Elliptic Curves and an Introduction to Abelian Varieties

Jennifer Paulhus 
Department of Mathematics and Statistics 
Grinnell College 
Abstract: 
From the proof of Fermat's Last Theorem to applications in cryptography and factorization, elliptic curves played pivotal roles in some fundamental mathematical research of the 20th century. We will begin this mini course by introducing elliptic curves and their history, describing the beautiful, natural group structure on them, and discussing the key results in the field. We will next explore how we might generalize elliptic curves, particularly focusing on abelian varieties over the complex field, and talk about which results carry over through these generalizations. As time permits, we may talk about special abelian varieties called Jacobian varieties. 
References: 
[1] J. H. Silverman and J. T. Tate: Rational Points of Elliptic Curves, SpringerVerlag New York, 1992. 
[2] D. Cox, J. Little and D. O'Shea: Ideals, Varieties, and Algorithms, SpringerVerlag New York, Third Edition, 2007. 
[3] C. Birkenhake and H. Lange: Complex Abelian Varieties, SpringerVerlag Berlin Heidelberg, 2004. 
[4] J. S. Milne: Abelian Varieties, Course Notes, 2008. http://www.jmilne.org/math/CourseNotes/AV.pdf 