Elliptic curves, modular forms, and the associated exponential sums
von Subham Bhakta
Datum der mündl. Prüfung:2023-03-27
Erschienen:2024-02-27
Betreuer:Prof. Dr. Harald Andrés Helfgott
Gutachter:Prof. Dr. Harald Andrés Helfgott
Gutachter:Prof. Dr. Jörg Brüdern
Dateien
Name:Subham_thesis.pdf
Size:1.42Mb
Format:PDF
Zusammenfassung
Englisch
This thesis focuses on the various arithmetic aspects of elliptic curves and modular forms. Employing the exponential sum estimates obtained from sum-product estimates for finite fields, we estimate exponential sums linked to linear recurrence sequences and unveil additive characteristics of the Fourier coefficients of congruence modular forms. Additionally, we present a method for analyzing these additive properties through associated Galois representations of composite moduli. In exploring the non-congruence aspect, we delve into modular forms within vector-valued settings, analyze the growth patterns of Fourier coefficients, and evaluate their impact on corresponding exponential sums. Towards the end, we study character sums with elliptic sequences, leading to a better understanding of the elliptic Wieferich primes.
Keywords: Elliptic curves; Exponential Sums; Modular forms; Galois representations; Automorphic forms