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There is now a large body of theory concerning algebraic varieties over finite fields, and many conjectures exist in this area that are of great interest to researchers in number theory and algebraic geometry. This book is concerned with the arithmetic of diagonal hypersurfaces over finite fields, with special focus on the Tate conjecture and the Lichtenbaum-Milne formula for the central value of the L-function. It combines theoretical and numerical work, and includes tables of Picard numbers. Although this book is aimed at experts, the authors have included some background material to help non-specialists gain access to the results.

Hypersurfaces. --- Finite fields (Algebra) --- Modular fields (Algebra) --- Algebra, Abstract --- Algebraic fields --- Galois theory --- Modules (Algebra) --- Hyperspace --- Surfaces

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This volume presents a lively introduction to the rapidly developing and vast research areas surrounding Calabi–Yau varieties and string theory. With its coverage of the various perspectives of a wide area of topics such as Hodge theory, Gross–Siebert program, moduli problems, toric approach, and arithmetic aspects, the book gives a comprehensive overview of the current streams of mathematical research in the area. The contributions in this book are based on lectures that took place during workshops with the following thematic titles: “Modular Forms Around String Theory,” “Enumerative Geometry and Calabi–Yau Varieties,” “Physics Around Mirror Symmetry,” “Hodge Theory in String Theory.” The book is ideal for graduate students and researchers learning about Calabi–Yau varieties as well as physics students and string theorists who wish to learn the mathematics behind these varieties.

Mathematics. --- Number Theory. --- Algebraic Geometry. --- Several Complex Variables and Analytic Spaces. --- Geometry, algebraic. --- Differential equations, partial. --- Number theory. --- Mathématiques --- Théorie des nombres --- Calabi-Yau manifolds --- String models --- Mathematical physics --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Calabi-Yau manifolds. --- String models. --- Models, String --- String theory --- Algebraic geometry. --- Functions of complex variables. --- Nuclear reactions --- Manifolds (Mathematics) --- Partial differential equations --- Algebraic geometry --- Geometry --- Number study --- Numbers, Theory of --- Complex variables --- Elliptic functions --- Functions of real variables

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This volume presents a lively introduction to the rapidly developing and vast research areas surrounding Calabi–Yau varieties and string theory. With its coverage of the various perspectives of a wide area of topics such as Hodge theory, Gross–Siebert program, moduli problems, toric approach, and arithmetic aspects, the book gives a comprehensive overview of the current streams of mathematical research in the area. The contributions in this book are based on lectures that took place during workshops with the following thematic titles: “Modular Forms Around String Theory,” “Enumerative Geometry and Calabi–Yau Varieties,” “Physics Around Mirror Symmetry,” “Hodge Theory in String Theory.” The book is ideal for graduate students and researchers learning about Calabi–Yau varieties as well as physics students and string theorists who wish to learn the mathematics behind these varieties.

Number theory --- Algebraic geometry --- Geometry --- Analytical spaces --- Mathematical analysis --- landmeetkunde --- analyse (wiskunde) --- complexe veranderlijken --- getallenleer