Diophantine Equations and Anticyclotomic Iwasawa Theory of Abelian Varieties with Complex Multiplication

Iwasawa theory is an area of Number Theory that was named after the fundamental work of Kenkichi Iwasawa in the late 1950s and onward. Classically, it concerns with the growth of arithmetically interesting objects, such as class groups, Mordell–Weil and Tate–Shafarevich groups, or more generally Selmer groups, in Zp-power-extensions of a number field (or in modern days, any p-adic families, such as the ones constructed by Hida and Coleman). In 1969, Iwasawa predicted a deep relationship between the Kubota–Leopoldt p-adic L-function and the ideal class groups of cyclotomic fields, which came to be called the Iwasawa Main Conjecture. This conjecture was proved by Mazur and Wiles in 1984 using advanced tools that dwell on the study of the geometry of modular curves. The classical Iwasawa Main Conjecture was later re-proved in the work of Rubin using Euler systems. In the early 70s Mazur, inspired by Iwasawa’s work, formulated analogous conjectures for an elliptic curve E defined over Q and for any prime p at which E has good ordinary reduction (in fact, more generally for abelian varieties over number fields that have good ordinary reduction at every place above p). One of the motivations behind Mazur’s Main Conjectures was to provide an approach to understanding the behavior of the Mordell–Weil groups of elliptic curves (more generally, of abelian varieties), and study the celebrated Birch and Swinnerton-Dyer conjecture (BSD). One of the main objectives of this thesis is to study the p-ordinary Iwasawa theory for Hecke characters attached to CM abelian varieties (and, more generally, to Hilbert modular forms) via the conjectural Rubin–Stark elements. More precisely, we prove a (g + 1)-variable Iwasawa main conjecture for abelian varieties at ordinary primes using the approach developed by Büyükboduk, which has an application in the arithmetic study of CM abelian varieties. Combining the results of Hsieh and Büyükboduk with Nekovář’s theory of Selmer complexes, we establish a version of the anticyclotomic Iwasawa main conjecture for CM Hilbert modular forms. In the second part of the dissertation, we focus on the modular approach to the ternary Diophantine equations xp + yp = z2 and xp + yp = z3 (joint work with Özman and Kara). Solving Diophantine equations, in particular, Fermat-type equations is one of the oldest and most widely studied topics in mathematics. After Wiles’ proof of Fermat’s Last Theorem (FLT) using his celebrated modularity theorem, several mathematicians have attempted to extend this approach to various Diophantine equations and number fields. For instance, in 2015, Freitas and Siksek proved the asymptotic FLT for certain totally real fields K. That is, they showed that there is a constant BK such that, for any prime p > BK , the only solutions to the Fermat equation ap + bp + cp = 0 where a, b, c ∈ OK are the ones satisfying abc = 0. Later, in 2018, Şengün and Siksek proved the asymptotic FLT for any number field K by assuming two deep but standard conjectures from the Langlands program. The second goal of the thesis is to adapt the strategy developed by Freitas – Siksek and Şengün – Siksek to obtain results regarding the solutions of the Fermat equations with signature (p, p, 2) and (p, p, 3). More precisely, we prove that there is a constant BK depending only on K such that for any prime exponent p > BK the Fermat type equation xp +yp = z2 with x, y, z ∈ OK does not have a certain type of solutions. Regarding the solutions of the Diophantine equation xp +yp = z3 over various number fields, we obtain two main results. The first outcome of our research is an asymptotic result for some of the solutions over general number fields and the second one is an explicit result for the solutions of the equation over Q(i), Q(√−7), Q(√−19), Q(√−43) and Q(√−67).