0
votes
0answers
51 views

Multivariable irreducible polynomials over finite fields [migrated]

It is not difficult to prove the following result, and it seems that it should be already proved. I would appreciate it if someone offer me some reference to it. For any $f(x_1,\dots, x_n)=\sum ...
5
votes
1answer
361 views

More expanders?

Having received several exhausting answers to my recent question about the expansion properties of a certain graph, I now wonder whether anything is known on the following graphs of a similar nature: ...
10
votes
2answers
320 views

An expander (?) graph

For a prime $p$, consider the graph on the vertex set ${\mathbb F}_p$, in which every vertex $z$ is adjacent to $z\pm 1$ and also to $z^{-1}$ (unless $z=0$). I was told that this graph is known to be ...
5
votes
3answers
339 views

Source for embedding multiplicative group of an algebraic closure of a finite field?

It's easy to embed the (cyclic) multiplicative group of a finite field into the multiplicative group of $\mathbb{C}$ (or other algebraically closed field of characteristic 0): assign to a generator of ...
1
vote
0answers
150 views

reference-request for connection between numerator of zeta-function and characteristic polynomial of Frobenius on hyperelliptic curves over finite fields

Let $H$ be a hyperellipitic curve of genus $g$ defined over $\mathbb{F}_q$. The Frobenius endomorphism operates on the divisor class group of $H$ and satisfies a characteristic polynomial ...
6
votes
1answer
571 views

Exponential sums over finite fields with even characteristic

I am looking for an elementary evaluation (if one exists) of the exponential sum $$ G_r(a,b) = \sum_{x \in \mathbb{F}_{2^r}} \psi(ax^2 + bx), $$ where $a,b \in \mathbb{F}_{2^r}^*$ are both units, ...
7
votes
1answer
677 views

étale cohomology with G_m coefficients

Most calculations of ├ętale cohomology in Milne's book deal with constructible or torsion sheaves. Are there references where the cohomology of varieties with $\mathbf{G}_m$ coefficients are ...