This book provides an introduction to the new field of -geometry and its applications to the construction of certain quotient spaces that cannot be approached within usual algebraic geometry, specifically quotients of Siegel moduli spaces by the action of Hecke correspondences. -geometry is a geometry obtained from usual algebraic geometry by 'adjoining' a Fermat quotient operator; the latter morally plays the role of an 'arithmetic differentiation' with respect to a prime integer. The book assumes some basic knowledge of the algebraic geometry of schemes, including abelian schemes, but it is otherwise self-contained.

Its intended audience includes mathematics graduate students and researchers interested in algebraic geometry and number theory.

Introduces the new field of -geometry Provides applications to the construction of quotient spaces that do not exist in usual algebraic geometry Gives an in-depth study of the invariant theory of tuples of quadrics

Autorentext

Alexandru Buium is a distinguished professor emeritus at the University of New Mexico (UNM Albuquerque, USA). Previously, he was a professor at UNM and at the University of Illinois (Urbana-Champaign). He has held visiting positions at Columbia University, the University of Paris (Orsay), the IAS (Princeton), the MPI (Bonn), and the IHES (Bures sur Yvette). He is a former Humboldt fellow and an AMS fellow. He has published 91 research papers, 6 research monographs, and 1 textbook. His fields of research are algebraic geometry and number theory.

Adrian Vasiu is a professor at SUNY Binghamton (Binghamton, USA). He was awarded the SUNY Chancellor s Award for Excellence in Scholarship and Creative Activities. He was a member of the IAS (Princeton) and a visiting scientist at MPI (Bonn), at the IHES (Bures sur Yvette), at the Tata Institute of Fundamental Research (Mumbai), at the Indian Institute of Science Education and Research (Thiruvananthapuram), and at the Institute for Mathematical Research (Zürich). He has published 38 research papers, 1 research monograph and 2 Oberwolfach Reports. His field of research is arithmetic algebraic geometry.

Inhalt

Chapter 1. Introduction.- Chapter 2. Main concepts and results.- Chapter 3. Proofs of the main results.- Chapter 4. Background and required results in GIT.