In the previous part of this paper, we constructed a large family of Hecke algebras on some classical groups G defined over p-adic fields in order to understand their admissible representations. Each Hecke algebra is associated to a pair (JΣ, ρΣ) of an open compact subgroup JΣ and its irreducible representation ρΣ which is constructed from given data Σ = (Γ, P′0, ϱ). Here, Γ is a semisimple element in the Lie algebra of G, P′0 is a parahoric subgroup in the centralizer of Γ in G, and ϱ is a cuspidal representation on the finite reductive quotient of P′0. In this paper, we explicitly describe those Hecke algebras when P′0 is a minimal parahoric subgroup, ϱ is trivial and ρΣ is a character.

We prove the Tate conjecture for codimension 2 cycles on an ordinary cubic fourfold over a finite field. The proof involves the construction of canonical coordinates on the formal deformation space via a crystalline period map.

We present a microlocal version of the Riemann–Hilbert correspondence for regular holonomic $\cal D$-modules. We show that a regular holonomic system of microdifferential equations is associated to a perverse sheaf concentrated in degree 0. Moreover, we show that this perverse sheaf can be recovered from the local system it determines on the complementary of its singular locus. We characterize the classes of perverse sheaves and local systems associated to regular holonomic systems of microdifferential equations.

We describe birational models and decide the rationality/unirationality of moduli spaces $\cal A$d (and $\cal A$levd) of (1, d)-polarized Abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of Abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi–Yau threefolds with Euler characteristic zero.

We show that 17.9% of all elliptic curves over Q, ordered by their exponential height, are semistable, and that there is a positive density subset of elliptic curves for which the root numbers are uniformly distributed. Moreover, for any α > 1/6 (resp. α > 1/12) the set of Frey curves (resp. all elliptic curves) for which the generalized Szpiro Conjecture |Δ(E)| [Lt]α NE12α is false has density zero. This implies that the ABC Conjecture holds for almost all Frey triples. These results remain true if we use the logarithmic or the Faltings height. The proofs make use of the fibering argument in the square-free sieve of Gouvêa and Mazur. We also obtain conditional as well as unconditional lower bounds for the number of curves with Mordell–Weil rank 0 and [ges ]2, respectively.

We compute relations of rational equivalence among special codimension 2 cycles on families of Abelian surfaces using elements of a higher Chow group. These relations are similar to those between Heegner points and special divisors obtained by Zagier, Van der Geer and others.

Stark conjectured that given a positive integer h, there are only a finite number of CM-fields L with class number equal to h. We prove this conjecture for fields L of degree [ges ]6 whose normal closure is solvable.

Let G be a real reductive Lie group, K its compact subgroup. Let A be the algebra of G-invariant real-analytic functions on T*(G/K) (with respect to the Poisson bracket) and let C be the center of A. Denote by 2ε(G,K) the maximal number of functionally independent functions from A\C. We prove that ε(G,K) is equal to the codimension δ(G,K) of maximal dimension orbits of the Borel subgroup B⊂G$^{\Bbb C}$ in the complex algebraic variety G$^{\Bbb C}$/K$^{\Bbb C}$. Moreover, if δ(G,K)=1, then all G-invariant Hamiltonian systems on T*(G/K) are integrable in the class of the integrals generated by the symmetry group G. We also discuss related questions in the geometry of the Borel group action.

Let M be a manifold of X = $\Bbb C$n, A a small analytic disc attached to M, zo a point of ∂A where A is tangent to M, z1 another point of ∂A where M extends to a germ of manifold M1 with boundary M. We prove that CR functions on M which extend to M1 at z1 also extend at zo to a new manifold M2. The directions M1 and M2 point to, are related by a sort of connection associated to A which is dual to the connection obtained by attaching ‘partial analytic lifts’ of A to the co-normal bundle to M in X.

Let M be a projective manifold, p: MG→M a regular covering over M with a free Abelian transformation group G. We describe the holomorphic functions on MG of an exponential growth with respect to the distance defined by a metric pulled back from M. As a corollary, we obtain Cartwright and Liouville-type theorems for such functions. Our approach brings together the L2 cohomology technique for holomorphic vector bundles on complete Kähler manifolds and the geometric properties of projective manifolds.

In the previous works of the first author, two completely different constructions of single valued Grassmannian trilogarithms were given. One of the constructions, in Math. Res. Lett. 2 (1995), 99–114, is very simple and provides Grassmannian n-logarithms for all n. However its motivic nature is hidden. The other construction in Adv. in Math. 114 (1995), 197–318, is very explicit and motivic, but the generalization for n>4 is not known. In this paper we will compute explicitly the Grassmannian trilogarithm constructed in Math. Res. Lett. 2 (1995), 99–114 and prove that it differs from the motivic Grassmannian trilogarithm by an explicitly given product of logarithms. We also derive some general results about the Grassmannian polylogarithms.

The connection between these Fano 3-folds and plane quartic curves is explained.

Let X be a curve of genus g[ges ]2 over a field of characteristic zero. Then X has at most finitely many torsion packets of size greater than 2. Moreover, X has infinitely many torsion packets of size 2 if and only if either g=2, or g=3 and X is both hyperelliptic and bielliptic.

We prove a generalisation of a theorem of Nagata on ruled surface to the case of the fiber bundle E/P → X, associated to a principal G-bundle E. Using this we prove boundedness for the isomorphism classes of semi-stable G-bundles in all characteristics.

We present an analogy of the famous formula that the square of the Pfaffian is equal to the determinant for an alternating matrix for the case where the entries are the generators of the orthogonal Lie algebras. This identity clarifies the relation between the two sets of central elements in the enveloping algebra of the orthogonal Lie algebras. We employ systematically the exterior calculus for the proof.