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...
Saad El Boukhari (2023)
Czechoslovak Mathematical Journal
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Let be a finite abelian extension of number fields with imaginary quadratic. Let be the ring of integers of and a rational integer. We construct a submodule in the higher odd-degree algebraic -groups of using corresponding Gross’s special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher “twisted” class number of , which is the cardinal of the finite algebraic -group .