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Let (X, O) be a germ of a normal surface singularity, π: → X be the minimal resolution of singularities and let A = (ai,j) be the n × n symmetrical intersection matrix of the exceptional set of In an old preprint Nash proves that the set of arcs on a surface singularity is a scheme , and defines a map from the set of irreducible components of to the set of exceptional components of the minimal resolution of singularities of (X,O). He proved that this map is injective and ask if it is surjective. In this paper we consider the canonical decomposition •For any couple (Ei,Ej) of distinct exceptional components, we define Numerical Nash condition (NN(i,j)). We have that (NN(i,j)) implies In this paper we prove that (NN(i,j)) is always true for at least the half of couples (i,j).•The condition (NN(i,j)) is true for all couples (i,j) with i ≠ j, characterizes a certain class of negative definite matrices, that we call Nash matrices. If A is a Nash matrix then the Nash map N is bijective. In particular our results depend only on A and not on the topological type of the exceptional set.•We recover and improve considerably almost all results known on this topic and our proofs are new and elementary.•We give infinitely many other classes of singularities where Nash Conjecture is true.The proofs are based on my old work [8] and in Plenat [10].

An algebraic group of type E7 over an algebraically closed field has an irreducible representation in a vector space of dimension 56 and is, in fact, the identity component of the automorphism group of a quartic form on the space. This paper describes the construction of the quartic form if the characteristic is ≠ 2, 3, taking into account a field of definition F. Certain F-forms of E7 appear in the automorphism groups of quartic forms over F, as well as forms of E6. Many of the results of the paper are known, but are perhaps not easily accessible in the literature.

The test ideal $\tau(R)$ of a ring $R$ of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal $\tau(\a^t)$ associated to a given ideal $\a$ with rational exponent $t \ge 0$. We first prove a key lemma of this paper, which gives a characterization of the ideal $\tau(\a^t)$. As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal $\tau(R)$. Moreover, we prove an analog of so-called Skoda's theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the "modified Brian\c{c}on--Skoda theorem.", Comment: 11 pages, AMS-LaTeX; v.2: minor changes, to appear in Nagoya Math. J

Let (G, ρ, X) be a prehomogeneous vector space with singular set S over an algebraic number field F. The main result of this paper is a proof for the convergence of the zeta fucntions Z(Φ, s) associated with (G, ρ, X) for large Re s under the assumption that S is a hypersurface. This condition is satisfied if G is reductive and (G, ρ, X) is regular. When the connected component of the stabilizer of a generic point x is semisimple and the group Πx of connected components of Gx is abelian, a clear estimate of the domain of convergence is given.Moreover when S is a hypersurface and the Hasse principle holds for G, it is shown that the zeta fucntions are sums of (usually infinite) Euler products, the local components of which are orbital local zeta functions. This result has been proved in a previous paper by the author under the more restrictive condition that (G, ρ, X) is irreducible, regular, and reduced, and the zeta function is absolutely convergent.

This paper is concerned with a finitely generated module M over a (commutative Noetherian) local ring R. In the case when R is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of M with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when R is universally catenary and such that all its formal fibres are Cohen–Macaulay.These formulae involve certain subsets of the spectrum of R called the pseudosupports of M; these pseudo-supports are closed in the Zariski topology when R is universally catenary and has the property that all its formal fibres are Cohen–Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.

In this paper we propose a new category Qcl of complex numbers which contains π, e and the set of algebraic numbers. In fact this category contains most of the numbers studied so far in number theory. An element of the category is here called a classical number. The category of the classical numbers forms an algebraically closed field and consists of countably many numbers. The definition depends on algebraic differential equations related with algebraic groups. Throughout the paper unless otherwise stated, we deal with functions of one variable and a differential equation is an ordinary differential equation. We are inspired of the Leçons de Stockholm of Painlevé [P].

Although class number one problem for imaginary quadratic fields was solved in 1966 by A. Baker [3] and by H. M. Stark [25] independently, the problem for real quadratic fields remains still unsettled. However, since papers by Ankeny–Chowla–Hasse [2] and H. Hasse [9], many papers concerning this problem or giving estimate for class numbers of real quadratic fields from below have appeared. There are three methods used there, namely the first is related with quadratic diophantine equations ([2], [9], [27, 28, 29, 31], [17]), and the second is related with continued fraction expantions ([8], [4], [16], [14], [18]).

