ZERO DENSITY ESTIMATES FOR FAMILIES OF L-FUNCTIONS, WITH APPLICATIONS TO EFFECTIVE CHEBOTAREV DENSITY THEOREMS AND l-TORSION IN CLASS GROUPS

Abstract

This thesis concentrates on several topics of number theory, including zero density estimates, the effective Chebotarev density theorem, and l-torsion in class groups.The study of the topics has a long history. In 1801, Gauss raised conjectures on class numbers. There were multiple breakthroughs in the 20th century on the so-called class number problem, but there was little progress on bounding the l- torsion in class groups. For the Chebotarev density theorem, Chebotarev proved it in 1926 and Lagarias and Odlyzko proved the first effective version in 1975. Here the goal is to improve the ability to effectively count small primes with prescribed splitting types in a fixed number field. This is a hard problem without assuming the Generalized Riemann Hypothesis (GRH). For zero density estimates, early pioneers include Bohr and Landau who proved zero density estimates for the Riemann zeta function. However, the theory for zero density estimates of automorphic L-functions is less developed, and is a major focus of ongoing research. In this thesis, we first prove an upper bound for l-torsion in class groups of almost all fields in certain families of D4-quartic fields. Our key tools are a new effective Chebotarev density theorem for these families of D4-quartic fields and a lower bound for the number of fields in the families. Second, we prove a generalized version of the Barban-Vehov problem, that is, an estimate for a sum over ideals involving Selberg sieve weights. The original version only holds over Q; the new version holds over an arbitrary number field k. Third, we prove a new zero density estimate for families of automorphic L-functions defined over a number field k. This work generalizes and sharpens the method of pseudo-characters and the large sieve developed by Kowalski and Michel. This work also applies the generalized Barban-Vehov problem mentioned above. Lastly, as an application, we demonstrate for a particular family of number fields of degree n over k (for any n) that an effective Chebotarev density theorem holds for almost all fields in the family. Hence, we deduce a nontrivial bound on l-torsion in class groups for all l and for almost all fields in the family.