Background and purpose:

Abelian varieties are one of main interests in arithmetic geometry. There are several ways of studies with abelian varieties – geometric (line bundles), arithmetic (Galois representations and L-fucntions) and their classifications. In this course we shall focus on the classification problem of some abelian varieties together with a polarization over finite fields. We shall first introduce abelian varieties over finite fields with sufficient background and then study [1] together with its possible variant with polarizations. The remaining part will study the works of Howe [2,3]. If time permits we will discuss recent works of Marselgia and his collaborators.    

 

Outline:

The first part is to introduce briefly the Honda-Tate theory for abelian varieties over finite fields. Then we shall study the work [1] of Poonen and others for description of abelian varieties isogenous to a product of elliptic curves together with its possible extension with polarizations. The remaining lectures will be on the works of Howe [2,3] for polarized ordinary abelian varieties. If time permits we will discuss recent computation results of Marselgia and his collaborators.

 

References:

[1] Jordan, Bruce W.; Keeton, Allan G.; Poonen, Bjorn; Rains, Eric M.; Shepherd-Barron, Nicholas; Tate, John T. Abelian varieties isogenous to a power of an elliptic curve. Compos. Math. 154 (2018), no. 5, 934–959.

[2] Howe, Everett W. Principally polarized ordinary abelian varieties over finite fields. Trans. Amer. Math. Soc. 347 (1995), no. 7, 2361–2401. 

[3] Howe, Everett W. Kernels of polarizations of abelian varieties over finite fields. J. Algebraic Geom. 5 (1996), no. 3, 583–608.

 

(2) 13:30 - 15:00