Elliptic curves, modular forms, and the associated exponential sums

Abstract

This chapter delves into the motivations, historical context, and pivotal findings of this thesis. The insights into exponential sums linked with linear recurrence sequences over finite fields were imparted to me by Bajpai and García. Additionally, Bajpai provided valuable insights into the background of vector-valued modular forms. While Theorem 1.4.2 is stated, its proof is omitted in this thesis. Nevertheless, the key methodology is elaborated on in detail on page 29. The contents of pages 29-30 are the outcome of collaborative efforts with Myerson, Loughran, and Nakahara. The remainder of the introduction represents my original contributions. Chapter 2I gained insight into the historical background of exponential sums over prime fields through discussions with García and Bajpai. They initially established the case ν = 1 in Theorem 2.2.1. Subsequently, I extended their work to cover the case ν = 2. Finally, through collaborative discussions, we resolved the case ν > 2 using one of Garaev's techniques. Corollary 2.3.1 follows immediately from the non-triviality of the exponential sums, as highlighted by Bajpai and García. Building on this observation, I proceeded to prove Theorem 2.3.2 and provided Example 2.3.3 in support. Chapter 3The suggestion from Bajpai and García to investigate exponential sums with a(p n ) by studying those associated with linear recurrence sequences