This paper is concerned with the problem of the determination of unknown scalars involved in the algorithm of computing the generalized Green functions of reductive groups G over a finite field. In the previous paper, we have treated the case where G = SLn. In this paper, we determine the scalars in the case where G is a classical group Sp2n or SON for arbitrary characteristic.

We compute the global log-canonical thresholds (lct) of del Pezzo surfaces of degrees ≥ 2 with du Val singularities.

Let S(n) be the matrix of a positive definite quadratic form and

Define

1.

Here the sum is over unimodular matrices which lie in a complete set of representatives for the equivalence relation with P unimodular and having block form

For a complex space X we consider the group Aut (X) of all automorphisms of X, where an automorphism means a holomorphic automorphism, i.e. an injective holomorphic mapping of X onto X itself with the holomorphic inverse. In 1935, H. Cartan showed that Aut (X) has a structure of a real Lie group if X is a bounded domain in CN([7]) and, in 1946, S. Bochner and D. Montgomery got the analogous result for a compact complex manifold X ([2] and [3]). Afterwards, the latter was generalized by R.C. Gunning ([11]) and H. Kerner ([16]), and the former by W. Kaup ([14]), to complex spaces. The purpose of this paper is to generalize these results to the case of complex spaces with weaker conditions. For brevity, we restrict ourselves to the study of σ-compact irreducible complex spaces only.

Let f(X, T1,…, Tm) be a polynomial over an algebraic number field k of finite degree. In his paper [2], T. Kojima proved

THEOREM. Let K = Q. if for every m integers t1, …, tm, there exists an r ∈ K such that f(r, t1, …, tm) =), then there exists a rational function g(T1,…,Tm) over Q such that

F(g(T1,…,Tm), T1,…,T)= 0.

In this paper, we improve a result of the third paper of this series, that is we show

THEOREM. Let K be a nilpotent extension of the rational number field Q with Galois group Γ, and G a Γ- stable finite subgroup of GLn(0K). Then G is of A-type.

Let be the space of cusp forms of weight k with respect to SL(2, Z). Let p be a prime number and let Tk(p) be the Hecke operator of degree p acting on as a linear endomorphism. Put Hk(X) = det (I – Tk(p)X + pk-lX2I), where I is the identity operator on . Hk(X) is a polynomial with coefficients of rational integers, which is called the Hecke polynomial.

In my recent paper [1], I associated a projectively invariant pseudo-distance dM to every affinely connected manifold M and proved the following

Theorem 1. Let M be a Riemannian manifold with metric and Ricci tensor RicM such that RicM ≤–c2. Let δM be the Riemannian distance defined by .

Our object in this paper is to prove the following result.

THEOREM. Let G be a finite group satisfying the following conditions:

• (*) G has subgroups L1, L2 such that L1 ≃ SL(2, q1), L2 ≃ SL(2, q2), [L1 L2] = 1, , where j is an involution, and |C(j): L1L2| = 2.

• (**) where n2 = 1,

Let Γ be a Fuchsian group (of finite type) acting on the upper half plane. To each parabolic cusp Ki (i = 1, …, h), corresponds a Eisenstein serie

If D ⊂ ℂn is a pseudoconvex domain and X ⊂ D a closed analytic subset, the famous theorem B of Cartan-Serre asserts, that the restriction operator r : (D) → (X) mapping each function F to its restriction F|X is surjective. A very important question of modern complex analysis is to ask what happens to this result if certain growth conditions for the holomorphic functions on D and on X are added.

We define weight changing operators for automorphic forms on Grassmannians, that is, on orthogonal groups, and investigate their basic properties. We then evaluate their action on theta kernels, and prove that theta lifts of modular forms, in which the theta kernel involves polynomials of a special type, have some interesting differential properties.

Tahara [1] has concretely determined the conjugate classes of finite subgroups of GL(3, Z). The group W5 in his list of groups of order 24 is in fact of order 12 and consists of the same matrices as the group W6 in his list of groups order 12. Hence, there are only 10 conjugate classes of subgroups of order 24 in GL(3, Z) and the total number of conjugate classes of finite subgroups is reduced to 73.

For prime l we study the structure of the 2-part of the ideal class group Cl of ℚ(ζl). We prove that Cl ⊗ ℤ2) is a cyclic Galois module for all l < 10000 with one exception and compute the explicit structure in several cases.

Let k be either a local or a global field, and K be a finite Galois extension of k with g = Gal (K/k). Let L be a Galois extension of K which is also Galois over k. Such an extension is called central if Gal(L/iT) lies inside the centre of Gal(L/K). Clearly L is abelian over K. Next set L* = L∩K · kab where kab is the maximal abelian extension of k in its algebraic closure. This is the genus field of L over K/k.

Let K be a Galois extension of an algebraic number field k of finite degree with Galois group g, be a congruent ideal class group of K, and M be the class field over K corresponding to . Assume that M is normal over k. Then g acts on as a group of automorphisms. Donote by lg the augmentation ideal of the group ring Zg over the ring of integers Z.

We prove that a finite-dimensional algebra $ \Lambda $ is $ \tau $ -tilting finite if and only if all the bricks over $ \Lambda $ are finitely generated. This is obtained as a consequence of the existence of proper locally maximal torsion classes for $ \tau $ -tilting infinite algebras.

In [I-S2], we gave an explicit form of zeta functions associated to the space of symmetric matrices. In this paper, the case of L-functions is treated. In the case of definite symmetric matrices, we show the ratinality of special values of these L-functions.

Probability measure valued diffusion processes have been discussed by many authors, in connection with population genetics. Most papers studying probability measure valued diffusions are mainly concerned with the ones describing single locus models. In this paper, we will discuss a measure valued diffusion describing an n locus model. Random sampling, mutation and gene conversion, a kind of interaction between loci, which was introduced and investigated by T. Ohta in [5], [6], will be taken into consideration.

In this paper, the commutators of singular integrals with rough kernels are considered. By the method of block decomposition for kernel function and Fourier transform estimates, some new results about the Lp(ℝn) boundedness for these commutators are obtained.