scholarly journals Massey products 〈y,x,x,…,x,x,y〉 in Galois cohomology via rational points

2017 ◽  
Vol 221 (7) ◽  
pp. 1845-1866 ◽  
Author(s):  
Kirsten Wickelgren
Keyword(s):  
Galois Cohomology ◽  
Rational Points ◽  
Massey Products

scholarly journals Vanishing of Massey Products and Brauer Groups

2015 ◽  
Vol 58 (4) ◽  
pp. 730-740 ◽  
Author(s):  
Ido Efrat ◽  
Eliyahu Matzri
Keyword(s):  
Prime Number ◽  
Galois Cohomology ◽  
Brauer Groups ◽  
Central Simple Algebras ◽  
Massey Products ◽  
Global Fields ◽  
Root Of Unity ◽  
Simple Algebras ◽  
Central Simple

AbstractLet p be a prime number and F a field containing a root of unity of order p. We relate recent results on vanishing of triple Massey products in the mod-p Galois cohomology of F, due to Hopkins, Wickelgren, Mináč, and Tân, to classical results in the theory of central simple algebras. We prove a stronger form of the vanishing property for global fields.

scholarly journals Two families of pro-𝑝 groups that are not absolute Galois groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Claudio Quadrelli
Keyword(s):  
Galois Cohomology ◽  
Galois Groups ◽  
Residue Theorem ◽  
Smoothness Property ◽  
Massey Products ◽  
Norm Residue

Abstract Let 𝑝 be a prime. We produce two new families of pro-𝑝 groups which are not realizable as absolute Galois groups of fields. To prove this, we use the 1-smoothness property of absolute Galois pro-𝑝 groups. Moreover, we show in these families, one has several pro-𝑝 groups which may not be ruled out as absolute Galois groups employing the quadraticity of Galois cohomology (a consequence of the norm residue theorem), or the vanishing of Massey products in Galois cohomology.

scholarly journals On the arithmetic of abelian varieties

2020 ◽  
Vol 2020 (762) ◽  
pp. 1-33
Author(s):  
Mohamed Saïdi ◽  
Akio Tamagawa
Keyword(s):  
Function Fields ◽  
Abelian Varieties ◽  
Galois Cohomology ◽  
Rational Points ◽  
Selmer Groups ◽  
Selmer Group ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Natural Objects ◽  
Weil Group

AbstractWe prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects “discrete Selmer groups” and “discrete Shafarevich–Tate groups”, and prove that they are finitely generated {\mathbb{Z}}-modules. Further, we prove that in the isotrivial case, the discrete Shafarevich–Tate group vanishes and the discrete Selmer group coincides with the Mordell–Weil group. One of the key ingredients to prove these results is a new specialisation theorem for first Galois cohomology groups, which generalises Néron’s specialisation theorem for rational points of abelian varieties.

scholarly journals Four-fold Massey products in Galois cohomology

2018 ◽  
Vol 154 (9) ◽  
pp. 1921-1959 ◽  
Author(s):  
Pierre Guillot ◽  
Ján Mináč ◽  
Adam Topaz
Keyword(s):  
Number Field ◽  
Galois Cohomology ◽  
Number Fields ◽  
Galois Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Massey Products ◽  
Necessary And Sufficient ◽  
Global Principle

In this paper, we develop a new necessary and sufficient condition for the vanishing of $4$-Massey products of elements in the modulo-$2$ Galois cohomology of a field. This new description allows us to define a splitting variety for $4$-Massey products, which is shown in the appendix to satisfy a local-to-global principle over number fields. As a consequence, we prove that, for a number field, all such $4$-Massey products vanish whenever they are defined. This provides new explicit restrictions on the structure of absolute Galois groups of number fields.

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

scholarly journals Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

2021 ◽  
Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Waring’S Problem ◽  
Waring Problem ◽  
Waring's Problem ◽  
Main Conjecture ◽  
Higher Dimensional ◽  
Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

scholarly journals Le complémentaire des puissances -ièmes dans un corps de nombres est un ensemble diophantien

2015 ◽  
Vol 151 (10) ◽  
pp. 1965-1980 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Jan Van Geel
Keyword(s):  
Prime Number ◽  
Rational Points ◽  
Parameter Family ◽  
Symmetric Power ◽  
Symmetric Powers

For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

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