a_0=1$, $a_n$ is the number of positive divisor of degree $n\geq 1$. $a_1$ in this case is the number of points of E, i.e., $a_1=1$ ...
- 41
2
votes
0
answers
76
views
Behaviour of densities of places of finitely generated fields under specialisation
This question is a follow-up on question 2, posed in:
On the distribution of roots modulo primes of an integral polynomial
In appendix B of [1] by Pink, and in [2,3] by Serre, there are definitions ...
- 83
10
votes
1
answer
633
views
Convergence of zeta functions for schemes of finite type over the integers
In his lecture "Zeta functions and $L$-functions", Serre presents a very elegant proof of the convergence of the zeta function
$ \zeta (X,s) = \prod_{x \in |X|} (1- N(x)^{-s})^{-1}$ in the half plane ...
user76025
12
votes
1
answer
594
views
Is it easy to prove that $\sum_n |X(\mathbb{F}_{q^n})| t^n$ is rational?
Background: Let $X$ be an algebraic variety over a finite field $\mathbb{F}_q$. One of the successes of Etale cohomology - previously achieved by Dwork- was proving the rationality of the Zeta ...
- 2,824
5
votes
2
answers
1k
views
Local factors of Hasse-Weil zeta function - what do they have in common?
Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of ...
- 5,637
8
votes
2
answers
833
views
is there a p-adic Borel theorem?
Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. The ...
- 81
4
votes
1
answer
431
views
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 variety, ...
- 643
16
votes
0
answers
878
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 ...
- 643
3
votes
0
answers
365
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 ...
- 6,348
14
votes
3
answers
2k
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}} , \...
- 5,637
14
votes
1
answer
1k
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 ...
- 5,637
9
votes
2
answers
656
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 ...
- 141
12
votes
4
answers
715
views
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 ...
- 671