[Math] Elliptic curves over finite fields

ag.algebraic-geometryalgebraic-number-theory

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?

Best Answer

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).

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