International Journal of Mathematics

We explicitly classify the nonharmonic biharmonic submanifolds of the unit three-dimensional sphere .

We give a construction of stationary discs and the indicatrix for manifolds of higher codimension which is a partial analog of L. Lempert's theory of stationary discs for strictly convex hypersurfaces. This leads to new invariants of the CR structure in higher codimension linked with the contact structure of the conormal bundle.

We give a lower bound estimate of the sum of the square norm of the sections of the pluricanonical bundles over a Riemann surface of genus greater than 2 and Gauss curvature -1. Such an estimate must depend on the injective radius of the Riemann surface. However, using this estimate, we give a uniform estimate of the corona problem on Riemann surface. Here "uniform" means that the estimate depends only on the genus of Rieman surface, not on the injective radius.

Let M_H(c₁, c₂) be a coarse moduli scheme parameterizing all rank-two H-μ-stable vector bundles with Chern classes (c₁, c₂) on a smooth projective surface X over an algebraically closed field. For fixed two ample line bundles H and H′, it is known that if c₂ is greater than some constant p(X, H, H′) depending on H and H′ then M_H(c₁, c₂) and M_H′(c₁, c₂) are birationally equivalent. In this paper we show that this constant p(X, H, H′) generally does depends on the choice of H and H′. More precisely, we give some example of surface (and c₁) on which, for any number K, there exists an integer c₂ with c₂≥K such that sup_H:ampledimM_H(c₁, c₂) = + ∞. This result is available also for normal surfaces.

This paper investigates some aspects of the arithmetic of a quintic threefold in Pr⁴ with double points singularities. Particular emphasis is given to the study of the L-function of the Galois action ρ on the middle ℓ-adic cohomology. The main result of the paper is the proof of the existence of a Hilbert modular form of weight (2, 4) and conductor 30, on the real quadratic field , whose associated (continuous system of) Galois representation(s) appears to be the most likely candidate to induce the scalar extension . The Hilbert modular form is interpreted as a common eigenvector of the Brandt matrices which describe the action of the Hecke operators on a space of theta series associated to the norm form of a quaternion algebra over and a related Eichler order.

We prove non-commutative reciprocity formulae for certain polynomials using Fox's free differential calculus. The abelianizations of these reciprocity formulae rediscover the polynomial reciprocity formulae of Carlitz and Berndt–Dieter. Further, many other reciprocity formulae related to Dedekind sums are rederived from our polynomial reciprocity formulae; these include, for instance, generalizations of the classical Eisenstein reciprocity formula.