I am particularly interested in elliptic curves over finite fields of prime order, so let $\mathbb{F}_{p}$ denote the finite field of order $p$ (where $p$ is prime) and let $E$ be the elliptic curve $y^2 = x^3 + ax + b$ over $\mathbb{F}_{p}$. Additionally, we will let $E(\mathbb{F}_{p})$ denote the set of solutions of $E$ over $\mathbb{F}_{p}$. Now we know that $E(\mathbb{F}_{p})$ is a finite abelian group and that for each divisor $d$ of $| E(\mathbb{F}_{p})|$, the order of $E(\mathbb{F}_{p})$, there exists a subgroup $H \leq E(\mathbb{F}_{p})$ of order $d$. My question is: Is there a geometric interpretation for these subgroups? Can they perhaps be thought of as a 'subcurve' in some sense?

I have read that if you have an elliptic curve $E$ over a field $K$ and $L$ is a subfield of $K$, then $E(L) \leq E(K)$, which is intuitive and I understand. However when working over $\mathbb{F}_{p}$, there are no nontrivial subfields, yet we still have nontrivial subgroups of $E(\mathbb{F}_{p})$ (provided that $|E(\mathbb{F}_{p})|$ is not prime). Further, since $E(\mathbb{F}_{p})$ is abelian and all of its subgroups are normal, is there a natural interpretation for quotients formed by considering $E(\mathbb{F}_{p})$ modulo some subgroup as we have described?

  • $\begingroup$ The first sentence in your second paragraph (which you say is intuitive and you understand) is meaningless. If you have an elliptic curve over a field $K$ it makes sense to talk about $E(L)$ for any overfields of $K$, but not in general for subfields of $K$. $\endgroup$ – Mohan Nov 7 '15 at 23:03
  • $\begingroup$ Consider the binary operation that makes the set of solutions an abelian group. $\endgroup$ – Taylor Nov 7 '15 at 23:15
  • $\begingroup$ I realize that we are often interested in examining an elliptic curve in an overfield, but I think this question falls outside of the scope of generality to which you referred because I am concerned with subgroups of an elliptic curve group. Also, I understand the group law, but I suppose I am curious as to whether a subgroup can be realized as an elliptic curve over the same field? Obviously, this won't always be the case, but are there instances where it can? $\endgroup$ – Oiler Nov 8 '15 at 2:10
  • $\begingroup$ If I understand your question correctly, the answer is No, there is no curve associated to a subgroup of $\Bbb E(\Bbb F_p)$. In particular, think of the case $b=0$ where you have a subgroup consisting of the origin and the point at infinity. Just two elements in the subgroup, but elliptic curves can never have so few as two points (if $p$ is reasonably large, that is). $\endgroup$ – Lubin Nov 10 '15 at 6:40
  • $\begingroup$ This was my thinking. Thank you. $\endgroup$ – Oiler Nov 10 '15 at 16:59

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