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A Course in Arithmetic
TLDR
In this article, the theorem on arithmetic progressions modular forms is proved for finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratics forms with discriminant +-1.Abstract:
Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on arithmetic progressions modular forms.read more
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Failure of the Hasse principle on general surfaces
TL;DR: In this paper, it was shown that the Brauer group of an algebraic surface can obstruct the Hasse principle and constructed a general general Brauer-Brauer group for algebraic surfaces.
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The maximum number of singular points on rational homology projective planes
TL;DR: In this paper, it was shown that a rational homology projective complex surface with quotient singularities has at most 4 singular points, which is the best known result for the case of the Enriques surfaces.
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Finite groups of automorphisms of locally soluble groups
TL;DR: In this article, the authors use the term rank in the sense of the Mal'cev special or Priifer rank, and show that a group X has finite rank n if every finitely generated subgroup of X can be generated by n elements, while some finitely generate subgroup X requires n generators.
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Right-angled Coxeter polytopes, hyperbolic six-manifolds, and a problem of Siegel
TL;DR: In this article, the first known examples of orientable hyperbolic six-manifolds having the smallest possible volume were constructed by gluing together the sides of eight copies of an all-right-aligned polytope.
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Simple cellular automata as pseudorandom m-sequence generators for built-in self-test
TL;DR: An extremely simple and explicit construction of cellular automata (CA) generating pseudorandom m-sequences, which consist of only one type of cells, which has advantages over the previous researches in the following points.