Abelian Varieties

Abelian varieties are central objects of arithmetic geometry, connecting algebraic geometry, complex analysis, automorphic forms, and number theory. They serve as relatively accessible but still nontrivial examples of general projective smooth varieties, while their moduli spaces form basic building blocks for Shimura varieties and consequently play a key role in the theory of automorphic representations and the Langlands program. The book focuses on four topics at the forefront of current research: the geometry of the moduli space of abelian varieties in positive characteristic, the Chow theory of abelian varieties, the Torelli locus within the moduli space of abelian varieties in positive characteristic, and the theory of heights on abelian varieties. This volume explores each of these aspects through the perspectives of four leading experts. It is based on the lecture notes prepared for the twenty-seventh Arizona Winter School held in March 2024 at the University of Arizona in Tucson. The speakers were Valentijn Karemaker, Ben Moonen, Rachel Pries, and Joseph H. Silverman. Written in a style accessible to students entering the field, the book offers advanced graduate students and researchers an excellent introduction to several important and active areas of modern research.