Hecke modules associated to certain ternary lattices of half-discriminant $p^2$ and $4p^2$

Source abstract

We consider Hecke modules of orthogonal modular forms attached to definite ternary lattices of half-discriminant $p^2$ with $p$ prime and $4p^2$ with $p$ an odd prime. Where possible, we prove explicit Jacquet-Langlands style correspondences with spaces of classical modular forms (in the spirit of Birch and Hein-Tornaria-Voight), and give conjectures for the remaining genera. This is done by applying sequences of Watson-style transformations that reduce the discriminant while preserving the adelic orthogonal group.