  We prove the existence of abelian varieties not isogenous to Jacobians over
characterstic $p$ function fields. Our methods involve studying the action of
degree $p$ Hecke operators on hypersymmetric points, as well as their effect on
the formal neighborhoods using Serre Tate co-ordinates. We moreover use our
methods to provide another proof over number fields, as well as proving a
version of this result over finite fields.
