We show that, with some technical conditions, an Abelian monoidal category admits a monoidal embedding into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.

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.

We calculate the Euler characteristics of the local systems Sk$\open V$ [otimes] S[ell ] Λ 2$\open V$ on the moduli space $\mathcal M$2 of curves of genus 2, where $\open V$ is the rank 4 local system R1π *$\open C$.

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.

In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature KM [ges ] 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than π/2 and contains a geodesic loop of length 2π. We also prove that if M is an n([ges ]3)-dimensional complete connected Riemannian manifold with KM [ges ] 1 and radius bigger than π/2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.

We give sharp, explicit estimates for linear forms in two logarithms, simultaneously for several non-Archimedean valuations. We present applications to explicit lower bounds for the fractional part of powers of rational numbers, and to the Diophantine equation (xn − 1)/(x − 1) = yq.

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 CSn be the flag manifold SO(2n)/U(n). We give a partial classification for the endomorphisms of the cohomology ring H*(CSn; Z) which is very close to a homotopy classification of all selfmaps of CSn. Applications concerning the geometry of the space are discussed.

We prove that, for an inner form of GLn, the number of classes of cuspidal automorphic representations with fixed central character and fixed factor at almost every place is finite. We also prove, in the local situation, relations between the level and the ε-factor of an irreducible smooth representation.

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.

Let Mg be the moduli space of smooth curves of genus g [ges ] 3, and M¯g the Deligne-Mumford compactification in terms of stable curves. Let M¯g[1] be an open set of M¯g consisting of stable curves of genus g with one node at most. In this paper, we determine the necessary and sufficient condition to guarantee that a $\open Q$-divisor D on M¯g is nef over M¯g[1], that is, (D · C) [ges ] 0 for all irreducible curves C on M¯g with C ∩ M¯g[1] ≠ [empty ].

Let VD be the Shimura curve over $\open Q$ attached to the indefinite rational quaternion algebra of discriminant D. In this note we investigate the group of automorphisms of VD and prove that, in many cases, it is the Atkin–Lehner group. Moreover, we determine the family of bielliptic Shimura curves over $\bar\open Q$ and over $\open Q$ and we use it to study the set of rational points on VD over quadratic fields. Finally, we obtain explicit equations of elliptic Atkin–Lehner quotients of VD.