9
votes
6answers
1k views

Proofs in the same vein as Ax-Grothendieck

I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...
4
votes
0answers
328 views

Given two linear operators A and B over a finite field, is there a third operator C whose kernel is the intersection of kernels of A and B?

Let $V$ be a finite dimensional linear space over a finite field $k$. Let $A$ and $B$ be two endomorphisms of $V$. Question 1. Is there an endomorphism $C$ of $V$, which is expressed in terms of ...
8
votes
1answer
535 views

Restriction theorems over finite fields

A short while ago, Dvir proved the Kakeya conjecture over finite fields. Does this have any implications for restriction theorems over finite fields? I am aware only of implications going in the ...
7
votes
1answer
306 views

algorithm for calculating the Chow groups of a variety over a finite field

Is there an algorithm for calculating the Chow groups of a variety over a finite field? It is know that $H^{2i,i}_\mathrm{mot}(X,\mathbf{Z}) = CH^i(X)$. In how many cases does this help us?
2
votes
2answers
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?
8
votes
4answers
1k views

Equivalent Statements of Riemann Hypothesis in the Weil Conjectures

In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with q elements says that: the eigenvalues of ...