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 On the ramification of étale cohomology groups

2019 ◽  
Vol 2019 (749) ◽  
pp. 295-304 ◽  
Author(s):  
Isabel Leal
Keyword(s):  
Positive Characteristic ◽  
Residue Field ◽  
Cohomology Groups ◽  
Generic Fiber ◽  
Codimension One ◽  
Etale Cohomology ◽  
Rank One ◽  
Étale Cohomology ◽  
Valuation Field ◽  
Dimension One

Abstract Let K be a complete discrete valuation field whose residue field is perfect and of positive characteristic, let X be a connected, proper scheme over \mathcal{O}_{K} , and let U be the complement in X of a divisor with simple normal crossings. Assume that the pair (X,U) is strictly semi-stable over \mathcal{O}_{K} of relative dimension one and K is of equal characteristic. We prove that, for any smooth \ell -adic sheaf \mathcal{G} on U of rank one, at most tamely ramified on the generic fiber, if the ramification of \mathcal{G} is bounded by t+ for the logarithmic upper ramification groups of Abbes–Saito at points of codimension one of X, then the ramification of the étale cohomology groups with compact support of \mathcal{G} is bounded by t+ in the same sense.

Hilbert 90 for KnM

2019 ◽  
pp. 42-53
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Exact Sequence ◽  
Galois Extension ◽  
Galois Cohomology ◽  
Inductive Step ◽  
Etale Cohomology ◽  
Étale Cohomology ◽  
Cyclic Galois Extension

This chapter formulates a norm-trace relation for the Milnor 𝐾-theory and étale cohomology of a cyclic Galois extension, herein called Hilbert 90 for 𝐾𝑀 𝑛. To begin, the chapter uses condition BL(n) to establish a related exact sequence in Galois cohomology. It then establishes that condition BL(n − 1) implies the particular case of condition H90(n) for 𝓁-special fields 𝑘 such that 𝐾𝑀 𝑛(𝑘) is 𝓁-divisible. This case constitutes the first part of the inductive step in the proof of Theorem A. The remainder of this chapter explains how to reduce the general case to this particular one. The chapter concludes with some background on the Hilbert 90 for 𝐾𝑀 𝑛.

scholarly journals Crystalline Cohomology of Towers of Curves

2018 ◽  
Vol 2020 (21) ◽  
pp. 7454-7488
Author(s):  
Jan Vonk
Keyword(s):  
Modular Curves ◽  
De Rham Cohomology ◽  
Crystalline Cohomology ◽  
Etale Cohomology ◽  
Étale Cohomology ◽  
De Rham ◽  
Integral Structures

Abstract We investigate the geometry of finite maps and correspondences between curves, and construct canonical trace and pullback maps between Hyodo–Kato integral structures on de Rham cohomology of curves, which are functorial for finite morphisms of the generic fibres. This leads to a crystalline version of the étale cohomology of towers of modular curves considered by Hida and Ohta, whose ordinary part satisfies $\Lambda $-adic control and Eichler–Shimura theorems.

Etale Cohomology of Rigid Spaces

2004 ◽  
pp. 239-258
Author(s):  
Jean Fresnel ◽  
Marius van der Put
Keyword(s):  
Etale Cohomology ◽  
Étale Cohomology
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