Higher Derivatives of L-series associated to Real Quadratic Fields

This text is a modified version of a chapter in a PhD thesis [21] submitted to Nottingham University in September 2006, which studied an approach to Hilbert’s twelfth problem inspired by Manin’s proposed theory of Real Multiplication [7]. In [20] we defined and studied a nontrivial notion of line bundles over Quantum Tori. In this text we study sections of these line bundles leading to a study concerning theta functions for Quantum Tori. We prove the existence of such meromorphic theta functions, and view their application in the context of Stark’s conjectures and Hilbert’s twelfth problem. Generalising the work of Shintani, we show that (modulo a Conjecture 5.7) we can write the derivatives of L-series associated to Real Quadratic Fields in terms of special values of theta functions over Quantum Tori.