Let f be a primitive Hilbert modular cusp form of arbitrary level and parallel weight k, defined over a totally real number field F. We define a finite set of primes $\cal S$ that depends on the weight and level of f, the field F, and the torsion in the boundary cohomology groups of the Borel–Serre compactification of the underlying Hilbert-Blumenthal variety. We show that, outside $\cal S$, any prime that divides the algebraic part of the value at s=1 of the adjoint L-function of f is a congruence prime for f. In special cases we identify the ‘boundary primes’ in terms of expressions of the form $N_{F/{\mathbb{Q}}}(\epsilon^{k-1} - 1)$, where ε is a totally positive unit of F.

Let U be a quantumgroup with divid d powers at root ofunity constructed froma root system R .Let u U b th small quantumgroup. The cohomology of u with trivial coefficients was computed by Ginzburg and Kumar. It turns out to be isomorphic to the functions algebra of the nilpotent cone of a semisimpl algebraic group with root system R. In this not we calculate cohomology of u with coefficients in simplest reducible tilting modul with nontrivial cohomology. It appears to b isomorphic to th functions algebra of th closure of the subregular nilpotent orbit.

Let Ω be an irreducible bounded symmetric domain and $\Gamma \subset {\rm Aut}(\Omega)$ be a torsion-free discrete group of automorphisms, $X: = \Omega/\Gamma$. We study the problem of algebro-geometric and differential-geometric characterizations of certain compact holomorphic geodesic cycles $S \subset X$. We treat special cases of the problem, pertaining to a situation in which S is a compact holomorphic curve, and to the case where Ω is a classical domain dual to the hyperquadric. In both cases we consider algebro-geometric characterizations in terms of tangent subspaces. As a consequence we derive effective pinching theorems where certain complex submanifolds $S \subset X$ are proven to be totally geodesic whenever their scalar curvatures are pinched between certain computed universal constants, independent of the volume of the submanifold S, giving new examples of the gap phenomenon for the characterization of compact holomorphic geodesic cycles.

Let p>q and let G be the group U(p, q) or Spin0(p, q). Let P=LN be the maximal parabolic subgroup of G with Levi subgroup $L\cong M\times U$ where$\openup2(M,U)=\left \{ \matrix{ ({\rm GL} _q({\mathbb C}),{\rm U}(\,p-q)),\hfill \mbox { if } G={\rm U}(\,p,q), \hfill\cr ({\rm GL} ^+_q({\mathbb R}),{\rm Spin}(\,p-q)), \mbox { if } G={\rm Spin}_0(\,p,q). }\right.$Let χ be a one-dimensional character of M and τμ an irreducible representation of U with highest weight μ. Let $\pi_{\chi,\mu}$ be the representation of P which is trivial on N and $\pi_{\chi,\mu}|_L=\chi\boxtimes \tau ^\mu$. Let $I_{p,q}$ be the Harish-Chandra module of the induced representation ${\rm Ind}_{P}^{G} \pi_{\chi,\mu}$. In this paper, we shall determine (i) the reducibility of $I_{p,q}$, (ii) the K-types of all the irreducible subquotients of $I_{p,q}$ when it is reducible, where K is the maximal compact subgroup of G, (iii) the module diagram of $I_{p,q}$ (from which one can read off the composition structure), and (iv) the unitarity of $I_{p,q}$ and its subquotients. Except in the cases $q=p-1$ and $q=1$, $I_{p,q}$ is not K-multiplicity free.

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain $n \geq b_2-2$ disjoint smooth rational curves with self-intersection −2, where $b_2$ is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with $p_g = 0$ and $K^{\,2} = 8, 9$ which admit an automorphism of order 2.