In his famous lectures [7] Painlevé investigates general solutions of algebraic differential equations which depend algebraically on some of arbitrary constants. Although his discussions are beyond our understanding, the rigorous and accurate interpretation to make his intuition true would be possible. Successful accomplishments have been done by some authors, for example, Kimura [1], Umemura [8, 9]. From differential algebraic viewpoint in [5] the author introduces the notion of rational dependence on arbitrary constants of general solutions of algebraic differential equations, and in [6] clarifies the relation between it and the notion of strong normality. Here we aim at generalizing to higher order case the result in [4] that in the first order case solutions of equations depend algebraically on those of equations free from moving singularities which are determined uniquely as the closest ones to the given. Part of our result can be seen in [7].

Let M be a complete simply-connected riemannian manifold of even dimension m. J. Dodziuk and I.M. Singer ([D1]) have conjectured that H2p(M) = 0 if p ≠ m/2 and dim H2m/2(M) = ∞, where H2*(M) is the space of L2-harmonic forms on M.

The aim of this note is to study birational transformations of three-folds with nef canonical classes. One would like to write any such map as a composite of certain simple and basic transformations.

A special case was first considered by Kulikov [Ku] and later extensively studied by several authors. The next main conceptual step was Reid’s study [R2] of small resolutions of terminal threefold singularities. He obtained the elementary birational transformations as follows: find a copy of such a small resolution inside the threefold and then replace it with another resolution. This approach was extended by Kawamata [K3] to canonical singularities.

Our objective is to prove that certain Dirichlet series (in our variable q−s), which are defined by infinite sums, can be expressed as a product of an explicit rational function in q−s times an unknown polynomial M in q−s Moreover we show that M(q−s) is 1 if a simple condition is met. The Dirichlet series appear in the Euler products of Fourier coefficients for Eisenstein series. The series discussed below generalize the functions α0(N, q−s) used by Shimura in [12], and the theorem is an extension of Kitaoka’s result [5].

Let k be a field and let S be a polynomial ring k[x1, x2,…, xn] over k in n variables. An S-module M is called a module with linear resolution if M has a free resolution;

where, after taking suitable bases of free modules, all fi’s are matrices consisting of linear forms of S. The reader should be referred to Eisenbud- Goto [2, Sections 0 and 1] for elementary facts concerning modules with linear resolution.

Given a semilocal 1-dimensional Cohen-Macauly ring A, J. Lipman in [10] gives an algorithm to obtain the integral closure Ā of A, in terms of prime ideals of A. More precisely, he shows that there exists a sequence of rings A = A0 ⊂ A1 ⊂… ⊂ Ai ⊂…, where, for each i, i ≥ 0, Ai+1 is the ring obtained from Ai by “blowing-up” the Jacobson radical ℛ i of Ai+ i.e. Ai+l = ∪n(ℛin:ℛin). It turns out that ∪ {Ai;i≥0} = Ā (cf. [10, proof of Theorem 4.6]) and, if Ā is a finitely generated A-module, the sequence {Ai; i ≥ 0} is stationary for some m and Am = Ā, so that

Let R be a regular noetherian ring. A central question concerning projective modules over polynomial R-algebras is the following.

(1.1) BASS-QUILLEN CONJECTURE ([2] Problem IX, [10]). Every finitely generated projective module P over a polynomial R-algebra R[T], T = (T1,…, Tn) is extended from R, i.e.

P≊R[T]⊗R P/(T)P.

