We investigate the following two problems on a hermitian form Φ over an algebraic number field: (1) classification of Φ over the ring of algebraic integers; (2) hermitian Diophantine equations. The same types of problems for quadratic forms were treated in the author’s previous articles. Here we discuss the hermitian case. Problem (2) concerns an equation ξΦ · ξ = Ψ , where Φ and Ψ represent he...

In these notes we will first define projective modules and prove some standard properties of those modules. Then we will classify finitely generated projective modules over Dedekind domains Remark 0.1. All rings will be commutative with 1. 1. Projective modules Definition 1.1. Let R be a ring and let M be an R-module. Then M is called projective if for all surjections p : N → N ′ and a map f : ...

A generalisedWeber function is wN (z) = η(z/N)/η(z) where η(z) is the Dedekind function and N is any integer (the original function corresponds to N = 2). We give the complete classification of cases where some power w N evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a ...

Given an elliptic fibration $f \colon X \to S$ over the spectrum of a complete discrete valuation ring with algebraically closed residue field, we use Hochschild--Serre spectral sequence to express torsion in $R^1f_\ast \mathscr{O}_X$ as first group cohomology $H^1(G,H^0(S^\prime, \mathscr{O}_{S^\prime}))$. Here, $G$ is Galois maximal extension $K^\prime / K$ such that normalization $X \times_S...

In this paper, I present a new decision procedure for the ideal membership problem for polynomial rings over principal domains using discrete valuation domains. As a particular case, I solve a fundamental algorithmic question in the theory of multivariate polynomials over the integers called “Kronecker’s problem”, that is the problem of finding a decision procedure for the ideal membership prob...

A generalised Weber function is given by wN (z) = η(z/N)/η(z), where η(z) is the Dedekind function and N is any integer; the original function corresponds to N = 2. We classify the cases where some power w N evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formu...

We prove a generalization of the Flat Cover Conjecture by showing for any ring R that (1) each (right R-) module has a Ker Ext(−, C)-cover, for any class of pure-injective modules C, and that (2) each module has a Ker Tor(−,B)-cover, for any class of left R-modules B. For Dedekind domains, we describe Ker Ext(−, C) explicitly for any class of cotorsion modules C; in particular, we prove that (1...

In this note we shall investigate a topological version of the problem of Kurosh: “Is any algebraic algebra locally finite?” Kaplansky’s theorem concerning the local nilpotence of nil PI-algebras is well-known. We will prove a generalization of Kaplansky’s theorem to the class of locally compact rings. We use in the proof a theorem of A. I. Shirshov [8] concerning the height of a finitely gener...

The hypersurface in C3 with an isolated quasi-homogeneous elliptic singularity of type Ẽr, r = 6, 7, 8, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type Er provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we ...

We study the amplitude of mesoscopic Aharonov-Bohm oscillations in quasi-one-dimensional (Q1D) diffusive rings. We consider first the lowtemperature limit of a fully coherent sample. The variance of oscillation harmonics is calculated as a function of the length of the leads attaching the ring to reservoirs. We further analyze the regime of relatively high temperatures, when the dephasing due t...