Let $p$ be prime, let $q$ be a power of $p$ and let $E/\mathbf{F}_q$ be an elliptic curve defined over the finite field $\mathbf{F}_q$. Let $\overline{\mathbf{F}_q}$ be the algebraic closure of $\mathbf{F}_q$.

Main Question: How might one conceive of $E/\overline{\mathbf{F}_q}$ both geometrically and as a group ? (And is there any benefit to having such a mental picture ?)

If my limited understanding is correct (and most of what follows I don't yet actually understand),

  1. $E/\overline{\mathbf{F}_q}$ is all torsion : it ought to be the union (colimit) of the $E/\mathbf{F}_{q^d}$ since $\overline{\mathbf{F}_q}$ is the union (colimit) of the $\mathbf{F}_{q^d}$,
  2. if $\gcd(m,p)=1$ then the $m$-torsion $(E/\overline{\mathbf{F}_q})[m]\simeq (\mathbb{Z}/m\mathbb{Z})\times(\mathbb{Z}/m\mathbb{Z})$,
  3. as one climbs up the ladder of powers of $q$, more points of the elliptic curve $E/\overline{\mathbf{F}_q}$ reveal themselves; it may happen that for some power $q^d$ the $m$-torsion of the curve $E/\mathbf{F}_{q^d}$, where $m$ is a power of a prime $\neq p$, forms a cyclic group $\mathbb{Z}/m\mathbb{Z}$ and only at a later stage the full $m$-torsion will appear,
  4. and $p$-torsion behaves differently : either ((for all $\nu$ the groups $(E/\overline{\mathbf{F}_q})[p^\nu]$ are trivial)), or ((for all $\nu$ the groups $(E/\overline{\mathbf{F}_q})[p^\nu]$ are cyclic $\simeq\mathbb{Z}/p^\nu\mathbb{Z}$)).

Proposal Is it a good mental picture to think of $E/\overline{\mathbf{F}_q}$ as a kind of quotient of $(\mathbb{Q}/\mathbb{Z})\times(\mathbb{Q}/\mathbb{Z})$ that would squeeze $(\mathbb{Z}[\frac1p]/\mathbb{Z})\times(\mathbb{Z}[\frac1p]/\mathbb{Z})$ either to a point or to a copy of $(\mathbb{Z}[\frac1p]/\mathbb{Z}$ ? That is, somewhat in analogy to complex elliptic curves, a kind of strange rational torus with some compression of the $p$-torsion?

Auxiliary Question: If such a mental picture is correct, how does one picture the process of successive unveiling of the points that are rational over the intermediary fields $\mathbf{F}_{q}\subset\mathbf{F}_{q^{a}}\subset\mathbf{F}_{q^{ab}}\subset\overline{\mathbf{F}_{q}}$ ? (And is there any point to it?)

  • $\begingroup$ I don't know if this might be relevant, but in the paper A note on elliptic curves over finite fields, Bull. Soc. Math. France 116 (1988), 455-458, Felipe Voloch gives the possible group structures for a non-supersingular elliptic curve over a finite field $\Bbb F_q$. $\endgroup$ – Watson Apr 27 '18 at 20:25

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