Elliptic curves over finite fields

Question

I have basic questions about elliptic curves over finite fields.

1. Where to find general references? Hartshorne for instance restricts to algebraically closed ground fields.

2. Over an arbitrary field $K$, is the right definition of an elliptic curve a smooth proper curve of genus 1 with a choice of $K$-rational point?

3. What is known about the structure of the group of $K$-rational points when $K$ is finite? In particular, how much does it depend on the curve?

4. Are there simple examples where you can explicitly see all of the $K$-rational points, $K$ finite?

Answer 1

Elliptic curves $E$ and $E'$ over a finite field $K$ are $K$-isogenous if and only if the orders of $E(K)$ and $E'(K)$ coincide. However, it may happen that the groups $E(K)$ and $E'(K)$ have the same order (and even isomorphic) but $E$ and $E'$ are not isomorphic over $K$. Even worse, there exist such a $K$ and non-isomorphic over $K$ elliptic curves $E$ and $E'$ such that if $\bar{K}$ is an algebraic closure of $K$ then the Galois modules $E(\bar{K})$ and $E'(\bar{K})$ are isomorphic. In particular, if $L$ is an arbitrary finite field containing $K$ then the groups $E(L)$ and $E'(L)$ are isomorphic. (Of course, $E(K)$ and $E'(K)$ are the subgroups of Galois invariants in $E(\bar{K})$ and $E'(\bar{K})$ respectively.) See arXiv:0711.1615 [math.AG].

An explicit description of all groups that can be realized as $E(K)$ (for a given $K$) was done by Misha Tsfasman (In: Theory of numbers and its applications, Tbilisi, 1985, 286--287; see also Sect. 3.3.15 of the book Algebraic geometric codes: basic notions by Tsfasman, Vladut and Nogin, AMS 2007). See also papers of René Schoof (J. Combinatorial Th. A 46 (1987), 183--211), Felipe Voloch (Bull. SMF 116 (1988), 455--458) and Sergey Rybakov (Centr. Eur. J. Math. 8 (2010), 282--288).

Answer 2

1) Silverman's book mentioned above is surely a very good reference. For the first steps you can also consider Silverman-Tate "Rational points on elliptic curves". As you already use Hartshorne, I assume that you are familiar with basic notations in algebraic geometry and scheme theory. So you may wish to switch to references on general abelian varieties at some point. (Elliptic curves are abelian varieties of dimension $1$.) For abelian varieties I suggest Milne's article in the Storrs book "Arithmetic geometry" and the wonderful "prebook" on ablian varieties written by Moonen and van der Geer. To my knowledge this prebook is not published yet, but it can be downloaded on the homepage of Ben Moonen.

2) This is true, as mentioned above. An alternative definition: An elliptic curve is an abelian variety of dimension $1$. Here an abelian variety over $K$ is a (geometrically) integral proper group scheme over $K$.

3) I can give a few basic informations on Mordell-Weil groups: Let $K$ be a field and $E/K$ an elliptic curve. For $n$ coprime to $char(K)$ you have $E(\overline{K})[n]\cong ({\mathbb Z}/n)^2$, hence you know that $E(K)[n]$ is always isomorphic to a subgroup of $({\mathbb Z}/n)^2$. If $char(K)=p>0$, then there is an integer $f\in\{0, 1\}$ such that $E(\overline{K})[p^i]\cong ({\mathbb Z}/p^i)^f$ (and consequently $E(K)[p^i]$ is isomorphic to a subgroup of $({\mathbb Z}/p^i)^f$) for all $i\ge 1$.

If $K$ is finitely generated (over its prime field), then it is known that $E(K)$ is a finitely generated ${\mathbb Z}$-module by the so called Mordell-Weil-Lang-Neron theorem. (Cf. The article of Brian Conrad "Chows $K/k$-trace and $K/k$-image, and the Lang-Neron theorem (via schemes)".)

If $K$ is finite with $|K|=q$, then clearly $E(K)$ is finite. In addition to the information above, you then have the Hasse-Weil bound on the size of $E(K)$: $$||E(K)|-q-1|\le 2\sqrt{q}.$$

4) I am not sure whether I interpret this question in the right way. But you can of course take your favorite elliptic curve $E$ over ${\mathbb F}_5$, given by an explicit Weierstrass equation, and and use a computer to make a list of the points in $E({\mathbb F}_5)$. (Just check which of the $31$ points in ${\mathbb P}_2({\mathbb F}_5)$ lie on $E$.)

Answer 3

As Xandi Tuni said, most of the answers to your questions can be found in standard references.

1. Silverman, Knapp's book on elliptic curves, Milne's book, many more (just google for elliptic curves).

2. Yes

3. The number of points is bounded by the Hasse-bound. Within that bound, the exact number depends a lot on the elliptic curve itself. The $l^n$-torsion for prime $l\neq \text{char }K$ is isomorphic to $\mathbb{Z}/l^n\mathbb{Z} \times \mathbb{Z}/l^n\mathbb{Z}$ over the algebraic closure of $K$. For $p= \text{char }K$, the $p$-primary torsion over the algebraic closure is either cyclic (then $E$ is called ordinary) or 0 (then $E$ is called supersingular). So this again depends on the curve.

4. I don't know what you mean by that question. You can always compute all the $K$-rational points by hand, since there are only finitely many values you have to try.

Answer 4

I completely agree with the earlier answers. Just two remarks...

• For question 3, If $K$ is finite of cardinality $q$, then $E(K)$ is isomorphic to $\mathbf Z/n\mathbf Z\times\mathbf Z/m\mathbf Z$, where $n$ divides gcd$(q-1,m)$.
• Concerning your last question, here is a simple example where you can explicitly 'see' all the $K$-rational points without direct computations; I hope it may interest you. Consider the elliptic curve $$E:Y^2=X^3+1$$ defined over the finite field $K=\mathbf F_p$, where $p=3n+2$ is a prime number. Then, there is a bijection $\varphi:K\to E(K)-\lbrace O\rbrace$ (where $O$ is the point at infinity) given by $$\varphi(t)=\left((t^2-1)^{2n+1},t\right).$$