In § 1 of this note we construct abelian varieties of dimension two defined over Fpn, n odd, which admit infinitely many distinct principal polarizations. These polarizations determine an infinite family of geometrically non-isomorphic complete singular curves defined and irreducible over Fpn which have isomorphic Jacobian varieties. In § 2 we calculate the zeta function of these curves.

Let M be an n-dimensional compact Kaehler manifold, TM its (holomorphic) tangent bundle and T*M its cotangent bundle. Given a complex vector bundle E over M, we denote its m-th symmetric tensor power by SmE and the space of holomorphic sections of E by Γ(E).

Throughout this paper, let p be an odd prime. Let k be a -adic number field and be the ring of all integers in k. Let K/k be a finite totally ramified Galois p-extension of degree pn with the Galois group G. Clearly the ring of all integers in K is an [G]-module. In the previous paper [4], we studied [G]-module structure of in a cyclic totally ramified p-extension, and we have obtained the condition for to be an indecomposable [G]-module.

Let k be an algebraic number field of finite degree. For a finite extension L/k we denote by L/k the different of L/k, and by L the integer ring of L. Let K1 and K2 be finite extensions of k.

The first order language ℒ that we consider has two nullary function symbols 0, 1, a unary function symbol –, a binary function symbol +, a unary relation symbol 0 <, and the binary relation symbol = (equality). Let ℒ′ be the language obtained from ℒ, by adding, for each integer n > 0, the unary relation symbol n| (read “n divides”).

In [M] Murthy asked if every subvariety V of kn (k-any field) with a trivial conormal bundle is a (scheme theoretic) complete intersection.

In [B] we were able to prove that all such subvarieties of kn are at least set theoretic complete intersections. Mohan Kumar has shown in [MK] that if one assumes moreover that n ≥ 2 dim V + 2 then V is a complete intersection. The aim of this paper is to prove the following result which extends the Mohan Kumar Theorem in the nonsingular (and connected) case and moreover sheds some light on his bound for n.

Let X be a projective non-singular variety and H an ample line bundle on X. The moduli space of H-stable vector bundles exists by Maruyama [4]. If X is a curve defined over C, the structure of the moduli space (or its compactification) M(X, d, r) of stable vector bundles of degree d and rank r on X is studied in detail. It is known that the variety M(X, d, r) is irreducible. Let L be a line bundle of degree d and let M(X, L, r) denote the closed subvariety of M(X, d, r) consisting of all the stable bundles E with det E = L.

In this paper we give a new characterization of weak normalization and we use it to discuss some questions about weakly normal varieties formulated by A. Andreotti and E. Bombieri in [1] and studied by S. Greco and C. Traverso in [8] for seminormality.

Let (φ1(u), …, φn(u)) be a system of holomorphic functions whose Wronskian does not vanish at origin, where holomorphic functions mean functions holomorphic around origin.

Let G be a finite subgroup of GL(n, C) (C is the field of complex numbers). Then G acts naturally on the polynomial ring S = C[X1, …, Xn]. We consider the following

Problem. When is the invariant subring SG a complete intersection?

In this paper, we treat the case where G is a finite Abelian group. We can solve the problem completely. The result is stated in Theorem 2.1.

In my paper [F3] I more or less explicitly conjectured that if A is a complete local integral domain with maximal ideal and if I = (t1, …, tn) is an ideal in A with n ≤ dim (A) – 2, then Spec (A/I) – is G3 in Spec (A) – . This will be proved in this paper.

A locally strongly convex hypersurface in the affine space Rn + 1 is called an affine hypersphere if the affine normals (§ 1) through each point of the hypersurface either all intersect at one point, called its center, or else are all mutually parallel. It is called elliptic, parabolic or hyperbolic according to whether the center is, respectively, on the concave side of the hypersurface, at infinity or on the convex side. This class of hypersurfaces was first studied systematically by W. Blaschke ([1]) in the frame of affine geometry. In his paper [3] E. Calabi redefined it and proposed a problem of determining all complete hyperbolic affine hyperspheres and raised a conjecture that these hypersurfaces are asymptotic to the boundary of a convex cone and every non-degenerate cone V determines a hyperbolic affine hypersphere, asymptotic to the boundary of V, uniquely by the value of its mean curvature.

Let X be a projective Gorenstein variety, Y ⊂ X a proper closed subscheme such that X is smooth at all points of Y, so that the formal completion of X along Y is regular.

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?

During 1934-1936, W. L. Ferrar investigated the relation between summation formulae and Dirichlet series with functional equations, inspired by Voronoi’s works (1904) on summation formula with respect to the numbers of divisors. In [11] A. Weil showed that the automorphic properties of theta series are expressed by generalized Poisson summation formulae with respect to the so-called Weil representation. On the other hand, T. Kubota, in his study of the reciprocity law in a number field, defined a generalized theta series and a generalized Weil type representation of SL(2, C) and obtained similar results to those of A. Weil (1970-1976, [5], [6], [7]). The methods, used by W. L. Ferrar and T. Kubota, to obtain (generalized Poisson) summation formulae depend similarly on functional equations of Dirichlet series (attached to the generalized theta series).

In the theory of integral quadratic forms or the related theory such as modular forms, it is important to give examples of forms with given discriminant for various computational purposes. However, little is known about this problem even in the case where the explicit formula for the class number has been given. Among the works that deal with satisfactorily many examples or give a constructive method of them, there are [1] for ternary forms, and [5] for quaternary forms with square discriminants. As for quaternary forms with prime discriminant, P. Ponomarev [8] gave an explicit way to construct such forms from ternary forms with discriminant 2q. It was based on the result of his previous work [7] and that of Y. Kitaoka [6], but was rather complicated.

In this paper, we study the composition series of certain principal series representations of the three-fold metaplectic covering group of SL(2, K), where K is a non-archimedean local field. These representations are parametrized by unramified characters μ(x) = |x|s of K× and characters ω of the group of third roots of unity.