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Factorization of matrices associated with classes of arithmetical functions

Shaofang Hong (2003)

Colloquium Mathematicae

Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix ( f ( x i , x i ) ) having f evaluated at the greatest common divisor ( x i , x i ) of x i and x i as its...

Factors of a perfect square

Tsz Ho Chan (2014)

Acta Arithmetica

We consider a conjecture of Erdős and Rosenfeld and a conjecture of Ruzsa when the number is a perfect square. In particular, we show that every perfect square n can have at most five divisors between √ n - ∜ n ( l o g n ) 1 / 7 and √ n + ∜ n ( l o g n ) 1 / 7 .

Failure of the Hasse principle for Châtelet surfaces in characteristic 2

Bianca Viray (2012)

Journal de Théorie des Nombres de Bordeaux

Given any global field k of characteristic 2 , we construct a Châtelet surface over k that fails to satisfy the Hasse principle. This failure is due to a Brauer-Manin obstruction. This construction extends a result of Poonen to characteristic 2 , thereby showing that the étale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteristic.

Faithfully quadratic rings - a summary of results

M. Dickmann, F. Miraglia (2016)

Banach Center Publications

This is a summary of some of the main results in the monograph Faithfully Ordered Rings (Mem. Amer. Math. Soc. 2015), presented by the first author at the ALANT conference, Będlewo, Poland, June 8-13, 2014. The notions involved and the results are stated in detail, the techniques employed briefly outlined, but proofs are omitted. We focus on those aspects of the cited monograph concerning (diagonal) quadratic forms over preordered rings.

Families of hypersurfaces of large degree

Christophe Mourougane (2012)

Journal of the European Mathematical Society

Grauert and Manin showed that a non-isotrivial family of compact complex hyperbolic curves has finitely many sections. We consider a generic moving enough family of high enough degree hypersurfaces in a complex projective space. We show the existence of a strict closed subset of its total space that contains the image of all its sections.

Families of modular forms

Kevin Buzzard (2001)

Journal de théorie des nombres de Bordeaux

We give a down-to-earth introduction to the theory of families of modular forms, and discuss elementary proofs of results suggesting that modular forms come in families.

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