Inseparable endomorphisms and rank-2 sublattices of the Gross lattice

We answer a question posed by Love asking about a correspondence between isogenies from a supersingular elliptic curve to its Frobenius base-change and rank-2 sublattices of its Gross lattice. We recast the question as one about the inseparable endomorphisms of the curve, and show that the correspondence holds when the trace of the endomorphism is zero, and may not hold otherwise.

💡 Research Summary

The paper addresses a question originally posed by Jonathan Love concerning a proposed correspondence between isogenies from a supersingular elliptic curve (E) over (\mathbb{F}_p) to its Frobenius base‑change (E^{(p)}) and rank‑2 sublattices of the Gross lattice (\operatorname{End}(E)^T). Love asked whether, for a positive integer (\ell), the existence of an isogeny (E\to E^{(p)}) of degree (\ell) is equivalent to the presence of a rank‑2 sublattice of determinant (4\ell p) inside (\operatorname{End}(E)^T). The authors recast this problem in terms of inseparable endomorphisms of (E) and prove that the equivalence holds precisely when the endomorphism has reduced trace zero.

The paper begins by recalling that for a supersingular curve (E) the geometric endomorphism ring (\operatorname{End}(E)) is isomorphic to a maximal order (\mathcal O) in the quaternion algebra (B_{p,\infty}) ramified at (p) and (\infty). The reduced trace (\operatorname{trd}) and reduced norm (\operatorname{nrd}) on (B_{p,\infty}) correspond respectively to the usual trace and degree of endomorphisms. The Gross lattice is defined by the linear map (\tau(x)=2x-\operatorname{trd}(x)); its kernel is (\mathbb Z) and its image on a lattice (\Lambda) is (\Lambda^T).

Forward direction (Proposition 2.2).
Assume a rank‑2 sublattice (\Lambda\subset\operatorname{End}(E)^T) with determinant (4\ell p) exists, with basis ({\gamma_1,\gamma_2}). Define \