Character Theory and Artin L-functions

Abstract

In the spirit of Artin, Brauer, and Heilbronn, we implement representation theory together with the Artin formalism to study L-functions in this thesis. One of the major themes is motivated by the work of Heilbronn and many others on classical Heilbronn characters. We define the arithmetic Heilbronn characters and apply them to study L-functions. In particular, we prove a theorem concerning the analytic ranks of elliptic curves as predicted by the Birch-Swinnerton-Dyer conjecture. In a different vein, we employ the theory of supercharacters introduced by Diaconis and Isaacs to derive a supercharacter-theoretic analogue of Heilbronn characters. Moreover, we generalise the effective Chebotarev density theorem due to M. R. Murty, V. K. Murty, and Saradha in the context of supercharacter theory. Lastly, we study the conjectures of Artin and Langlands via group theory and extend the previous work of Arthur and Clozel. For instance, we introduce the notion of near supersolvability and near nilpotency, and show that Artin's conjecture holds if $\Gal(K/k)$ is nearly supersolvable. As a consequence, the Artin conjecture is true for any solvable Frobenius Galois extension. Also, we derive the automorphy for every nearly nilpotent group. Furthermore, the Langlands reciprocity conjecture has been established for Galois extensions of number fields of either square-free degree or odd cube-free degree as well as all non-$A_5$ extensions of degree at most 100.