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

Office: Office: WSC-O-3.50

Email: xiaoyu.zhang - at -

WS2023-24: Seminar on number theory: Elliptic Curves

On Monday, 16-18h

In this seminar, we will study basic properties of elliptic curves (over complex numbers, over p-adic fields, over finite fields), their arithmetic properties, the Selmer groups and Tate-Shafarevich groups and prove the Mordell-Weil theorem.


Prerequisite: algebra, analysis and topology on advanced undergraduate level.


Tentative program

  1. Introduction and distribution of talks;
  2. [M] I.1-2, Bezout's theorem and rational points on plane curves;
  3. [M] I.3-4, group law on a cubic curve and Riemann-Roch theorem;
  4. [M] II.1-2, definition of elliptic curves and Weierstrass equations;
  5. [M] II.3-4, elliptic curves over p-adic fields and reduction modulo p;
  6. [M] II.5-6, torsion points on elliptic curves and Neron models;
  7. [M] III.1-2, doubly periodic functions;
  8. [M] III.3, elliptic curves as Riemann surfaces;
  9. [M] IV.1-2, group cohomology, Selmer groups and Tate-Shafarevich groups;
  10. [M] IV.3, the finiteness of Selmer groups;
  11. [M] IV.4, heights;
  12. [M] IV.5-6, the rank of Mordell-Weil group;
  13. [M] IV.7+I.5, cohomology groups and Jacobians;
  14. [M] IV.9, elliptic curves over finite fields;
  15. [M] IV.10, the conjecture of Birch and Swinnerton-Dyer



  1. [M] Milne, Elliptic curves (here is a link to the pdf);
  2. Silverman, The arithmetic of elliptic curves;
  3. Silverman and Tate, Rational points on elliptic curves.


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] Appendix B, continuous group cohomology;
  3. [R] Appendix A and C, Herbrand quotients and Galois cohomology;
  4. [R]1.1-1.3, Galois cohomology of local fields;
  5. [R]1.4, duality of Galois cohomology of local fields;
  6. [R]1.5-1.6, Galois cohomology of global fields and Selmer groups;
  7. [R]1.7, Poitou-Tate duality;
  8. [R]2.1-2.2, definition of Euler systems and bounding Selmer groups over K;
  9. [R]2.3-2.4, bounding Selmer groups over K_∞ and twists by characters;
  10. [R] Appendix D, preliminary computations in cyclotomic fields;
  11. [R]3.4, Example of Euler system: Stickelberger elements;
  12. [R]3.5, Example of Euler system using elliptic curves;


[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, Teaching and Conferences/Talks