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1: Suppose $A_3 $ is the moduli space of abelian varieties of dimension 3 .Is the union of all one dimension shimura varieties in $A_3 $ connected? 2: Given a Shimura curve (explicit construction), …

Is there a classification of embeddings of SL_2 into SP_6 as algebraic groups over Q and R respectively? see also the link:mathoverflow.net/questions/36762,

For $k$ a finite field , $A,A'$ an abelian varieties over $k$, $G$ the Galois group of $k$, $l$ a prime number different from the characteristic of $k$ . Tate has proved that: $Q_l\otimes Hom_k(A,A' …

Let $k$ be an algebraically closed field of char $p\neq 0$, $W_2(k)$ the witt vector of length 2. $C_1$ a smooth projective curve over $W_2(k)$, and $H_1$ a vector bundle over $C_1$. We denote $C_0$ …

Let $X$ be a smooth projective curve defined over a finite field of char $p$, let $G[1]$ be a truncated Barsotti-Tate grop of level-1. My question is : can $G[1]$ be extended to a truncated Barsott …

Suppose we have a PEL type $(H,\phi ,*;T,O,V)$ where H is a rational nonsplit quaternion algebra, $\phi$is an embedding of Q-algebra $\phi : H-->M(2,R)$, and * is a positive anti involution of H; O i …

$K=Q(\sqrt{d} ) , d<0 $, $\Gamma $ an arithmetic subgroup of $G=SU(2,1)(K)$ . Is $\cup_{g\in G}(g^{-1}\Gamma g)$ dense in G for the complex topology?

Suppose $R=Z_p[t]$ , and $\hat{R}$ its p-adic completion, suppose we have Endormorphism $\Phi$ of $\hat{R}$, whose redution mop p is just the absolute Frobenius of $\hat{R}/p\hat{R}$. And $R_{\infty} …

Suppose K is a local field , G is its galois group, V a fine dimensional Vector space over F, which is a sub field of K, and totally ramified over $Q_p$. Consdider the linear action of G on V (V is no …

Let $K$ be a local field with residue field of char $p$, denote $G$ its Galois group. Is it possible that we have two Abelian varieties $A_1$ and $A_2$, defined over $K$, such that they are not isog …

Let $K=Q(\sqrt{-3})$ , is $SU(2,1)(K)$ dense in $SU(2,1)(C)$ for the complex topology?

In the paper "Travaux de Shimura" section 6, Deligne had defined a PEL- type shimura variety, for the following datum $(F,E,D,\psi)$, with $F$ a totally real cubic field, and $E$ a imaginary qudrat …

Suppose $H$ is a indefinite quaternion algebra over $\mathbb{Q}$. Are there infinitely many quadratic fields that can be embedded into $H$?