## Departmental PhD Thesis Exam – Zavosh Amir-Khosravi

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.

Thursday, April 11, 2013 @ 3:00 PM

BA 6183, 40 St. George Street

PhD Candidate:           Zavosh Amir-Khosravi

PhD Thesis Title:        MODULI OF ABELIAN SCHEMES AND SERRE’S TENSOR CONSTRUCTION

Abstract:

In this thesis we study moduli stacks $\cal M_\Phi^n$, indexed by an integer $n>0$ and a CM-type $(K,\Phi)$, which parametrize abelian schemes equipped with action by ${\cal O}_{K}$ and an ${\cal O}_{K}$-linear principal polarization, such that the representation of ${\cal O}_K$ on the relative Lie algebra of the abelian scheme consists of $n$ copies of each character in $\Phi$. We do this by systematically applying Serre’s tensor construction, and for that we first establish a general correspondence between polarizations on abelian schemes $M\otimes_R A$ arising from this construction and polarizations on the abelian scheme $A$, along with positive definite hermitian forms on the module $M$. Next we describe a tensor product of categories and apply it to the category $\text{Herm}_n({\cal O}_{K})$ of finite non-degenerate positive-definite
${\cal O}_{K}$-hermitian modules of rank $n$ and the category fibred in groupoids $\cal M_\Phi^1$ of principally polarized CM abelian schemes. Assuming $n$ is prime to the class number of $K$, we show that Serre’s tensor construction provides an identification of this tensor product with a substack of the moduli space $\cal M_\Phi^n$, and that in some cases, such as when the base is finite type over $\Bbb C$ or an algebraically closed field of characteristic zero, this substack is the entire space. We then use
this characterization to describe the Galois action on $\cal M_\Phi^n(\overline{\Bbb Q})$, by using the description of the action on $\cal M_\Phi^1(\overline{\mathbb Q})$ supplied by the main theorem of complex multiplication.