Lang-Trotter Questions on the Reductions of Abelian Varieties

Abstract

Let A be a geometrically simple g-dimensional abelian variety over the rationals. This thesis investigates the behavior of the reductions A_p of A modulo its primes p of good reduction. Questions about these reductions are called "questions of Lang-Trotter type" after the 1976 memoir of S. Lang and H. Trotter. This thesis studies two aspects of the reductions A_p in particular: the "Frobenius fields," End(A_p) tensor Q, when A_p is simple and ordinary, and the primality (or failure thereof) of the number of rational points, #A_p(F_p). Our questions and conjectures generalize the study of the "fixed-field" Conjecture of Lang-Trotter and the Koblitz Conjecture on elliptic curves, and our work generalizes the work by previous authors to this higher-dimensional context: through sieve-theoretic arguments and the use of explicit error bounds for the Chebotarev Density Theorem, we produce various conditional and unconditional upper and lower bounds on the number of primes p at which A_p has a specified behavior.