UCSD Number Theory Seminar (Math 209)

Thursday 2-3pm, AP&M 7421

Winter Quarter 2015

For previous quarters' schedule, click here.

April 2


April 9

Xinwen Zhu (Caltech)
On some Tate cycles on Shimura varieties

I will first describe certain conjectural Tate classes in the middle dimensional etale cohomology of many Shimura varieties over finite fields (e.g. Hilbert and Picard modular surfaces at inert primes). According to the Tate conjecture, there should exist corresponding algebraic cycles. Surprisingly, we find that these cycles are provided by the supersingular (or more precisely basic) loci of these Shimura varieties. This is based on a joint work with Liang Xiao.

April 16

René Schoof (Università degli Studi di Roma Tor Vergata)
Finite group schemes and abelian varieties with good reduction outside one prime

The Jacobian $J_0(23)$ of the modular curve $X_0(23)$ is a semi-stable abelian variety over $\Bbb Q$ with good reduction outside $23$. It is simple. We prove that every simple semi-stable abelian variety over $\Bbb Q$ with good reduction outside $23$ is isogenous over $\Bbb Q$ to $J_0(23)$.

April 23

Majid Hajian (Caltech)
On a motivic method in Diophantine geometry

By studying variation of motivic path torsors associated to a variety, we show how certain nondensity assertions in Diophantine geometry can be translated to problems concerning K-groups. Concrete results then follow based on known (and conjectural) vanishing theorems.

April 30

Grzegorz Banaszak (Poznan)
The algebraic Sato-Tate group and Sato Tate conjecture

Let $K$ be a number field and let $A$ be an abelian variety over $K.$ In an effort of proper setting of the Sato-Tate conjecture concerning the equidistribution of Frobenius elements in the representation of the Galois group $G_K$ on the Tate module of $A$, one of attempts is the introduction of the algebraic Sato-Tate group $AST_{K}(A)$. Maximal compact subgroups of $AST_{K}(A)(\mathbb{C})$ are expected to be the key tool for the statement of the Sato-Tate conjecture for $A$. At the lecture, following an idea of J-P. Serre, an explicit construction of $AST_{K}(A)$ will be presented based on P. Deligne's motivic category for absolute Hodge cycles. I will discuss the arithmetic properties of $AST_{K}(A)$ along with explicit computations of $AST_{K}(A)$ for some families of abelian varieties. I will also explain how this construction extends to absolute Hodge cycles motives in the Deligne's motivic category for absolute Hodge cycles. This is joint work with Kiran Kedlaya.

May 7


May 14


May 21


May 28


June 4


June 11