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field. Let $(\unicode[STIX]{x1D70B}_{n})_{n\geqslant 0}$ be a system of $p$-power roots of a uniformizer $\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}_{0}$ of $K$ with $\unicode[STIX]{x1D70B}_{n+1}^{p}=\unicode[STIX]{x1D70B}_{n}$, and define $G_{s}$ (resp. $G_{\infty }$) the absolute Galois group of $K(\unicode[STIX]{x1D70B}_{s})$ (resp. $K_{\infty }:=\bigcup _{n\geqslant 0}K(\unicode[STIX]{x1D70B}_{n})$). In this paper, we study $G_{s}$-equivariantness properties of $G_{\infty }$-equivariant homomorphisms between torsion crystalline representations.

Letkbe a-adic number field with the ring 0 of all integers ink, andKbe a finite normal extension with Galois groupG.∏denotes a prime element of the ringof all integers inK. Then, an ideal (∏a) ofis an 0G-module. E. Noether showed that ifK/kis tamely ramified,is a free 0G-module. A. Fröhlich generalized E. Noether’s theorem as follows:is relatively projective with respect to a subgroupSofGif and only ifS⊇G1whereG1is the first ramification group ofK/k. Now we define the vertexV(∏a) of (∏a) as the minimal normal subgroupSofGsuch that (∏a) is relatively projective with respect to a subgroupSofG(cf. § 1). Then, the above generalization by A. Fröhlich impliesV() =G1. In the previous paper, we provedG1⊇ V(∏a) ⊇ G2, (whereG2is the second ramification group ofK/k(cf. Theorem 5). Further, we dealt with the case whereG=G1is of orderp2, and proved that ifV(∏a) ≠G1thena≡ 1(p2) andt2≡ 1(p2) for the second ramification numbert2ofK/k(cf. Theorems 15 and 21). The purpose of this paper is to prove the similar theorem for the wildly ramified p-extension of degreepn(Theorem 7).

Let A = R/, where R is a regular local ring of arbitrary dimension and is an ideal of R. If A is a Gorenstein ring and if height = 2, it is easily proved that A is a complete intersection, i.e., is generated by two elements (Serre [5], Proposition 3). Hence Gorenstein rings which are not complete intersections are of embedding codimension at least three. An example of these rings is found in Bass’ paper [1] (p. 29). This is obtained as a quotient of a three dimensional regular local ring by an ideal which is generated by five elements, i.e., generated by a regular sequence plus two more elements. In this paper, suggested by this example, we prove that if A is a Gorenstein ring and if height = 3, then is minimally generated by an odd number of elements. If A has a greater codimension, presumably there is no such restriction on the minimal number of generators for , as will be conceived from the proof.

Let (R,m) be a local ring, and let M and N be finite R-modules. In this paper we give a formula for the degree of the polynomial giving the lengths of the modules ExtiR(M,N/mnN). A number of corollaries are given, and more general filtrations are also considered.

In the first half of this paper, all the limits of irreducible characters of Gn = 𝔖n(T) as n → ∞ are calculated. The set of all continuous limit functions on G = 𝔖 ∞(T) is exactly equal to the set of all characters of G determined in [HH6]. We give a necessary and sufficient condition for a series of irreducible characters of Gn to have a continuous limit and also such a condition to have a discontinuous limit. In the second half, we study the limits of characters of certain induced representations of Gn which are usually reducible. The limits turn out to be characters of G, and we analyse which of irreducible components are responsible to these limits.

In this paper, the characterization of domains in ℂn by their noncompact automorphism groups are given.

We prove an analog of the classical Hartogs extension theorem for CR L2 functions defined on boundaries of certain (possibly unbounded) domains on coverings of strongly pseudoconvex manifolds. Our result is related to a question formulated in the paper of Gromov, Henkin and Shubin [GHS] on holomorphic L2 functions on coverings of pseudoconvex manifolds.

