Soham Karwa

Étale Homotopy Study Group

This term, Jesse and I are organising an étale homotopy study group with a focus on obstructions to rational points on algebraic varieties. Our main goal is to define the étale homotopy type of a variety and then study the étale homotopy obstruction to rational points. The main reference we will follow is the the paper of Harpaz and Schlank. The notes of Schlank and Skorobogatov are an excellent, rapid introduction to the world of étale homotopy. A breakdown of the topics and talks can be found here.

The seminar will be held every Monday at 10AM BST over Zoom. Each session will run for two hours (excluding the first talk) and we aim to cover two “list items” below in this time; to do this we are looking for two speakers each week. If you would like to give a talk, shoot us an email! The Zoom link for each week can be found here.

Schedule:

  1. Obstruction to Rational Points (28/09) – Soham Karwa
    • An introduction to the theory of obstruction of rational points
    • A overview of the different type of obstructions (functorial, torsors and etale homotopy)
    • The étale Brauer obstruction
  2. Simplicial Homotopy I (5/10) – TBD
    • Model Categories
    • Simplicial Sets
  3. Simplicial Homotopy II (5/10) – TBD
    • Skeleton/Coskeleton
    • Homotopy of simplicial sets
  4. Étale Homotopy Type (12/10) – Jef Laga
    • Hypercoverings
    • The Homotopy Category
    • Definition of étale homotopy type
  5. Relative étale homotopy type (12/10) – Jesse Pajwani
    • Definition of relative étale homotopy type
  6. Homotopy Fixed Points (19/10/20) – TBD
    • Definition of set of “homotopy fixed points”
    • Theorems involving this set
  7. p-adic Homotopy Fixed Points, Adelic Homotopy Fixed Points, Étale Homotopy Obstruction (19/10/20) – TBD
    • Some awkward theorems (more details to follow)
  8. Homotopy Fixed point theorems for profinite groups I (26/10/20) – TBD
    • Proof of Lemma 4.1 and 4.2 in [HS13]
    • Statement of Theorem 4.3 with the remarks that follows
  9. Homotopy Fixed Point theorems for profinite groups II (26/10/20) – TBD
  10. Homotopy Fixed point theorems for profinite groups III (2/11/20) – TBD
  11. “An Auxiliary theorem” (2/11/20) – TBD
  12. Finite Descent I (9/11/20) – TBD
    • Pages 70-72 [HS13] (more detail to follow)
  13. Finite Descent II (9/11/20) – TBD
    • Pages 73-79 in [HS13] (more detail to follow)
  14. Main Theorems I (16/11/20) – TBD
    • The  Étale Brauer obstruction is stronger than the  Étale homotopy obstruction
    • Pages 99-102 in [HS13]
  15. Main Theorems II (16/11/20) – TBD
    • The  Étale homotopy obstruction is stronger than Étale Brauer I
    • Pages 103-105 in [HS13]
  16. Main Theorems III (23/11/20) – TBD
    • The  Étale homotopy obstruction is stronger than Étale Brauer II
    • Applications
    • Pages 106-110 in [HS13]