Let $p$ be a prime, $E$ be a supersingular elliptic curve defined over $\bar{ \mathbb{ F}}_p$, and $\mathscr{ O}$ be its (geometric) endomorphism ring. Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima of the Gross lattice of $\mathscr{ O}$ characterize the isomorphism class of $\mathscr{ O}$. In this paper, we refine and extend this work and show that the value of the third successive minimum $D_3$ of the Gross lattice gives necessary and sufficient conditions for the curve to have its $j$-invariant in the field $\mathbb{ F}_p$ or in the set $\mathbb{ F}_{ p^2} \setminus \mathbb{ F}_p$, as well as finer information about the endomorphism ring of $E$ when its $j$-invariant belongs to $\mathbb{ F}_p$ and $p \equiv 3 \pmod{ 4}$. Finally, in the case where $j(E)$ belongs to $\mathbb{ F}_p$, we define a new invariant of the Gross lattice, the Gram matrix of a normalized successive minimal basis, and develop an algorithm to compute it explicitly given the value of the first successive minimum of the Gross lattice.
Categories: math.NT