Let L = K be the composite of two imaginary quadratic fields and K. Suppose that the discriminants of and K are relatively prime. For any primitive ray class character χ of L, we shall compute L(1, χ) for the Hecke L-function in L. We write for the conductor of χ and C for the ray class modulo . Let c ε C be any integral ideal prime to . We write as g-module where g, n and ϑL are, respectively, the ring of integers in k, an ideal in k and the differente of L. Let where T(χ) is the Gaussian sum and, as in (3.2),

For an integer m > 2, we denote by C(m) and H(m) the ideal class group and the class-number of the field

K = Q(ζm + ζm−1)

respectively, where ζm is a primitive m-th root of unity. Let q be a prime and /Q be a real cyclic extension of degree q. Let C() and h() be the ideal class group and the class-number of . In this paper, we give a relation between C() (resp. h()) and C(m) (resp. H(m)) in the case that m is the conductor of (Main Theorem). As applications of this main theorem, we give the following three propositions. In the previous paper [4], we showed that there exist infinitely many square-free integers m satisfying n|H(m) for any given natural number n. Using the result of Nakahara [2], we give first an effective sufficient condition for an integer m to satisfy n|H(m) for any given natural number n (Proposition 1). Using the result of Nakano [3], we show next that there exist infinitely many positive square-free integers m such that the ideal class group C(m) has a subgroup which is isomorphic to (Z/nZ)2 for any given natural number n (Proposition 2). In paper [4], we gave some sufficient conditions for an integer m to satisfy 3|H(m) and m≡l (mod 4). In this paper, using the result of Uchida [5], we give moreover a sufficient condition for an integer m to satisfy 4|H(m) and m ≡ 3 (mod 4) (Proposition 3). Finally, we give numerical examples of some square-free integers m satisfying 4|H(m) and m ≡ 3 (mod 4).

Weyl algebra is an associative algebra generated by two elements â and a over R such that the generating relation is given by

âa — aâ = 1,

which is isomorphic to the algebra of differential operators

Let R be a commutative Noetherian ring and suppose q is a prime ideal of R. A fundamental problem is to decide when powers qn of q are primary (that is qn is its own primary decomposition). If q is generated by a regular sequence then powers of q are always primary, because G(q, R) (the associated graded ring of R with respect to q) is an integral domain (see [12 page 98] and also [5 (2.1)]). Let qn) denote the nth symbolic power of q-defined by q(n) = {rεR|there exists sεR\q such that sr ε qn}. Then qn is primary if and only if qn = q(n) If q is generated by a regular sequence then we call it a complete intersection prime ideal, so if q is a complete intersection prime ideal then qn ≠ q(n) for all n ≥ 1. If q is not a complete intersection then powers need not be primary. If R is a three-dimensional regular local ring and q is a non-complete intersection height two prime ideal for example, then Huneke showed [11 Corollary (2.5)] that qn = q(n) for all n ≥ 2. Thus, for such a prime q it is impossible for qn to occur in the primary decomposition of any ideal. This phenomena increases the difficulty in finding a primary decomposition for an ideal having q as an associated prime.

The general solution of an algebraic differential equation depends on the initial conditions, though it is in general too difficult to make explicit the shape of the relationship. Painlevé studied in [8] algebraic differentia] equations of second order with the general solutions depending rationally on the initial conditions and the solvability of such equations. Giving the precise definition of the notion “rational dependence on the initial conditions”, Umemura [10] revived and generalized rigorously the discussion of Painlevé in the language of modern algebraic geometry. The theorem of Umemura is as follows; Let K be a differential field extension of complex number field C generated by a finite number of meromorphic functions on some domain in C. Let y be the general solution of a given algebraic differential equation over K. Suppose that y depends rationally on the initial conditions. Then it is contained in the terminal Km of a finite chain of differential field extensions: K = K0 ⊂ K1 ⊂… ⊂Km such that each Ki is strongly normal over Ki−1.

Let X = On/i be an analytic singularity where ṫ is an ideal of the C-algebra On of germs of analytic functions on (Cn, 0). Let denote the maximal ideal of X and A = Aut X its group of automorphisms. An abstract subgroup equipped with the structure of an algebraic group is called algebraic subgroup of A if the natural representations of G on all “higher cotangent spaces” are rational. Let π be the representation of A on the first cotangent space and A1 = π(A).