134,492 research outputs found

    Averages and moments associated to class numbers of imaginary quadratic fields

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    For any odd prime β„“\ell, let hβ„“(βˆ’d)h_\ell(-d) denote the β„“\ell-part of the class number of the imaginary quadratic field Q(βˆ’d)\mathbb{Q}(\sqrt{-d}). Nontrivial pointwise upper bounds are known only for β„“=3\ell =3; nontrivial upper bounds for averages of hβ„“(βˆ’d)h_\ell(-d) have previously been known only for β„“=3,5\ell =3,5. In this paper we prove nontrivial upper bounds for the average of hβ„“(βˆ’d)h_\ell(-d) for all primes β„“β‰₯7\ell \geq 7, as well as nontrivial upper bounds for certain higher moments for all primes β„“β‰₯3\ell \geq 3.Comment: 26 pages; minor edits to exposition and notation, to agree with published versio

    Simultaneous Integer Values of Pairs of Quadratic Forms

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    We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.Comment: 63 page

    Higher Descent Data as a Homotopy Limit

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    We define the 2-groupoid of descent data assigned to a cosimplicial 2-groupoid and present it as the homotopy limit of the cosimplicial space gotten after applying the 2-nerve in each cosimplicial degree. This can be applied also to the case of nn-groupoids thus providing an analogous presentation of "descent data" in higher dimensions.Comment: Appeared in JHR
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