Factorization of natural numbers in algebraic number fields
Factorization problems in class number two
Factorization problems in semigroups.
Factorizations of distinct lengths in algebraic number fields
Factors of a perfect square
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 and .
Factors of alternating sums of products of binomial and q-binomial coefficients
Factors of generalized Rudin-Shapiro sequences. (Facteurs des suites de Rudin-Shapiro généralisées.)
Factors of sums of powers of binomial coefficients
Failure of the Hasse principle for Châtelet surfaces in characteristic
Given any global field of characteristic , we construct a Châtelet surface over 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 , thereby showing that the étale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteristic.
Faisceaux pervers, homomorphisme de changement de base et lemme fondamental de Jacquet et Ye
Faithfully quadratic rings - a summary of results
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.
Fake CM and the stable model of .
Falling factorials, generating functions, and conjoint ranking tables.
Families of curves having each an integer point
Families of elliptic curves of high rank with nontrivial torsion group over ℚ
Families of hypersurfaces of large degree
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
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.
Families of nets of low and medium strength.
Families of Pisot numbers with negative trace
Families of rational numbers with predictable Engel product expansions.