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Universität Duisburg-Essen,
Fakultät für Mathematik,
Thea-Leymann-Straße 9
45127 Essen

Office: Office: WSC-O-3.50

Email: xiaoyu.zhang - at -

SS2023: Master Seminar on number theory: Euler Systems


On Monday, 16-18h, WSC-S-U-3.03


If you are interested in this seminar, you can write to me directly or enrol yourself in the moodle page (the title of the course is: Master Seminar on number theory: Euler Systems). The enrolment code is: ES2023


In this seminar, we will learn Euler systems, an important tool in Iwasawa theory used to bound the size of Selmer groups and eventually to prove (one direction of) Iwasawa Main Conjectures. We will cover basic properties of Galois cohomology of global and local fields, the notion of Euler systems and how they are related to Selmer groups.


Tentative program (note that we will have only 12 sessions for this seminar as there will be 3 holidays on Monday during this semester):

  1. Introduction and overview;
  2. [R]1.1-1.3, Galois cohomology of local fields;
  3. [R]1.4, duality of Galois cohomology of local fields;
  4. [R]1.5-1.6, Galois cohomology of global fields and Selmer groups;
  5. [R]1.7, Poitou-Tate duality;
  6. [R]2.1-2.2, definition of Euler systems and bounding Selmer groups over K;
  7. [R]2.3-2.4, bounding Selmer groups over K_∞ and twists by characters;
  8. [R]3.4, Example of Euler system: Stickelberger elements;
  9. [R]3.5, Example of Euler system using elliptic curves;
  10. [R]4.1-4.3, universal Euler systems and their properties;
  11. [R]4.4-4.5, Construction of Kolyvagin’s derivative classes and statement of their properties;
  12. [R]5.1-5.2, Bounding the order of Selmer groups over K;


[B] Joel Bellaïche, An introduction to Bloch-Kato’s conjecture (two lectures at the Clay Mathematical Institute Summer School, Honolulu, Hawaii), 2009.

[LZ] David Loeffler and Sarah Zerbes, Euler systems (Arizona Winter School 2018 notes), 2018.

[M] James Milne, Arithmetic duality theorems, Charleston: BookSurge, 2006.

[R] Karl Rubin, Euler systems, No. 147. Princeton University Press, 2000.








I am in Prof.Bertolini's group. I got my PhD from Université Paris 13 in 2019 under the supervision of Jacques Tilouine.

Research and Teaching