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In earlier work, we established the a.e. convergence of certain Eisenstein series on arithmetic quotients of loop groups. In this article we prove that these series converge everywhere, and uniformly on certain bounded sets
Let be a primitive positive integral binary quadratic form of discriminant , and let be the number of representations of by up to automorphisms of . In this article, we give estimates and asymptotics for the quantity for all and uniformly in . As a consequence, we get more-precise estimates for the number of integers which can be written as the sum of two powerful numbers
We generalize Griffiths's theorem on the Hodge filtration of the primitive cohomology of a smooth projective hypersurface using the local Bernstein-Sato polynomials, the -filtration of Kashiwara  and Malgrange  along the hypersurface, and the Brieskorn module of the global defining equation of the hypersurface (see )
A. W. Reid [R, Theorem 2.1] showed that if and are arithmetic lattices in or in which give rise to isospectral manifolds, then and are commensurable (after conjugation). We show that for and or for , the situation is quite different; there are arbitrarily large finite families of isospectral noncommensurable compact manifolds covered by . The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but different invariants
Let denote a random symmetric (-matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables (which take values and with probability ). We prove that is nonsingular with probability for any fixed . The proof uses a quadratic version of Littlewood-Offord-type results concerning the concentration functions of random variables and can be extended for more general models of random matrices
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