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We study the stable category of the graded Cohen–Macaulay modules of the factor algebra of the preprojective algebra associated with an element w of the Coxeter group of a quiver. We show that there exists a silting object M(w)| of this category associated with each reduced expression w of w and give a sufficient condition on w such that M(w) is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of M(w)⁠. Moreover, we compare it with a triangle equivalence given by Amiot–Reiten–Todorov for a cluster category. [ABSTRACT FROM AUTHOR]

We prove that the Kloosterman sum S(1, 1; c) changes sign infinitely often as c runs over squarefree numbers with at most 7 prime factors, which improves the previous results of Fouvry and Michel, Sivak-Fischler, Matomäki, and the author. [ABSTRACT FROM AUTHOR]

We prove that the continuous version of the Connes-Shlyakhtenko first L2-cohomology for II1 factors, as proposed by Thom, always vanishes. [ABSTRACT FROM AUTHOR]

Given a polynomial f (z)= zd + c ∈ K[z] over a global field K and α0 ∈ K, we study the density of prime ideals of K dividing at least one element of the orbit of α0 under f . We show that for many choices of d and c this density is zero for all α0, assuming K contains a primitive dth root of unity. The proof relies on several new results, including some giving criteria to ensure the number of irreducible factors of the nth iterate of f remains bounded as n grows, and others on the ramification above certain primes in iterated extensions. Together these allow for nearly complete information when K is a global function field or when K =Q(ξd). [ABSTRACT FROM AUTHOR]