Let K be a local non-archimedian field, F = K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical and Iwahori Hecke algebras for representations of the group G = G(F) and its central extension Ĝ. For instance our spherical Hecke algebra corresponds to the subgroup G (A) ⊂ G(F) where A ⊂ F is the subring OK((t)) where OK ⊂ K is the ring of integers. It turns out that for generic level (cf. [4]) the spherical Hecke algebra is trivial; however, on the critical level it is quite large. On the other hand we expect that the size of the corresponding Iwahori-Hecke algebra does not depend on a choice of a level (details will be considered in another publication).

We denote by FPMC the class of all non-singular projective algebraic surfaces X over ℂ with finite polyhedral Mori cone NE(X) ⊂ NS(X) ⊗ ℝ. If ρ(X) = rk NS(X) ≥ 3, then the set Exc(X) of all exceptional curves on X ∈ FPMC is finite and generates NE(X). Let δE(X) be the maximum of (-C2) and pE(X) the maximum of pa(C) respectively for all C ∈ Exc(X). For fixed ρ ≥ 3, δE and pE we denote by FPMCρ,δE,pE the class of all algebraic surfaces X ∈ FPMC such that ρ(X) = ρ, δE(X) = δE and pE(X) = pE. We prove that the class FPMCρ,δE,pE is bounded in the following sense: for any X ∈ FPMCρ,δE,pE there exist an ample effective divisor h and a very ample divisor h′ such that h2 ≤ N(ρ, δE) and h′2 ≤ N′(ρ, δE, pE) where the constants N(ρ, δE) and N′(ρ, δE, pE) depend only on ρ, δE and ρ, δE, pE respectively.

One can consider Theory of surfaces X ∈ FPMC as Algebraic Geometry analog of the Theory of arithmetic reflection groups in hyperbolic spaces.

Let f(z) be a cusp form of type (l,ε) on Γ0(N) which is a common eigenfunction of all Hecke operators. For such f(z), Deligne and Serre [1] proved that there exists a linear representation

such that the Artin L-function for p is equal to the L-function associated to f(z).

Following the analogies between 3-manifolds and number rings in arithmetic topology, we study the homology of branched covers of 3-manifolds. In particular, we show some analogues of Iwasawa’s theorems on ideal class groups and unit groups, Hilbert’s Satz 90, and some genus-theory–type results in the context of 3-dimensional topology. We also prove that the 2-cycles valued Tate cohomology of branched Galois covers is a topological invariant, and we give a new insight into the analogy between 2-cycle groups and unit groups.

The classical subject of surfaces containing a hyperelliptic curve (here a double cover of P1) among their hyperplane sections was settled some years ago by the third author and Van de Ven [SV] (see also [Se], [Io]). This paper is devoted to answering the following general question arising very naturally from that problem.

The basic properties of associative rings R satisfying a polynomial identity p[x1…, xn] = 0 were obtained under the assumptions that the ring was an algebra [e.g., [4] Ch. X], or with rather strong restrictions on the ring of operators ([1]). But it is desirable to have these properties for arbitrary rings, and the present paper is the first of an attempt in this direction. The problem is almost trivial for prime or semi-prime rings but quite difficult in arbitrary rings. The known proofs for algebras have to be modified and in some cases new proofs have to be obtained as the existing proofs fail to exploit the known structure. In the present paper we extend the results of [1] on the nil subalgebras of a ring with an identity for arbitrary multiplicative nil semi-groups of the ring and for arbitrary rings.

We show that isomorphisms between arithmetic fundamental groups of hyperbolic curves over p-adic local fields preserve the decomposition groups of all closed points (respectively, closed points arising from torsion points of the underlying elliptic curve), whenever the hyperbolic curves in question are isogenous to hyperbolic curves of genus zero defined over a number field (respectively, are once-punctured elliptic curves [which are not necessarily defined over a number field]). We also show that, under certain conditions, such isomorphisms preserve certain canonical “integral structures” at the cusps [i.e., points at infinity] of the hyperbolic curve.

Introduction. Let S be the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and let G denote the Galois group of the quotient field extension. Auslander and Rim have shown in [3] that the trivial crossed product Δ (1, S, G) is an hereditary order if and only if 5 is a tamely ramified extension of R. And the author has proved in [7] that if the extension S of R is tamely ramified then the crossed product Δ(f, 5, G) is a Π-principal hereditary order for each 2-cocycle f in Z2(G, U(S)). (See Section 1 for the definition of Π-principal hereditary order.) However, the author has exhibited in [8] an example of a crossed product Δ(f, S, G) which is a Π-principal hereditary order in the case when S is a wildly ramified extension of R.

For an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection, we classify thick subcategories of finitely generated modules over strict local complete intersections and produce generators for the category of coherent sheaves on a separated Noetherian scheme with an ample family of line bundles.

