# UCSD Number Theory Seminar (Math 209)

### Thursday 2-3pm, AP&M 7421

#### Winter Quarter 2015

 April 2 NO SEMINAR 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 Yuri Zarhin (Penn State University) Tate and Weil classes on abelian varieties over finite fields We discuss nontrivial multiplicative relations among eigenvalues of Frobenius endomorphisms of abelian varieties over finite fields. (The trivial relations are provided by the Riemann Hypothesis that was proven by A. Weil.) We classify all abelian varieties over finite fields of a certain dimension which admit the nontrivial relations and give an explicit construction of corresponding exotic Tate classes. May 14 NO SEMINAR May 21 Piotr Krason (Szczecin University, Poland) On arithmetic in Mordell-Weil groups We will describe the problem of detecting linear dependence of points in Mordell-Weil groups A(F) of abelian varieties. This is done via reduction maps. We determine the sufficient conditions for the reduction maps to detect linear dependence in A(F). We also show that our conditons are very close to be or perhaps are the best possible. In particular we try to determine the conditions for detecting linear dependence in Mordell-Weil groups via finite number of reductions. The methods combine applications of transcedental, l-adic and mod v techniques. This is joint work with G. Banaszak. May 28 TBA June 4 TBA June 11 Francesco Baldassarri (Padova) A $p$-adically entire function with integral values on $\Q_p$ and the exponential of perfectoid fields We give an essentially self-contained proof of the fact that a certain $p$-adic power series $$\Psi= \Psi_p(T) \in T + T^{2}\Z[[T]]\;,$$ which trivializes the addition law of the formal group of Witt $p$-covectors $\widehat{\rm CW}_{\Z}$, is $p$-adically entire and assumes values in $\Z_p$ all over $\Q_p$. We also carefully examine its valuation and Newton polygons. We will recall and use the isomorphism between the Witt and hyperexponential groups over $\Z_p$, and the properties of $\Psi_p$, to show that, for any perfectoid field extension $(K,|\,|)$ of $(\Q_p,|\,|_p)$, and to a choice of a pseudo-uniformizer $\varpi = (\varpi^{(i)})_{i \geq 0}$ of $K^\flat$, we can associate a continuous additive character $\Psi_{\varpi}: \Q_p \to 1+K^{\circ \circ}$, and we will give a formula to calculate it. The character $\Psi_{\varpi}$ extends the map $x \mapsto \exp \pi x$, where $$\pi := \sum_{i\geq 0} \varpi^{(i)} p^i + \sum_{i<0} (\varpi^{(0)})^{p^{-i}} p^i \in K\;.$$ I will also present numerical computation of the first coefficients of $\Psi_p$, for small $p$, due to M. Candilera.