# Dr. Xiaoyu Zhang

**Address**:

Universität Duisburg-Essen,

Fakultät für Mathematik,

Mathematikcarrée

Thea-Leymann-Straße 9

45127 Essen

Office: Office: WSC-O-3.50

Email: xiaoyu.zhang - at - uni-due.de

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

**Teachings:**

**WS2024-25: Modular Forms-1 (joint with Prof. Dr. Massimo Bertolini)**

This course is an introduction to modular forms and complex multiplication. In the first part of the course, we will introduce the notion of modular forms, modular curves, dimension formula, Hecke operators and L-functions. In the second part of the course, we will introduce the theory of complex multiplication for elliptic curves.

The name of the course on Moodle page is "Modular Forms 1" (here) and the enrolling key is "ModularForms1".

Room: Lectures: Wednesday 12h15-13h45, Friday 12h15-13h45; Exercise sessions: Friday 14h15-15h45 all in WSC-S-U-3.03

References:

- Borel, Chowla, Herz and Serre, Seminar on complex multiplication
- Diamond and Shurman, A first course in modular forms
- Lang, Modular forms
- Miyake, Modular forms

**SS2024: Modular Forms-2 (joint with Dr. Jie LIN)**

You can find the course on Moodle page by searching keywords "MF2 - 24". The name of the course is "Modular Forms 2 - 24" and the enrolling key is "MF2SS24".

Room: Lectures: Monday 10h15-11h45, Friday 12h-13h30; Exercise sessions: Friday 14h30-16h all in WSC-S-U-3.02

For more details, see https://www.esaga.uni-due.de/jie.lin/teaching/

**SS2024: Seminar on number theory: abelian l-adic representations (joint with Dr. Jie LIN)**

You can find the course on Moodle page by searching "Seminar on number theory: abelian l-adic representations". The enrolement key is "Abelian".

On Monday, 16-18h (**the first meeting will be April 15, 2024, not April 8, 2024**)

Room: WSC-S-U-3.01

In this seminar, we will study abelian l-adic representations, in particular those arising from Tate modules attached to CM elliptic curves (topic of last semester's seminar). We will consider arithmetic/analytic properties of L-functions attached to (a compatible family of) l-adic representations, local algebraicity and open image problem of such representations.

Prerequiste: basic notions of algebraic number theory and elliptic curves

Tentative program

- Introduction and distribution of talks (
**April 15, 2024**) - [S] I.1.1-1.2.4, l-adic representations
- [S] I.2.5-I.A.3, L-functions and Cebotarev density theorem
- [S] II.1.1-II.2.2, torus T and groups T_m, S_m
- [S] II.2.3-II.2.5, l-adic representations valued in S_m and its representations
- [S] II.3.1-II.3.4, structure of T_mand applications
- [S] III.1.1-III.1.2, locally algebraic representations (local case)
- [S] III.2.1-III.2.4, locally algebraic representations (global case)
- [S] III.3.1-III.3.4, local algebraicity for the case of composites of quadratic fields
- [S] IV.1.1-IV.2.1, Irreducibility theorem for Galois representation G_l on Tate module of elliptic curves
- [S] IV.2.2-IV.2.3, Open image theorem for G_l

References:

- [S]Serre, Abelian l-adic representations and elliptic curves, McGill University lecture notes, (1968).
- Cassels-Frölich, Algebraic Number Theory, Academic Press (1967).

**WS2023-24: Modular Forms-1 (joint with Dr. Jie LIN)**

You can find the course on Moodle page by searching keywords "MF1 - 23/24". The name of the course is "Modular Forms 1 - 23/24" and the enrolling key is "MF1WS2324".

Room: Lectures: Monday 10h-12h WSC-S-U-3.02, Wednesday 12h-14h WSC-N-U-3.04; Exercise sessions: Friday 14h-16h WSC-S-U-3.02.

For more details, see https://www.esaga.uni-due.de/jie.lin/teaching/.

**WS2023-24: Seminar on number theory: Elliptic Curves ****(joint with Dr. Jie LIN)**

You can find the course on Moodle page by searching "Seminar on number theory: ellptic curves". The enrolement key is "EC".

On Monday, 16-18h

Room: WSC-S-U-3.01

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

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

References:

- [M] Milne, Elliptic curves (here is a link to the pdf);
- Silverman, The arithmetic of elliptic curves;
- 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):

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

References:

[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.