All Questions
Tagged with finite-fields analytic-number-theory
14 questions
8
votes
1
answer
366
views
Why do we have fewer distinct Gauss sums over a field of characteristic $2$?
Let $p$ be a prime number and $q$ be a power of $p$. Fix a non-trivial additive character $\psi\colon \mathbb{F}_q\to \mathbb{C}^\times$ and, for each non-trivial multiplicative character $\chi\colon \...
- 711
0
votes
0
answers
107
views
Cubic monic polynomial over z_p
Let
$$
f_{a}(x)=x^3+(u-2-a)x^2+ax+1,
$$
where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all $a\in\mathbb{Z}_p$ such that $f_{a}(x)$ factor linearly. Then what is the cardinality ...
user126352
1
vote
1
answer
468
views
Artin's conjecture for polynomials and rational functions over finite fields
Artin's conjecture on primitive roots over the integers states that a given integer
$0\ne h\in \mathbb{Z}$ that is neither a square number nor $-1$ is a primitive root modulo infinitely many primes $p$...
- 11
14
votes
1
answer
285
views
Lower bounds for class number of function fields with fixed $q$, growing $g$
Let $X$ be a smooth project curve of genus $g$ over the finite field with $q$ elements. Let $h$ be $\# \mathrm{Pic}^0(X)(\mathbb{F}_q)$. Weil showed that $h \geq (\sqrt{q}-1)^{2g}$. Lachaud and Martin-...
- 156k
4
votes
0
answers
134
views
$\delta$-equidistributed polynomials over finite fields
I'm trying to show that a polynomial over finite (prime) field is "close enough" to being equidistributed over its range. A polynomial $p(\cdot)$ from $\mathbb{F}^n$ to $\mathbb{F}$ is $\...
- 301
2
votes
1
answer
600
views
Density of rational points over finite fields, an estimate of Lang-Weil constant
Let $X\hookrightarrow\mathbb P^n_{\mathbb F_q}$ be a geometrically integral hypersurface over the finite field $\mathbb F_q$ of degree $\delta$. In order to estimate the number of its rational points, ...
- 403
2
votes
0
answers
87
views
Variation of Gauss/Jacobi sums on a variety
Let $V \subset \mathbb P^n$ be a nice (smooth, projective?) variety over a finite field $\mathbb F_q$. Let $\chi_0,\chi_1,\dots,\chi_{n}: \mathbb F_q^\times \to \mathbb Q(\mu_{q-1})$ be multiplicative ...
- 7,746
2
votes
1
answer
277
views
Is there an analogue of the Balazard-Saias-Yor criterion for the Riemann Hypothesis for finite fields?
The Balazard-Saias-Yor criterion for the Riemann Hypothesis states that the latter is equivalent to the statement that
$$\int_{\Re(s)=1/2} \frac{\log|\zeta(s)|}{|s^2|}|ds|=0$$ where $\zeta$ denotes ...
- 105
2
votes
0
answers
249
views
An exponential sum like the Kloosterman sums
I encounter a tricky sum like the Kloosterman sum
$${\sum_{x \mod P}}^\ast e\left(\frac{ax+\overline{x}}{P}+\frac{lx^2}{P^2}\right),$$
where $l$ is a positive integer co-prime with $P$ and here $P$ ...
- 353
8
votes
2
answers
563
views
Distribution of primitive roots, as p varies
For a prime number $p$, let $\Phi(p)$ be the subset of $\{ 1, 2, \ldots, p-1 \}$ consisting of primitive roots modulo $p$. (Thus $\# \Phi(p) = \phi(p-1)$, where $\phi$ denotes the totient.)
I am ...
- 13.3k
5
votes
1
answer
328
views
Does a Kloosterman sum composed with a rational function exhibit square root cancellation?
Denote the classical Kloosterman and Salié sums, respectively, as $KL(a,b) = \sum_{r \in F_*} e(ar+\frac{b}{r})$ and $SL(a,b) =\sum_{r \in F_*} \chi(r) e(ar+\frac{b}{r})$, where $\chi(\cdot)$ is the ...
- 13k
7
votes
3
answers
911
views
Does the equation $x^2+x=a$ have an integer solution?
I am writing a paper on the topological structure of the Golomb space (defined here) and arrived to the following question:
Question 1. Is it true that for a number $a\in\mathbb N$ the equation $x^2+...
- 41.9k
2
votes
1
answer
260
views
Fixed points of $g^x$ (modulo a prime)
In an explicit construction in combinatorics I need to study the following problem: assume we pick a odd prime number $p$, a generator $g$ of the multiplicative group $(Z/pZ)^{\ast}$.
Question 1: ...
- 1,561
4
votes
1
answer
307
views
When the Kloosterman sum is an integer?
Let $q$ be a power of prime $p$ and $\zeta_p$ be the complex $p$ th root
of unity. We denote by
$\mathbb{F}_q$ the finite field of $q$ elements and by $Tr$ the absolute trace function $\mathbb{F}_q\...
- 255