We study the automorphisms of the nonsplit Cartan modular curves $X_{\text{ns}}(p)$ of prime level $p$. We prove that if $p\geqslant 29$ all the automorphisms preserve the cusps. Furthermore, if $p\equiv 1~\text{mod}~12$ and $p\neq 13$, the automorphism group is generated by the modular involution given by the normalizer of a nonsplit Cartan subgroup of $\text{GL}_{2}(\mathbb{F}_{p})$. We also prove that for every $p\geqslant 29$ the existence of an exceptional rational automorphism would give rise to an exceptional rational point on the modular curve $X_{\text{ns}}^{+}(p)$ associated to the normalizer of a nonsplit Cartan subgroup of $\text{GL}_{2}(\mathbb{F}_{p})$.

Let be a stochastic process, where T is a finite interval. The ε-entropy is defined as the following quantity:

(1)

Several years ago, Huneke and Leuschke proved a theorem solving a conjecture of Schreyer. It asserts that an excellent Cohen-Macaulay local ring of countable Cohen-Macaulay type which is complete or has uncountable residue field has at most a one-dimensional singular locus. In this paper, it is verified that the assumption of the excellent property can be removed, and the theorem is considered over an arbitrary local ring. The main purpose of this paper is to prove that the existence of a certain prime ideal and a certain totally reflexive module implies the existence of an uncountably infinite number of isomorphism classes of indecomposable totally reflexive modules.

Throughout this note, p denotes a fixed prime number and f denotes a fixed natural number prime to p.

It is easy to see and more or less known that for any natural number n, there exists an elliptic curve over p whose j-invariant is of degree n over Fp and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field. In this note, we consider a more precise problem: for any natural number n, decide whether or not there exists an elliptic curve over p whose j-invariant is of degree n over Fp and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field with conductor f.

Throughout the paper, a scheme means a noetherian scheme. By a curve C over a scheme S of genus g, we mean a proper and smooth S-scheme with irreducible curves of genus g as geometric fibres. In the previous paper [15], the author showed that the field of moduli for a non-hyperelliptic curve over a field coincides with the one for its canonically polarized jacobian variety, and in [16], he gave a partial result on the coincidence of the fields of rationality for a hyperelliptic curve and for its canonically polarized jacobian variety. In the present paper, we will discuss the isomorphy of the isomorphism schemes of two curves over a scheme and of their canonically polarized jacobian schemes, by using Oort-Steenbrink’s result [12].

We present results on the Watanabe–Yoshida conjecture for the Hilbert–Kunz multiplicity of a local ring of positive characteristic. By improving on a “volume estimate” giving a lower bound for Hilbert–Kunz multiplicity, we obtain the conjecture when the ring has either Hilbert–Samuel multiplicity less than or equal to 5 or dimension less than or equal to 6. For nonregular rings with fixed dimension, a new lower bound for the Hilbert–Kunz multiplicity is obtained.

The purpose of this paper is to prove the following three theorems which were announced in [2].

QF-3 algebras R are classified according to their second commutator algebras R′ with respect to the minimal faithful module, which satisfy dom.dim. R′ ≧ 2. The class C(S) of all QF-3 algebras whose second commutator is S, contains besides S only algebras R with dom.dim. R = 1. C(S) contains a unique (up to isomorphism) minimal algebra which can be represented as a subalgebra S0 of S describable in terms of the structure of S, and C(S) consists just of the algebras S0 ⊂ R ⊂ S (up to isomorphism).

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.

In [1] ∼ [6] the following question was treated: Let k be a totally real Galois extension of the rational number field Q, O the maximal order of k and G a finite subgroup of GL(n, O) which is stable under the operation of G(k/Q). Then does G ⊂ GL(n, Z) hold?

Es gibt zwei verschiedene Arten von Summen, die gewöhnlich Gausssche Summen genannt werden. Die eine Summe wird von einer Zahl ω ≠ 0 eines algebraischen Zahlkörpers F bestimmt.

Let us consider the following two problems.

(A) Does either

or

hold for the not identically vanishing solution of the equation

for x ∈ Ω ⊂ Rn (n ≥ 3), where λ is a constant satisfying λ > E0 and V(x) is a 2-body or many-body potential?

