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7
votes
2answers
483 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 ...
2
votes
2answers
431 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
528 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 ...
13
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
933 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 ...