scholarly journals On the Galois cohomology groups of CK/DK

1982 ◽  
Vol 8 (2) ◽  
pp. 407-415 ◽  
Author(s):  
Shin-ichi KATAYAMA
Keyword(s):  
Galois Cohomology ◽  
Cohomology Groups

scholarly journals On Higher Special Elements of p-adic Representations

2020 ◽  
Author(s):  
David Burns ◽  
Takamichi Sano ◽  
Kwok-Wing Tsoi
Keyword(s):  
Galois Cohomology ◽  
Euler System ◽  
Cohomology Groups ◽  
Exterior Power ◽  
Concrete Application ◽  
Natural Generalisation ◽  
Higher Rank ◽  
Derivatives Of

Abstract As a natural generalisation of the notion of “higher rank Euler system”, we develop a theory of “higher special elements” in the exterior power biduals of the Galois cohomology of $p$-adic representations. We show, in particular, that such elements encode detailed information about the structure of Galois cohomology groups and are related by families of congruences involving natural height pairings on cohomology. As a first concrete application of the approach, we use it to refine, and extend, a variety of existing results and conjectures concerning the values of derivatives of Dirichlet $L$-series.

Genus of abstract modular curves with level-ℓ structures

2019 ◽  
Vol 2019 (752) ◽  
pp. 25-61 ◽  
Author(s):  
Anna Cadoret ◽  
Akio Tamagawa
Keyword(s):  
Galois Cohomology ◽  
Fundamental Groups ◽  
Modular Curves ◽  
Cohomology Groups ◽  
Arbitrary Characteristic ◽  
Linear Representations ◽  
Etale Cohomology ◽  
Étale Cohomology

Abstract We prove – in arbitrary characteristic – that the genus of abstract modular curves associated to bounded families of continuous geometrically perfect {\mathbb{F}_{\ell}} -linear representations of étale fundamental groups of curves goes to infinity with {\ell} . This applies to the variation of the Galois image on étale cohomology groups with coefficients in {\mathbb{F}_{\ell}} in 1-dimensional families of smooth proper schemes or, under certain assumptions, to specialization of first Galois cohomology groups.

scholarly journals DERIVED HECKE ALGEBRA AND COHOMOLOGY OF ARITHMETIC GROUPS

2019 ◽  
Vol 7 ◽  
Author(s):  
AKSHAY VENKATESH
Keyword(s):  
Hecke Algebra ◽  
Galois Cohomology ◽  
Arithmetic Group ◽  
Arithmetic Groups ◽  
Cohomology Groups ◽  
Single Degree ◽  
Cohomology Of Arithmetic Groups ◽  
Favorable Conditions

We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group $\unicode[STIX]{x1D6E4}$ . Under favorable conditions, the cohomology is freely generated in a single degree over this graded Hecke algebra. From this construction we extract an action of certain $p$ -adic Galois cohomology groups on $H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q}_{p})$ , and formulate the central conjecture: the motivic $\mathbf{Q}$ -lattice inside these Galois cohomology groups preserves $H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q})$ .

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 A Note on Galois Cohomology Groups of Algebraic Tori

1969 ◽  
Vol 34 ◽  
pp. 121-127 ◽  
Author(s):  
Kazuo Amano
Keyword(s):  
Galois Group ◽  
Galois Extension ◽  
Valuation Ring ◽  
Galois Cohomology ◽  
Splitting Field ◽  
Cohomology Groups ◽  
Algebraic Tori ◽  
Usual Manner ◽  
Algebraic Torus ◽  
Complete Field

Let k be a complete field of characteristic 0 whose topology is defined by a discrete valuation and let T be an algebraic torus of dimension d defined over k. As is well known, T has a splitting field K which is a finite Galois extension of k with Galois group . For a ring R, denote by TR the subgroup of R-rational points of T. Then TK and T0K, DK being a valuation ring of K, become -modules in the usual manner.

scholarly journals COHOMOLOGY AND OVERCONVERGENCE FOR REPRESENTATIONS OF POWERS OF GALOIS GROUPS

2019 ◽  
pp. 1-61
Author(s):  
Aprameyo Pal ◽  
Gergely Zábrádi
Keyword(s):  
Galois Cohomology ◽  
Galois Groups ◽  
Cohomology Groups ◽  
Direct Power ◽  
Equivalence Of Categories

We show that the Galois cohomology groups of $p$ -adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ can be computed via the generalization of Herr’s complex to multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules. Using Tate duality and a pairing for multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules we extend this to analogues of the Iwasawa cohomology. We show that all $p$ -adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ are overconvergent and, moreover, passing to overconvergent multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.

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