Deep hole lattices and isogenies of elliptic curves (Lenny Fukshansky, CMC)

For 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 the furthest removed vector from the lattice contained in the triangle with sides corresponding to the shortest basis vectors. We study the geometric and arithmetic properties of deep hole lattices, which turn out to be quite interesting. In particular, we construct sequences of deep hole lattices corresponding to elliptic curves over a fixed number field. 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. Joint work with Pavel Guerzhoy and Tanis Nielsen.