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11
votes
4answers
547 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 ...
71
votes
7answers
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 ...
6
votes
2answers
1k views

What is the Beilinson regulator?

Trying to understand answer to this question. What is the (Beilinson) higher regulator of a number field?
29
votes
10answers
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
2answers
635 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 ...