Let \(L\) be a lattice in \(\mathbb{R}^2\) with successive minima \(\lambda_1 < \lambda_2\) and corresponding minimal basis vectors \(\mathbf{x}_1\) and \(\mathbf{x}_2\). A point in \(\mathbb{R}^2\) is a deep hole of \(L\) if the distance to \(L\) is as large as possible. There is a unique deep hole \(\mathbf{z}\) of \(L\) contained in the triangle with vertices \(\mathbf{0}\) and the endpoints of \(\mathbf{x}_1, \mathbf{x}_2\). The authors define the deep hole lattice of \(L\) to be \(H(L) = \operatorname{span}_{\mathbb{Z}}\{\mathbf{x}_1, \mathbf{z}\}\). They study the geometry and arithmetic of these lattices. For example, they investigate when \(H(L)\) is well-rounded. The authors study not just individual deep hole lattices, but sequences of these lattices.
The authors also investigate connections to elliptic curves. One can think of an isomorphism class of an elliptic curve in terms of the complex torus \(\mathbb{C}/\Lambda_\tau\) where \(\Lambda_\tau = \operatorname{span}_\mathbb{Z}\{1,\tau\} \subset \mathbb{C}\). When one starts from an arithmetic lattice \(\Lambda_\tau\), which means that \(\tau\) is a quadratic irrationality, the authors show that every lattice in the deep hole sequence corresponds to a CM elliptic curve and that all of these curves are isogenous to each other. They prove a bound on the degree of these isogenies. They also consider a kind of inverse problem where they study the set of similarity classes of lattices that give a particular deep hole lattice.
Some of the arguments involve ideas from plane geometry and the final section uses some ideas from analytic number theory about counting points of bounded primitive height. The authors also apply some results from the recent paper [M. Forst and L. Fukshansky, Monatsh. Math., 203, No. 3, 613–634 (2024; Zbl 07807401)].