Geometric methods in algebra and number theory : [... Conference Geometric Methods in Algebra and Number Theory in Miami December 2003] The transparency and power of geometric constructions has been a source of inspiration for generations of mathematicians; their applications to problems in algebra and number theory date back to Diophantus if not earlier. Although more sophisticated and subtle constructions have replaced the Greek techniques of intersecting lines and conics what remains unchallenged is the beauty and persuasion of pictures communicated in words or drawings. This volume focuses on the following topics: * moduli spaces Shimura varieties D modules * p adic methods (motivic integration) * number theoretic applications (rational points) All papers are strongly influenced by geometric ideas and intuition. The collection as a whole gives a representative sample of modern results and problems in algebraic and arithmetic geometry and the text can serve as an intense introduction for graduate students and others wishing to pursue research in these areas. Contributors include: V. Alexeev; L. Berger; J. B. Bost; M. Brion; C. L. Chai; T. Hausel; F. Loeser;P. Swinnerton Dyer; L. Szpiro; and Y. Zarhin TOC:Preface * Bost: Extension groups in Arakelov geometry * Chai: Hecke orbits * Loeser: Ax Kochen type theorems for p adic integrals * Szpiro: Mahler measure for dynamical systems * Zarhin: Jacobians endomorphisms and permutation groups * Alexeev: Toric Torelli map * Berger: De Rham representations and universal norms * Brion: The homogeneous coordinate ring of a spherical variety * Swinnerton Dyer: Counting points on cubic surfaces * Hausel: Mirror symmetry Langlands duality and representations of finite groups of Lie type