In math.RT/0205144 we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters. The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character as sheaves on the partial flag variety corresponding to the singularity of the character. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators, but is actually larger. We discuss translation functors and intertwining functors. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of D-modules and coherent sheaves., some minor corrections and expansions

In this paper, we give explicit combinatorial descriptions for toric extremal contractions under the relative setting, where varieties do not need to be complete. Fujino's completion theorem is the key to the main result. As applications, we can generalize some of Mustata's results related to Fujita's conjecture on toric varieties for the relative case., 8 pages

We present in this paper sufficient conditions for the topological triviality of families of germs of functions defined on an analytic variety V. The main result is an infinitesimal criterion based on a convenient weighted inequality, similar to that introduced by T. Fukui and L. Paunescu in [8]. When V is a weighted homogeneous variety, we obtain as a corollary, the topological triviality of deformations by terms of non negative weights of a weighted homogeneous germ consistent with V. Application of the results to deformations of Newton non-degenerate germs with respect to a given variety is also given.

In this paper we first prove some criteria for the propagation of algebraic dependence of dominant meromorphic mappings from an analytic finite covering space X over the complex m-space into a projective algebraic manifold. We study this problem under a condition on the existence of meromorphic mappings separating the generic fibers of X. We next give applications of these criteria to the uniqueness problem of meromorphic mappings. We deduce unicity theorems for meromorphic mappings and also give some other applications. In particular, we study holomorphic mappings into a smooth elliptic curve E and give conditions under which two holomorphic mappings from X into E are algebraically related.

This paper is concerned with mod p Morita-Mumford classes of the mapping class group Γg of a closed oriented surface of genus g ≥ 2, especially triviality and nontriviality of them. It is proved that is nilpotent if n ≡ − 1 (mod p − 1), while the stable mod p Morita-Mumford class is proved to be nontrivial and not nilpotent if n ≢ −1 (mod p − 1). With these results in mind, we conjecture that vanishes whenever n ≡ − 1 (mod p − 1), and obtain a few pieces of supporting evidence.

Let K/k be a Galois extension of a number field of degree n and p a prime number which does not divide n. The study of the p-rank of the ideal class group of K by using those of intermediate fields of K/k has been made by Iwasawa, Masley et al., attaining the results obtained under respective constraining assumptions. In the present paper we shall show that we can remove these assumptions, and give more general results under a unified viewpoint. Finally, we shall add a remark on the class numbers of cyclic extensions of prime degree of Q.

There is in the literature a lot of determinant formulae involving the relative class number of an imaginary abelian field. Usually such a formula contains a factor which is equal to zero for many fields and so it gives no information about the class number of these fields. The aim of this paper is to show a way of obtaining most of these formulae in a unique fashion, namely by means of the Stickelberger ideal. Moreover some new and non-vanishing formulae are derived by a modification of Ramachandra’s construction of independent cyclotomic units.

It is completely known the characterization of all Hilbert functions and all graded Betti numbers for 3-codimensional arithmetically Gorenstein subschemes of ℙr (works of Stanley [St] and Diesel [Di]). In this paper we want to study how geometrical information on the hypersurfaces of minimal degree containing such schemes affect both their Hilbert functions and graded Betti numbers. We concentrate mainly on the case of general hypersurfaces and of irreducible hypersurfaces, for which we find strong restrictions for the Hilbert functions and graded Betti numbers of their subschemes.

In this paper we prove Pesin’s entropy formula for general C2 (or C1+α) (non-invertible) endomorphisms of a compact manifold preserving a smooth measure.

The author continues his previous work with the purpose of establishing a theory of newforms in the case of half-integral weight. In this paper, the author formulates and proves a complete theory of newforms for Kohnen space. Kohnen space is an important canonical subspace in the space of cusp forms of half-integral weight k + 1/2 (k > 0). Every Hecke common eigenform in Kohnen space corresponds to a primitive form of integral weight 2k and of odd level via Shimura Correspondence.These newforms for Kohnen space satisfy the Strong multiplicity One theorem. Moreover, we explicitly determine the corresponding primitive form (of weight 2k) to each newform for Kohnen space. The space of oldforms is also explicitly described.In order to find all newforms for Kohnen space, the author needs a certain non-vanishing property of Fourier coefficients of cusp forms. Such property proves by using representation theory of finite linear groups. The method of proof of newform theory is mainly based on trace formulae and trace relations.