Let X be the set of all natural numbers and let be the group of all finite permutations of X. The group equipped with the discrete topology, is called the infinite symmetric group. It was discussed in F. J. Murray and J. von Neumann as a concrete example of an ICC-group, which is a discrete group with infinite conjugacy classes. It is proved that the regular representation of an ICC-group is a factor representation of type II1. The infinite symmetric group is, therefore, a group not of type I. This may be the reason why its unitary representations have not been investigated satisfactorily. In fact, only few results are known. For instance, all indecomposable central positive definite functions on , which are related to factor representations of type IIl, were given by E. Thoma. Later on, A. M. Vershik and S. V. Kerov obtained the same result by a different method in and gave a realization of the representations of type II1 in. Concerning irreducible representations, A. Lieberman and G. I. Ol’shanskii obtained a characterization of a certain family of countably many irreducible representations by introducing a particular topology in However, irreducible representations have been studied not so actively as factor representations.

In the previous papers [3], [4], [5] the author gave some results on the uniqueness of meromorphic maps of Cn into the N-dimensional complex projective space PN(C) which have the pre-assigned inverse images for some hyperplanes in PN(C). Relating to these results, we attempt in this paper to generalize the following Cartan-Nevanlinna’s theorem to the case of meromorphic maps into PN(C).

By the coadjoint representation of a connected Lie group G with the Lie algebra g we mean the representation CoAd(g) = tAd(g-1) in the dual space g*. Imitating Chevalley’s argument for complex semi-simple Lie algebras, we shall show that the CoAd (G)-invariant polynomial algebra on g* is finitely generated by algebraically independent polynomials when G is the inhomogeneous linear group IU(n) or ISO(n). In view of a well-known theorem [8, p. 183] our results imply that the centers of the enveloping algebras for the (or the complexified) Lie algebras of these groups are also finitely generated. Recently much more inhomogeneous groups have been studied in a similar context [2]. Our results, however, are further reaching as far as the groups IU(n) and ISO(n) are concerned [cf. 3, 4, 6, 7, 9].

It is proved that the central values of symmetric square L-functions of normalized Hecke eigenforms for the full modular group on average satisfy an analogue of the Lindelöf hypothesis in weight aspect, under the assumption that these values are non-negative.

Let Γ be a fuchsian group of the first kind and assume that Γ contains the element and let x be a unitary representation of Γ of degree 1 such that X(—I) = — 1. Let S1(Γ,X) be the linear space of cusp forms of weight one on the group Γ with character X. We shall denote by d1 the dimension of the linear space S1(Γ, X). It is not effective to compute the number dl by means of the Riemann-Roch theorem. Because of this reason, it is an interesting problem in its own right to determine the number d1 by some other method (for example,).

The order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.

It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.

We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

In this paper we are concerned with stochastic control problems of the following kind. Let Y(t) be a d’-dimensional Brownian motion defined on a probability space (Ω, F, Ft, P) and u(t) an admissible control. We consider the Cauchy problem of stochastic partial differential equations (SPDE in short)

where L(y, u) is the 2nd order elliptic differential operator and M(y) the 1st order differential operator.

Let E be a compact set in the z-plane and let Ω be its complement with respect to the extended z-plane. Suppose that E is of capacity zero. Then Ω is a domain and we shall consider a single-valued meromorphic function w = f(z) on Ω which has an essential singularity at each point of E. We shall say that a value w is exceptional for f(z) at a point ζ ∈ E if there exists a neighborhood of C where the function f(z) does not take this value w.

Let k be a field of characteristic p and G a finite subgroup of GL(V) where V is a finite dimensional vector space over k. Then G acts naturally on the symmetric algebra k[V] of V. We denote by k[V]G the subring of k[V] consisting of all invariant polynomials under this action of G. The following theorem is well known.

Theorem 1.1 (Chevalley-Serre, cf. [1, 2, 3]). Assume that p = 0 or (|G|, p) = 1. Then k[V]G is a polynomial ring if and only if G is generated by pseudo-reflections in GL(V).

Expansions of generalized functions have been investigated by many authors. Korevaar [11], Widlund [20], Giertz [8], Walter [19] developed procedures for expanding generalized functions of Korevaar [12], Temple [17], and Lighthill [13], Expansions of certain Schwartz distributions [15] into series of orthonormal functions were given by Zemanian [23] (see also Zemanian [24]) and thereby he extended a number of integral transforms to distributions. The method involved in his work is very much related to the Hilbert space technique and is of somewhat different character from those used in most of the works on integral transforms such as [24, chapters 1-8]. Other works that discuss orthogonal series expansions involving generalized functions are by Bouix [1, chapter 7], Braga and Schönberg [2], Gelfand and Shilov [7, vol. 3, chapter 4] and Warmbrod [21].

Let f(z) be a holomorphic and unbounded function in | z | < 1, with the property that it remains bounded on some spiral S in | z | < 1 which approaches | z | = 1 asymptotically. The existence of such functions was first established by G. Valiron. Accordingly, we shall refer to such functions as functions of class (V) relative to S. More recently, F. Bagemihl and W. Seidel obtained examples of functions holomorphic and unbounded in | z | < 1 which approach prescribed finite or infinite values as | z | → 1 on any given enumerable set of disjunct spirals which approach | z | =1 asymptotically, as well as on certain sets of such spirals having the power of the continuum.