Elliptic curves with same number of points as field order

Question

Is it possible for an elliptic curve defined over a finite field $\mathrm{GF}(p^k)$ to have $p^k$ points? If not, what is the highest possible $m$ such that there is an elliptic curve over $\mathrm{GF}(p^k)$ with number of points divisible by $p^m$?

Answer 1

The following reference gives a detailed study for such curves defined over $\Bbb F_q$, having $q$ $\Bbb F_q$-rational points:

CM-CURVES WITH GOOD CRYPTOGRAPHIC PROPERTIES, Neal Koblitz, in J. Feigenbaum (Ed.): Advances in Cryptology - CRYPT0 '91, LNCS 576, pp. 279-287, 1992. (@) Springer-Verlag Berlin Heidelberg 1992

Koblitz, following Mazur, calls them anomalous.

The $\Bbb F_q$-points are closely related to the action of the corresponding Frobenius morphism, their number is then related to its characteristic polynomial, and the trace of the Frobenius is the coefficient needed, if it is one we have an anomalous elliptic curve. A search with the favorite search engine for "anomalous elliptic curves" produces numerous hits.

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