Arithmetic Properties of Abelian Varieties

Let $A$ be an abelian variety of dimension $g$, defined over a number field $K$. Denote by $N_A$ the norm of the conductor ideal of $A$. For each prime $\fp$ in $K$ such that $\fp
mid N_A$, denote by $\overline{A}_{\fp}$ the reduction of $A$ modulo $\fp$, defined over the finite field $\F_\fp$. Questions around the distribution of certain properties $\overline{A}_\fp$ have been extensively studied over the years, such as the Sato-Tate Conjecture from the 1960s, the Lang-Trotter Conjecture from the 1970s, and the Murty-Patankar Conjecture from the 2000s. This thesis revolves around this theme of research, as follows. When $K$ is the field of rationals, we focus on a higher-dimensional generalization of a conjecture formulated by S. Lang and H. Trotter about the Frobenius trace of an elliptic curve (i.e., an abelian variety of dimension 1). Under the assumption of the Generalized Riemann Hypothesis, we prove nontrivial upper bounds for the number of primes $p\leq x$ for which $\overline{A}_p$ has a prescribed Frobenius trace. These bounds improve prior results obtained by A.C. Cojocaru, R. Davis, A. Silverberg, and K.E. Stange in 2016 and H. Chen, N. Jones, and V. Serban in 2020. The bounds also recover the best-known results obtained for $g=1$ by M.R. Murty, V.K. Murty, and N. Saradha in 1988, combined with a refinement in the power of $\log x$ by D. Zywina in 2015. By restricting to an absolutely simple abelian surface with a commutative endomorphism ring, we prove nontrivial upper bounds for the number of primes $\fp$ with norm bounded by $x$, for which $\overline{A}_{\fp}$ splits into a product of abelian varieties of smaller dimensions. These bounds improve prior results obtained by J. Achter in 2012 and D. Zywina in 2014.