April 2 |
NO SEMINAR |
April 9 |
Xinwen Zhu (Caltech) 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) 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) 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) 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 |
TBA |
May 14 |
TBA |
May 21 |
TBA |
May 28 |
TBA |
June 4 |
TBA |
June 11 |
TBA |