Galois sections for abelian varieties over number fields MIRELA ÇIPERIANI AND JAKOB STIX Abstract — For an abelian variety A over a number field k we discuss the space of sections of its fundamental group extension π

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GALOIS THEORY AT WORK: CONCRETE EXAMPLES KEITH CONRAD 1. Examples √ √ Example 1.1. The field extension Q( 2, 3)/Q is Galois of degree 4, so its Galois√group

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QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday SeptemberDaya) Prove that the Galois group G of the polynomial X 6 + 3 over Q is of order 6.

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LIFTING GLOBAL REPRESENTATIONS WITH LOCAL PROPERTIES BRIAN CONRAD 1. Introduction Let k be a global field, with Galois group Gk and Weil group Wk relative to a choice of separable closure ks /k. Let Γ be either Gk or Wk

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QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday SeptemberDaya) Prove that the Galois group G of the polynomial X 6 + 3 over Q is of order 6.

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The pro-étale fundamental group Wouter Zomervrucht, December 16, Infinite Galois theory We develop an ‘infinite’ version of Grothendieck’s Galois theory. It was introduced first by Noohi [3] and slightly m

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Homework 6 √ 1. Does there exist a normal extension L ⊃ Q( 3) ⊃ Q with cyclic Galois group Gal(L/Q) = Z/4Z? 2. Determine all subfields of Q(ζ11 ).

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885 Documenta Math. Change of Selmer Group for Big Galois Representations and Application to Normalization

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Lifting Galois covers of algebraic curves Frans Oort Intercity number theory seminar, Utrecht, June 8, 2012. Introduction

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• Algebraic geometry
• Geometry
• Algebra
• Algebraic curves
• Niels Henrik Abel
• Algebraic surfaces
• Abelian variety
• Abelian group
• tale fundamental group
• Conjectures
• Curve
• ArtinSchreier curve

CANONICAL SUBGROUPS VIA BREUIL-KISIN MODULES FOR p = 2 SHIN HATTORI Abstract. Let p be a rational prime and K/Qp be an extension of complete discrete valuation fields. Let G be a truncated Barsotti-Tate group of level n,

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• Algebra
• Abstract algebra
• Mathematics
• Ring theory
• Functions and mappings
• Algebraic number theory
• Matrix theory
• Unipotent
• Semilinear map
• Valuation ring
• Galois module
• Polar coordinate system