Conjectures in arithmetic algebraic geometry

The book explores various aspects of number theory and algebraic geometry through a structured approach. It begins with the zero-dimensional case, focusing on number fields, class numbers, Dirichlet L-functions, and the Class Number Formula, extending to both abelian and non-abelian number fields with Artin L-functions. The one-dimensional case is addressed through elliptic curves, detailing their general features, varieties over finite fields, and L-functions, including complex multiplication and the arithmetic of elliptic curves, as well as the Tate-Shafarevich group and higher genus curves. The text then delves into the formalism of L-functions, incorporating Deligne cohomology and Poincaré duality theories, alongside the Standard Conjectures. It discusses Riemann-Roch, K-theory, and motivic cohomology, covering Grothendieck-Riemann-Roch, Adams operations, and higher algebraic K-theory. Subsequent sections investigate regulators, including Borel's and Beilinson's conjectures, and delve into Beilinson's second and third conjectures, focusing on arithmetic intersections and Hilbert modular surfaces. The book also addresses absolute Hodge cohomology, the Hodge and Tate conjectures, and Abel-Jacobi maps, alongside mixed realizations and motives, culminating in examples and results that revisit key conjectures and explore modular curves and linear varieties.