Hecke stated in [1] Kronecker’s limit formula for CM-fields without proof, in which “Die zu log η(z) analog Funktion” appears, and he investigated in [3] the behaviors of this function under modular substitutions. S. Konno also discussed Kronecker’s limit formula for CM-fields in [4].

We introduce here the notion of (stochastically) differentiable process with respect to a fixed conformal martingale and compute the remainder term of the Taylor expansion of the given process (Definition 1 and Proposition 3). An a-priori estimate in the L2-norm of the above mentioned remainder term is given and consequently a power series representation of smooth conformal martingales is obtained (Theorem 4).

We are interested in whether there is a Cantor set E admitting no exceptionally ramified or normal meromorphic functions with E as the set of essential singularities. As for an exceptionally ramified meromorphic function, we [2] have recently given the following result.

All rings in this paper are assumed to be commutative with identity, and they will generally also be Noetherian.

In several recent papers the asymptotic theory of ideals in Noetherian rings has been introduced and developed. In this new theory the roles played in the standard theory by associated primes, R-sequences, classical grade, and Cohen-Macaulay rings are played by, respectively, asymptotic prime divisors, asymptotic sequences, asymptotic grade, and locally quasi-unmixed Noetherian rings. And up to the present time it has been shown that quite a few results from the standard theory have a valid analogue in the asymptotic theory, and a number of interesting and useful new results concerning the asymptotic prime divisors of an ideal in a Noetherian ring have also been proved. In fact the analogy between the two theories is so good that a very useful (but not completely valid) working guide is: results from the standard theory should have a valid analogue in the asymptotic theory. And, although asymptotic sequences are coarser than R-sequences (for example, they behave nicely when passing to R/z with z a minimal prime ideal in R), the converse of this working guide has also proved useful.

We study differential operators on an affine algebraic variety, especially a hypersurface, in the context of Nakai’s Conjecture. We work over a field k of characteristic zero. Let X be a reduced affine algebraic variety over k and let A be its coordinate ring. Let be the A-module of differential operators of A over k of order ≤ n. Nakai’s Conjecture asserts that if is generated by for every n ≥ 2 then A is regular. In 1973 Mount and Villamayor [6] proved this in the case when X is an irreducible curve. In the general case no progress seems to have been made on the conjecture, except for a result of Brown [2], where the assertion is proved under an additional hypothesis. An interesting result proved by Becker [1] and Rego [8] says that Nakai’s Conjecture implies the Conjecture of Zariski-Lipman, which is still open in the general case and which asserts that if the module of k-derivations of A is A-projective then A is regular.

In the previous papers [6], [7], we show that the set of an algebraic homogeneous space G/P fixed under the action of a maximal torus T can be canonically identified with the coset W1 = W/W1 of Weyl group W. We find a T invariant Zariski open set near each element w ∊ W1 and introduce a very nice local coordinate system such that we can express the maximal torus action explicitly. As a result, we become able to apply the study of J. B. Carrell and D. Lieberman [2], [3] to the space G/P and investigate the numerical properties of its characteristic classes and cycles.

What we call the homological conjectures on commutative Noetherian local rings were first collected and partially settled by C. Peskine and L. Szpiro [PS1]. The subsequent remarkable progress was made by M. Hochster [H1] who conjectured the existence of big Cohen-Macaulay modules and solved it in the affirmative for equicharacteristic local rings. It is, however, still open in general setting.

Soit h(m) le nombre des classes de , m > 0. Dans [1], [3] et [5] des conditions suffisantes pour que h(m) > 1 sont établies, lorsque m = 4q2+ 1, m = q2 + 4 et m + q2 ± 2. Dans le présent travail on dégage, par l’algorithme de Chatelet, des cycles principaux canoniques d’idéaux réduits pour ces cas. On retrouve alors les résultats de [1], [3] et [5] et on démontre des critères de divisibilité de h(m) dont certains se trouvent dans [4].

The problem to determine the Gevrey index of solutions of a given hypoelliptic partial differential equation seems to be not yet well investigated. In this paper, we shall show the Gevrey indices of solutions of the equations of Grushin type, [6], are determined by a rather simple application of a straightforward extension of the results given in [7], [8] and [13]. For simplicity to construct left parametrices in the operator valued sense, we shall consider the equations under the stronger condition than that of [6] (cf. Condition 1 of Section 3). Typical examples of Grushin type are given by which will be discussed in Section 4. We remark that our approach may be compared with the one to a similar problem discussed in [17] by using suitable L2-estimates constructed in [16].

Consider a homogeneous linear differential equation of the second order whose coefficients are rational functions of the independent variable x over the field C of complex numbers. We assume that the coefficient of the first order derivative vanishes:

Local densities of quadratic forms are important invariants in the theory of quadratic forms and they appear in Fourier coefficients of Eisenstein series. But it is not easy to evaluate them. To study their properties, it is desirable to look for relations among them, and it is known that there are many relations [3], but they are not concise. We consider a different kind of relations here and improve a result of Zharkovskaja [8, 9] in the case of Eisenstein series as an application.