Modular Forms and L-Functions. A Mathematical Journey

Research Paper (postgraduate) , 2026 , 11 Pages , Grade: A

Mathematics

This manuscript presents a structured introduction to modular forms and L-functions, two central themes in modern number theory. Modular forms are highly symmetric holomorphic functions on the upper half-plane, while L-functions encode their arithmetic information through Dirichlet series and Euler products. The exposition develops the foundational definitions, Fourier expansions, Hecke operators, analytic continuation, and the modularity theorem, and then connects these ideas to elliptic curves, cryptography, and mathematical physics. The goal is to provide a mathematically clear, book-style narrative suitable for a chapter-length or short monograph treatment.

• Grade: A
• Author: Fazal Rehman (Author)
• Publication Year: 2026
• Pages: 11
• Catalog Number: V1718264
• ISBN (PDF): 9783389186831
• Language: English
• Tags: modular forms, L-functions, upper half-plane, SL(2, Z), cusp forms, Eisenstein series, Hecke eigenforms, Fourier q-expansion, Ramanujan tau function, Euler products, modularity theorem, elliptic curves, elliptic curve cryptography, Langlands program, automorphic forms, string theory, arithmetic geometry