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5
votes
2answers
738 views

Elementary theory of finite fields

I read on Ax's article that the elementary theory of finite fields is decidable if one assumes the continuum hypothesis to be true. What about if one assumes the hypothesis to be false?
21
votes
3answers
605 views

A hypersurface with many points

Ok, it's time for me to ask my first question on MO. Consider the affine curve $Y+Y^q=X^{q+1}$ over the finite field $\mathbf{F}_q$. It's interesting because it has the largest number of points over ...
2
votes
2answers
695 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?
4
votes
1answer
436 views

Counting solutions to x^{p+1}=y^4 in a finite field

I need to compute the number of solutions to the equation $x^{p+1} = y^4$ in the field with $p^2$ elements (for p sufficiently large). The form of the equation suggests to me that the solution would ...
7
votes
2answers
486 views

Richardson varieties over finite fields

Let me start with some background to set the notation before I ask my question. Let G be a semisimple algebraic group over some algebraically closed field K, and suppose we have fixed a Borel ...
3
votes
2answers
433 views

Induction from split and non-split tori for GL_2 over a finite field

Let k be a finite field, G the k-points of GL_2, T1, T2 the k-points of the split and non-split tori of G. Then the G-representations C[G/T1] and C[G/T2] are almost the same. More precisely, they ...
11
votes
4answers
532 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 ...
9
votes
2answers
815 views

A comprehensive overview of finite fields

I've read numerous introductions to finite fields, but I feel like my intuition about them is fairly lacking. Considering that finite fields are the the most "inert" objects in algebraic geometry, I ...
14
votes
4answers
1k views

Exhibit an explicit bijection between irreducible polynomials over finite fields and Lyndon words.

Let $q$ be a power of a prime. It's well-known that the function $B(n, q) = \frac{1}{n} \sum_{d | n} \mu \left( \frac{n}{d} \right) q^d$ counts both the number of irreducible polynomials of degree ...
7
votes
2answers
937 views

What is an example of a smooth variety over a finite field F_p which does not lift to Z_p?

Somebody answered this question instead of the question here, so I am asking this with the hope that they will cut and paste their solution.
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 ...