L-values and Extensions

This lecture covers de Rham theory, L-values, and non-split extensions. It delves into Riemann's C-function, Euler product, analytic continuation, functional equation, and shared features of L-functions. The significance of certain L-values and extensions is explored, shedding light on mixed Hodge structures, p-adic Galois representations, and mixed motives. Special value formulas for quadratic characters and an extension to singular cohomology are discussed, along with examples related to Hecke characters, modular forms, and symmetric squares of modular forms. The lecture concludes with a discussion on Euler systems, Galois representations, and future research directions.