The dimension of affine Deligne-Lusztig varieties in the affine Grassmannian of unramified groups

In this paper we calculate the dimension of affine Deligne-Lusztig varieties in the affine Grassmannian of unramified groups. We also examine the irreducible components of ADLVs in the superbasic case and the $J_b(F)$-action on them.Comment: 24 pages, v2: Corrected typos and added details in the calculation

On the Tate Conjecture in Codimension One for Varieties with h2,0=1h^{2, 0} = 1 in Positive Characteristics

We prove that the Tate conjecture in codimension $1$ is ``generically true'' for mod $p$ reductions of complex projective varieties with $h^{2, 0} = 1$, under a mild assumption on moduli. By refining this general result, we prove in characteristic $p \ge 5$ the BSD conjecture for a height $1$ elliptic curve $\mathcal{E}$ over a function field of genus $1$, assuming that the singular fibers in its minimal compactification are all irreducible. We also prove the Tate conjecture for algebraic surfaces with $p_g = K^2 = 1$, $q = 0$ and an ample canonical bundle in characteristic $p \ge 5$.Comment: 69 pages. Major Revision. Generalized the results to finitely generated fields. Considerations of conjugate Shimura varieties were replaced by a simpler trick involving the Grothendieck restriction functor. A minor mistake in the previous treatment of the non-maximal monodromy case was correcte