Arizona Winter School 2024: Abelian Varieties

AWS 2024 will be held March 2-6, 2024 at the University of Arizona in Tucson, AZ.

Registration for students is now closed; the application/registration deadline for students was November 15th, 2023.

Senior participants (postdocs, faculty, and anyone post phd) should register here. The application deadline for senior participants is December 15th, 2023.

For more information go here.

Courses:

• SHORTTITLE  Valentijn Karemaker: Geometry and arithmetic of moduli spaces of abelian varieties in positive characteristic; Course and project outline

• SHORTTITLE  Ben Moonen: Algebraic cycles on abelian varieties; Course and project outline

• SHORTTITLE  Rachel Pries: Torelli locus in the moduli space of abelian varieties, with applications to Newton polygons of curves; Course and project outline

• SHORTTITLE  Joe Silverman: Canonical heights on abelian varieties; Course and project outline

With Clay lecturerSHORTTITLE  Barry Mazur

Organizers: Isabel Vogt, Hang Xue, David Zureick-Brown (main program) with Renee Bell, Alina Bucur, Brandon Levin, and Anthony Várilly-Alvarado.

Funded by the National Science Foundation and organized in partnership with the Clay Mathematics Institute. Standard Cloud hosting for online discussions sponsored by Zulip.

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Preliminary Arizona Winter School 2023: Elliptic Curves and Abelian Varieties

Registration is now closed; the deadline to apply was July 15th, 2023. Videos, notes, and problem sets will be available at this page for anyone who missed the registration deadline.

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.

PAWS 2023 will be held October 2nd — November 10th, 2023 and will consist of two concurrent six-week lecture series.

• SHORTTITLE  Lassina Dembele: Abelian varieties over finite fields

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.

Background: Please take a look at PSet0, which is meant as a guide to prepare for the start of the course. It is not expected that you understand all the material on day one of the course. As preparation, it might be a good idea to review any abstract algebra that you have taken, especially the subject of finite fields.

• SHORTTITLE  Wanlin Li: Elliptic curves with complex multiplication

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.

Background: Please take a look at PSet0, which is meant as a guide to prepare for the start of the course. It is not expected that you understand all the material on day one of the course. As preparation, it might be a good idea to review any abstract algebra or complex analysis that you have taken as these subjects in particular will be useful for the course.

For more information go here.

• Organizers: Renee Bell, Alina Bucur, and Anthony Várilly-Alvarado (main organizers), with Brandon Levin, Hang Xue, Isabel Vogt, and David Zureick-Brown.


• Funded by the National Science Foundation.
• Standard Cloud hosting for online discussions sponsored by Zulip.