1
vote
1answer
67 views

E: $y^2+y=x^3$ an elliptic curve over $F_{2}$. How to prove the number of $E(F_{2^n})$ = $2^n+1$ if n is odd, …

Let E be the elliptic curve $y^2 + y = x^3$ over $F_2$. Prove $ $#E($F_{2^n})$$ = \left\{ \begin{array}{ll} 2^n+1 & \quad n=odd \\ 2^n+1-2(-2)^{n/2} & \quad ...
0
votes
1answer
35 views

Why is the answer set limited here?

This question is based on pp $67$ - $68$ of Ash and Gross's "Elliptic Tales". Here the authors discuss points on a curve in the projective plane. We have an equation $f(x,y) = x^2+y^2$ We can ...
8
votes
0answers
76 views

What is the connection between Weil's character bound and Riemann Hypothesis over finite fields

Weil's character bound states that: Let $\mathbb{F}_{q}$ be a finite field of size $q$. Let $\chi$ be a multiplicative character of order $m$. Let $f(x)$ be a polynomial of degree $d$ such that $f(x) ...
7
votes
2answers
129 views

Does there exist an elliptic curve $E$ such that $\#E(\Bbb{F}_{q^2})=(q+1)^2$ for all prime powers $q$?

The following (paraphrased) question is a homework exercise for a course on elliptic curves: Let $p\not\equiv1\pmod{12}$ be a prime number and let $q=p^k$. Show that there exists an elliptic curve ...
3
votes
1answer
90 views

Genus over finite fields

Is there a way of computing the genus of a parametrized curve over a finite field? For instance I am interested in the genus of the following space curve in the m-dimensional space over $F_{q^k}$ ...
5
votes
1answer
178 views

Translation of a Paper of Tate

I'm just wondering if anyone knows if the paper "Classes d'isogenie des varietes abeliennes sur un corps fini" by John Tate has been translated into English. My French is not that good and I found it ...
4
votes
1answer
113 views

Help with a bilinear form

Let $a,b\in\mathbb{F}_{2^{m}}$ (a field of characteristic $2$, m is odd) I need to prove that ...
8
votes
2answers
137 views

Principal divisors

How can i calculate the principal divisor $(f)$ where $$f = \frac{(x^{3}-1)}{(x^{4}-1)}$$ with $f\in\mathbb{F}_2(x)$. I am recently reading about the subject, so i am looking for a simple solution ...
3
votes
0answers
124 views

when do two rational elliptic curves have identical size when reduced mod $p$ for all primes $p$?

If $E_1$ and $E_2$ are two elliptic curves over $\mathbb{Q}$ such that $|E_1(\mathbb{F}_p)|=|E_2(\mathbb{F}_p)|$ for all primes $p$, what does this tell us about the relationship between $E_1$ and ...
2
votes
1answer
78 views

Representation of a subset of a finite affine space as a variety

It is simple to see that every subset of a an affine space over a finite field is a variety - for example, it follows from the fact that finite subsets are closed in the Zariski Topology of every ...
3
votes
0answers
71 views

Does composing the Frobenius with an automorphism give another Frobenius

Let $X_0$ be a variety over $\mathbf{F}_q$. Consider the Frobenius $F_0:X_0\to X_0$. Let $X= X_0\times \bar{\mathbf{F}_q}$ and let $F:X\to X$ be $F_0 \times \textrm{id}$. Let $f:X\to X$ be an ...
-9
votes
1answer
685 views

A Hunt for a Mathematical Machine That Gives Points

The central question is : Is there any method for Producing the global Points on the curve (any cubic curve, or at least a Degree-2 curve ) , if we have local Part with us ? Explanation: ...