In this paper, we consider a kind of multilinear operators related to oscillatory singular integrals with rough kernels and give a criterion of certain boundedness for this kind of operators.

In the past few years there has been an increasing interest in a certain class of stochastic differential equations (SDE’s) in Hilbert spaces for applications in quantum mechanics (measurements continuous in time [1-5]) and in quantum optics (photon-detection theory and numerical simulations of master equations [6-10]). Part of the mathematical theory of these equations has been developed in [11], where also “structural properties” of this class of equations have been studied. In the paper [11] it has been shown that such equations are connected with certain semigroups of linear operators and the form of the generator of semigroups related to such SDE’s has been established.

Affine Kac-Moody algebras represent a well-trodden and well-understood littoral beyond which stretches the vast, chaotic, and poorly-understood ocean of indefinite Kac-Moody algebras. The simplest indefinite Kac-Moody algebras are the rank 2 Kac-Moody algebras (a) (a ≥ 3) with symmetric Cartan matrix , which form part of the class known as hyperbolic Kac-Moody algebras. In this paper, we probe deeply into the structure of those algebras (a), the e. coli of indefinite Kac-Moody algebras. Using Berman-Moody’s formula ([BM]), we derive a purely combinatorial closed form formula for the root multiplicities of the algebra (a), and illustrate some of the rich relationships that exist among root multiplicities, both within a single algebra and between different algebras in the class. We also give an explicit description of the root system of the algebra (a). As a by-product, we obtain a simple algorithm to find the integral points on certain hyperbolas.

Let p be an odd prime and d be a positive integer prime to p such that d ≢ 2 mod 4. For technical reasons, we also assume that . For each integer n ≥ 1, we choose a primitive nth root ζn of 1 so that whenever n | m. Let be its cyclotomic Zp-extension, where is the nth layer of this extension. For n ≤ 1, we denote the Galois group Ga\(Kn/K0) by Gn, the unit group of the ring of integers of Kn by En, and the group of cyclotomic units of Kn by Cn. For the definition and basic properties of cyclotomic units such as the index theorem, we refer [6] and [7]. In this paper we examine the injectivity of the homomorphism between the first cohomology groups induced by the inclusion Cn → En.

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.

Our setting for this paper is projective 3-spaceover an algebraically closed fieldK. By a curveC⊂is meant a 1-dimensional, equidimensional projective algebraic set, which is locally Cohen-Macaulay. Letbe the Hartshorne-Rao module of finite length (cf. [R]). HereZis the set of integers andℐcthe ideal sheaf ofC. In [GMV] it is shown that, whereis the homogeneous ideal ofC,is the first local cohomology module of theR-moduleMwith respect to. Thus there exists a smallest nonnegative integerk∊Nsuch that, (see also the discussion on the 1-st local cohomology module in [GW]). Also in [GMV] it is shown thatk= 0 if and only ifCis arithmetically Cohen-Macaulay andCis arithmetically Buchsbaum if and only ifk≤ 1. We therefore have the following natural definition.

This paper deals with a decomposition problem for some classes of distributions. Let D be a given class of distribution on R1, which we are interested in. After showing that the class D is closed under convolution, our purpose is to give an answer to the inverse problem: if the convolution of two distributions μ1 and (μ2 belongs to D, then do μ1 and μ2 belong to D?

In our recent papers [3, 4, 5], we defined some new D-invariants for any square-free positive integer D and considered their properties and interrelations among them. Especially, as an application of it, we discussed in [5] the characterization of real quadratic field Q() of so-called Richaud-Degert type in terms of these new D-invariants.

