Deep hole lattices and isogenies of elliptic curves

Given a lattice $L$ in the plane, we define the affiliated deep hole lattice $H(L)$ to be spanned by a shortest vector of $L$ and a deep hole of $L$ contained in the triangle with sides corresponding to the shortest basis vectors. We study the geometric and arithmetic properties of deep hole lattices. In particular we investigate conditions on $L$ under which $H(L)$ is well-rounded and prove that $H(L)$ is defined over the same field as $L$. For the period lattice corresponding to an isomorphism class of elliptic curves, we produce a finite sequence of deep hole lattices ending with a well-rounded lattice which corresponds to a point on the boundary arc of the fundamental strip under the action of ${\mathrm{SL}}_2(\mathbb{Z})$ on the upper halfplane. In the case of CM elliptic curves, we prove that all elliptic curves generated by this sequence are isogenous to each other and produce bounds on the degree of isogeny. Finally, we produce a counting estimate for the planar lattices with a prescribed deep hole lattice.