Towards Uniform Finiteness Results on Heavenly Elliptic Curves

In connection to a long-standing question of Ihara, the arithmetic of certain abelian varieties with highly constrained torsion, called heavenly abelian varieties, are of special interest. In earlier joint work with Tamagawa, we conjectured only finitely many examples of such varieties can exist when the dimension and field of definition are specified. The conjecture is a theorem under assumption of GRH and has been established unconditionally in certain cases.

We report on work in progress, joint with Cam McLeman (University of Michigan, Flint), towards uniform bounds on heavenly elliptic curves over quadratic number fields. We also describe surprising similarities between the Frobenius trace behavior of such elliptic curves and certain elliptic curves possessing complex multiplication and explain the phenomenon in terms of a possible generalization of the Frey-Mazur Conjecture.