Supersingular elliptic curve

An elliptic curve $E$ over a field $k$ of characteristic $p>0$ is supersingular if

E[p](\overline{k})=\{O\},

where $\overline{k}$ is an algebraic closure and $E[p](\overline{k})=\{P:pP=O\}$ .

Quaternionic endomorphisms

Its geometric endomorphism ring $\operatorname{End}_{\overline{k}}(E)$ is a maximal order in the rational quaternion algebra ramified exactly at $p$ and the real place. Tensoring this ring with $\mathbb Q$ gives that quaternion algebra.

The geometric qualifier is essential: the ring of endomorphisms defined over the original field $k$ can be smaller.

Terminology

The definition uses geometric points; the finite group scheme $E[p]$ still has degree $p^2$ .

“Supersingular” does not mean that the curve has a singular point. Every elliptic curve is smooth. The complementary case, in which the geometric $p$ -torsion has $p$ points, is called ordinary.

References

Andrew V. Sutherland, 18.783 Elliptic Curves, Fall 2023, Lecture 13, opening definition, Warning 13.1, and Theorems 13.18–13.19.
John Voight, Quaternion Algebras, §42.1, Proposition 42.1.7 and Main Theorem 42.1.9.