The purpose of this note is to study regular actions of simple algebraic groups on projective threefolds as an application of the theory of algebraic threefolds, especially Mori Theory and the theory of Fano threefolds (cf. Mori [11], Iskovskih [7, 8]). The motivation for this study is as follows. In a series of papers, Umemura, in part jointly with Mukai, has classified maximal connected algebraic subgroups of the Cremona group of three variables and also constructed minimal rational threefolds which correspond to such subgroups (cf. Umemura [16-19], Mukai-Ume-mura [12]). In particular, Umemura and Mukai studied in [12] the SL(2, C)-equivariant smooth projectivization of SL(2, C)/G, where G is a binary icosahedral or octahedral subgroup of SL(2, C). The study of equivariant smooth projectivization of SL(2, C)/G for any finite subgroup G has been completed along their lines in Nakano [14]. The main trick of these studies is the investigation of equivariant contraction maps of extremal rays in the context of Mori Theory [11]. In this note, we apply a similar idea to projective threefolds with a regular action of a simple algebraic group and determine which simple algebraic groups can act regularly and nontrivially on projective threefolds and in which fashion. We also need some standard (but difficult) facts from the theory of Fano threefolds. For the precise statement, see Theorem 1 in the main text. For the proof of this theorem, we need a classification of closed subgroups of simple algebraic groups of codimension 1 and 2, which could be derived easily from the classical work of Dynkin [4]. However, we shall give a geometric proof independent of [4] which leads up directly to the proof of Theorem 1. On the whole, we shall establish by geometric methods the scarcity of closed subgroups of small codimension in simple algebraic groups, which is implied in Dynkin [4].

For an integer d > 0 (resp. d < 0) let hd denote the number of Sl2(Z)-equivalence classes of primitive (resp. primitive positive-definite) integral binary quadratic forms of discriminant d. For where t and u are the smallest positive integral solutions of the equation t2 − du2 = 4 if d is a non-square and εd = 1 if d is a square.

Let (G, V) be an irreducible prehomogeneous vector space defined over a number field k, P ∈ k[V] a relative invariant polynomial, and χ a rational character of G such that . For , let Gx be the stabilizer of x, and the connected component of 1 of Gx. We define L0 to be the set of such that does not have a non-trivial rational character. Then we define the zeta function for (G, Y) by the following integral

where Φ is a Schwartz-Bruhat function, s is a complex variable, and dg” is an invariant measure.

Let M be an n-dimensional hypersurface immersed in the (n + 1)-dimensional unit sphere Sn+1 with the standard metric by an immersion f. We denote by A the second fundamental form of the immersion / which is considered as a symmetric linear transformation of each tangent space TXM, i.e. for an arbitrary point x of M and the unit normal vector field ξ defined in a neighborhood of x, A is given by where is the covariant differentiation in Sn+i and Thus, A depends on the orientation of the unit normal vector field ξ and, in general, it is locally defined on M.

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.

This short note gives the generalized sum theorem for Lebesgue dimension of paracompact Hausdorff spaces. Our theorem, though it is a generalization of Mr. Morita’s sum theorem for fully normal spaces [3, Theorem 3. 2] which is essentially based on his generalized sum theorem for normal spaces [3, Theorem 3.1], is obtained by very brief arguments, using only the usual sum theorem for normal spaces.

Let k be an algebraic number field and K be a finite extension of k. Recently, T. Ono defined positive rational numbers E(K/k) and E′(K/k) for K/k. In [7], he investigated some relations between E(K/k) and other cohomological invariants for K/k. He obtained a formula when K is a normal extension of k. In our paper [3], we obtained a similar formula for E′(K/k) in the case of normal extensions K/k. Both proofs essentially use Ono’s results on the Tamagawa number of algebraic tori, on which the formulae themselves do not depend. Hence, in [8], T. Ono posed a problem to give direct proofs of these formulae.

Let Rn+1 = Rn X R denote the (n + l)-dimensional Euclidian space (n ≧ 1). For X ε Rn+1 we write X = (x, t) with x ε Rn and t ε R. For an α with 0 < α < 1, we write

where Δ is the Laplacian on Rn and (– Δ)α is the α-fractional power of - Δ on R*