The Preliminary Arizona Winter School (PAWS) is a virtual program on topics related to the upcoming AWS, with an intended audience of advanced undergraduate students and junior graduate students.
Abelian varieties which are higher dimensional analogues of elliptic curves are important objects in arithmetic geometry and number theory. For example, abelian varieties defined by equations over the rational numbers pose interesting arithmetic problems concerning their rational points. To understand these abelian varieties over Q, a key tool is to reduce them modulo primes numbers and study the resulting abelian varieties over finite fields. The aim of this course is to provide an introduction to the theory of abelian varieties over finite fields.
If the endomorphism ring of an elliptic curve contains an order of a quadratic imaginary field, we say this elliptic curve has complex multiplication. Elliptic curves with complex multiplication have many special properties and are of great importance in the research of number theory and arithmetic geometry. The theory of complex multiplication played a crucial rule in the class field theory of imaginary quadratic fields. Moreover, the theory of complex multiplication is in the center of current research on L-functions, Galois representations, and Shimura varieties. In this course, we will start with the complex theory of elliptic curves and the definition of complex multiplication. We will discuss class field theory of imaginary quadratic fields; the L-functions and Galois representations of CM elliptic curves; the modular curve and the CM points on them; the Shimura-–Taniyama theorem and the use of CM elliptic curves in the proof of Elkies’ theorem.