Affine differential geometry developed by Blaschke and his school [B] has been reorganized in the last several years as geometry of affine immersions. An immersion f of an n-dimensional manifold M with an affine connection ∇ into an (n + 1)-dimensional manifold Ḿwith an affine connection ∇ is called an affine immersion if there is a transversal vector field ξ such that ∇xf*(Y) = f*(∇xY) + h(X,Y)ξ holds for any vector fields X, Y on Mn. When f: Mn→ Rn+1 is a nondegenerate hypersurface, there is a uniquely determined transversal vector field ξ, called the affine normal field, an essential starting point in classical affine differential geometry. The new point of view allows us to relax the non-degeneracy condition and gives us more freedom in choosing ξ; what this new viewpoint can accomplish in relating affine differential geometry to Riemannian geometry and projective differential geometry can be seen from [NP1], [NP2], [NS] and others. For the definitions and basic formulas on affine immersions, centroaffine immersions, conormal (or dual) maps, projective flatness, etc., the reader is referred to [NP1]. These notions will be generalized to codimension 2 in this paper.

In a recent paper [HI2] the first named author and S. Ikeda have investigated Gorenstein properties under blowing up of height two equimultiple ideals from the arithmetical point of view. The problem is to what extent does the Gorenstein property of the Rees algebra R(a) = of an equimultiple ideal a of a local ring (A, m) describe the structure of A and a. One result is that if A is a generalized Cohen-Mac aulay ring (i.e. a ring of finite local cohomology) with dim (A) ⊇ 4 and if there is an height two equimultiple prime ideal p such that R(p) is Gorenstein, then A is Gorenstein and p is generated by a regular sequence (see [HI2], Theorem 2.6). That has led to the question whether this result is still valid for dim (∧) = 3. ∧ partial answer was given in [HI2], Proposition 2.10, where “equimultiplicity” was replaced by the stronger condition “p/p2 is flat over A/p”

In this paper we study the ideal boundaries of surfaces admitting total curvature as a continuation of [Sy2] and [Sy3]. The ideal boundary of an Hadamard manifold is defined to be the equivalence classes of rays. This equivalence relation is the asymptotic relation of rays, defined by Busemann [Bu]. The asymptotic relation is not symmetric in general. However in Hadamard manifolds this becomes symmetric. Here it is essential that the manifolds are focal point free.

This letter concerns our papers [4], [5] and its aim is to give a simplification to the proof of the General Néron Desingularization (see [5] (2.4) or here below) together with a small reparation; as T. Ogoma pointed out in [3], our Lemma (9.5) from [4] does not hold in the condition iii2) (this is true because the “changing” from line 5 from down the page 123 [4] may not preserve iii2)). However our results were not affected in characteristic zero (they use just iii1) from [4] (9.5)). In [3] Ogoma gives a nice simplification of our proof.

Hartman proved that a diffeomorphism is topologically conjugate to a linear map on a neighbourhood of a hyperbolic fixed point ([3]). In this paper we study the topological conjugacy problem of a diffeomorphism on a neighbourhood of a hyperbolic set, and prove that for any hyperbolic set there is an arbitrarily slight extension to which a sub-shift of finite type is semi-conjugate.

In his paper [11] Shimura defined a correspondence between modular forms of half integral weight and modular forms of integral weight. To each pair (t, f(z)), consisting of a square-free integert≥ 1 and a cusp formof weightk/2 (kodd, ≥ 3), levelN(divisible by 4) and characterϰ, he associated a certain functionf(t)(z) (Ft(z) in Shimura’s notation).

Let X = (X(t); t ∈ R) be a real stationary mean continuous Gaussian process with expectation zero which is purely nondeterministic. In this paper we shall investigate the structure of splitting fields of X having finite multiple Markovian property using the results in [6]. We follow the notations and terminologies in [6].

The main result of the present paper is the theorem 1, which describes the topological support of an arbitrary Gaussian measure in a separable Banach space. This theorem will be proved after some discussion of the notion of support itself. But we begin with the reminder of the notion of covariance operator of a probability measure. This notion has a great importance not only for the description of support of Gaussian measures but also for the study of other problems in the theory of probability measures in linear spaces (c.f. [1], [2]).

