**5**

votes

**2**answers

534 views

### Zeta function for curves in a manifold

Motivation
In the analogy between prime numbers and knots, the prime number is thought sometimes as the circle of length $l([p]) = \text{log}\,p$. This is so you can express the zeta function as
$$ \...

**45**

votes

**8**answers

7k views

### Why does the Gamma-function complete the Riemann Zeta function?

Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or ...

**21**

votes

**5**answers

2k views

### What is the difference between a zeta function and an L-function?

I've been learning about Dedekind zeta functions and some basic L-functions in my introductory algebraic number theory class, and I've been wondering why some functions are called L-functions and ...

**10**

votes

**2**answers

793 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?

**2**

votes

**2**answers

724 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?

**19**

votes

**4**answers

3k views

### The class number formula, the BSD conjecture, and the Kronecker limit formula

If K is a number field then the Dedekind zeta function Zeta_K(s) can be written as a sum over ideal classes A of Zeta_K(s, A) = sum over ideals I in A of 1/N(I)^s. The class number formula follows ...

**8**

votes

**3**answers

848 views

### Is the maximum domain to which a Dirichlet series can be continued always a halfplane?

Let $f(s)=\sum_n a_n n^{-s}$ be a Dirichlet series whose coefficients satisfy $\lvert a_n\rvert\leq n^{C}$. Then $f(s)$ converges absolutely in some halfplanes, and is conditionally convergent in (...

**3**

votes

**2**answers

739 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, ...

**11**

votes

**4**answers

555 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 ...

**72**

votes

**7**answers

7k 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 one-...

**6**

votes

**2**answers

2k views

### What is the Beilinson regulator?

Trying to understand answer to this question.
What is the (Beilinson) higher regulator of a number field?

**30**

votes

**10**answers

4k views

### Why are functional equations important?

People who talk about things like modular forms and zeta functions put a lot of emphasis on the existence and form of functional equations, but I've never seen them used as anything other than a ...

**5**

votes

**2**answers

690 views

### What's the correct notion of determinant of a bilinear pairing?

By a pairing on a vector space $V$, I mean a linear map $A : V \otimes V \to R$. If $V$ is $n$-dimensional ($n < \infty$), then I can define the determinant of $A$ by considering the canonical ...