Positivity of quadratic base change -functions
Hervé Jacquet, Chen Nan (2001)
Bulletin de la Société Mathématique de France
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We show that certain quadratic base change -functions for are non-negative at their center of symmetry.
Hervé Jacquet, Chen Nan (2001)
Bulletin de la Société Mathématique de France
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We show that certain quadratic base change -functions for are non-negative at their center of symmetry.
I. D. Shkredov (2014)
Acta Arithmetica
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We describe all sets which represent the quadratic residues in the sense that R = A + A or R = A ⨣ A. Also, we consider the case of an approximate equality R ≈ A + A and R ≈ A ⨣ A and prove that A is then close to a perfect difference set.
Brandon Hanson (2013)
Acta Arithmetica
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An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → 1,...,r of the natural numbers, there exist x,y ∈ ℕ such that c(xy) = c(x+y) apart from the trivial solution x = y = 2. More generally, one could replace x+y with a binary linear form and xy with a binary quadratic form. In this paper we examine the analogous problem in a finite field . Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size...
Xiaoli Liu, Zhishan Yang (2022)
Czechoslovak Mathematical Journal
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Let be a quadratic field over the rational field and be the number of nonzero integral ideals with norm . We establish Erdős-Kac type theorems weighted by and of quadratic field in short intervals with . We also get asymptotic formulae for the average behavior of and in short intervals.
Kalyan Chakraborty, Azizul Hoque (2020)
Czechoslovak Mathematical Journal
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Let be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form whose ideal class group has an element of order . This family gives a counterexample to a conjecture by H. Wada (1970) on the structure of ideal class groups.
Noam D. Elkies, Daniel M. Kane, Scott Duke Kominers (2013)
Journal de Théorie des Nombres de Bordeaux
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In this note, we give simple examples of sets of quadratic forms that have minimal -universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh [KKO05] in the negative.
Adam Mohamed (2014)
Publications mathématiques de Besançon
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Let be an imaginary quadratic field and its ring of integers. Let be a non-zero ideal and let be a rational inert prime in and coprime with . Let be an irreducible finite dimensional representation of . We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in already lives in the cohomology with coefficients in for some ; except possibly in some few cases.
Georges Gras (2014)
Journal de Théorie des Nombres de Bordeaux
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From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields having isomorphic absolute Abelian Galois groups , we study any such issue for arbitrary number fields . We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some -adic obstructions coming from the global units of . By restriction to the -Sylow subgroups of and assuming the Leopoldt conjecture we...