Let H1, H2, …, HN+2 be hyperplanes in PN(C) located in general position and v1v2, … νN+2 divisors on Cn. We consider the set ℱ(Hi, νi) of all non-degenerate meromorphic maps of Cn into PN(C) such that the pull-backs ν(f, Hi) of the divisors (Hi) on PN(C) by f are equal to νi for any i = 1, 2, …, N + 2. In the previous paper [6], the author showed that =:= ℱ(Hi, νi) cannot contain more than N+ 1 algebraically independent maps. Relating to this, the following theorem will be proved.

Letkbe a finite algebraic number field and letℓbe a fixed odd prime number. In this paper, we shall prove the equivalence of certain rather strong conditions on the following four things (1) ~ (4), respectively :(1) the class number of the cyclotomicZℓ-extension ofk,(2) the Galois group of the maximal abelianℓ-extension ofkwith given ramification,(3) the number of independent cyclic extensions ofkof degreeℓ, which can be extended to finite cyclic extensions ofkof anyℓ-power degree, and(4) a certain subgroupBk(m, S) (cf. § 2) ofk×/k×)ℓmfor any natural numberm(see the main theorem in §3).

Let f be a meromorphic mapping of the n-dimensional complex plane Cn into the N-dimensional complex projective space PN(C). We denote by T(r,f) the characteristic function of f and by N(r,f*H) the counting function for a hyperplane H ⊂ PN(C). The purpose of this paper is to establish the following results.

It is interesting to know what kinds of primes are the factors of the class number of an algebraic number field, and especially to find ones being prime to the degree. About this matter it is desirable to construct the unramified Abelian extensions plainly. In this paper we shall show some of them for the prime cyclotomic number field and its quadratic extensions using the units of subfields.

Letpbe a prime. If one adjoins toQallpn-th roots of unity forn= 1,2,3, …, then the resulting field will contain a unique subfieldQ∞such thatQ∞is a Galois extension ofQwith Gal (Q∞/Q)Zp, the additive group ofp-adic integers. We will denote Gal (Q∞/Q) byΓ. In a previous paper [6], we discussed a conjecture relatingp-adicL-functions to certain arithmetically defined representation spaces forΓ. Now by using some results of Iwasawa, one can reformulate that conjecture in terms of certain other representation spaces forΓ. This new conjecture, which we believe may be more susceptible to generalization, will be stated below.

This work is inspired by the recent paper of Mytilinaios and Slaman [9] on the infinite injury priority method. It may be considered to fall within the general program of the study of reverse recursion theory: What axioms of Peano arithmetic are required or sufficient to prove theorems in recursion theory? Previous contributions to this program, especially with respect to the finite and infinite injury priority methods, can be found in the works of Groszek and Mytilinaios [4], Groszek and Slaman [5], Mytilinaios [8], Slaman and Woodin [10]. Results of [4] and [9], for example, together pinpoint the existence of an incomplete, nonlow r.e. degree to be provable only within some fragment of Peano arithmetic at least as strong as P- + IΣ2. Indeed an abstract principle on infinite strategies, such as that on the construction of an incomplete high r.e. degree, was introduced in [4] and shown to be equivalent to Σ2 induction over the base theory P- + IΣ0 of Peano arithmetic.

Let f(X, T1,…, Tm) be a polynomial over an algebraic number field k of finite degree. In his paper [2], T. Kojima provedTHEOREM. 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 thatF(g(T1,…,Tm), T1,…,T)= 0.

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.

We defined a prime decomposition symbol [d1 d2, p] in a previous paper [3], and characterized in [4] the set of rational primes p which are decomposed completely in a non-abelian central extension which is of degree 8 in substance. The explicit value of the symbol was determined by using a solution of certain ternary quadratic diophantine equation. The solution corresponds to a square root of an ideal class of the principal genus of a quadratic field. This is translated to a problem in classes of integral quadratic forms, namely to find a form whose duplication is a given one contained in a principal genus. An explicit method to find the form is given by Gauss in [5, Art. 286], which is due to his ternary form reduction.

In this paper, we shall give explicitZ-basis of certain maximal orders of definite quaternion algebras over the rational number fieldQ(See Theorems below). We shall also give some remarks on symmetric maximal orders in Ponomarev [9] and Hashimoto [6] (Proposition 4.3). More precise contents are as follows. LetDbe a division quaternion algebra overQ.