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:

SS2025: Modular forms 2 (joint with Dr. Gautier Ponsinet)

The course will introduce automorphic forms on GL_2(A_Q). We will first introduce adeles A_Q over the rational numbers Q and Fourier analysis on it. Then we will introduce the automorphic forms on GL_1(A_Q) (the ideles over Q), which is a generalisation of Dirichlet characters. After that we will move to GL_2(A_Q). We will generalise the notion of modular forms to automorphic functions and study Maass forms and their Fourier expansions. Hecke operators will then be defined in an adelic way. Then we will introduce automorphic forms on GL_2(A_Q) and automorphic representations of GL_2(A_Q).

Prerequisites: the course Modular Forms-1 (chapters 1-5 of Diamond-Shurman, A first course in modular forms; in particular the definitions of modular forms and Hecke operators).

Time and Room: Lectures: Tuesday 10h00-11h30 in WSC-S-U-3.02 and Friday 12h30-14h00 in WSC-S-U-3.01; Exercise Session: Friday 14h30-16h00 in WSC-S-U-3.01 (pay attention to the rescheduled time slots!)

 

The name of the course on Moodle page is "Modular forms 2" (here) and the enrolling key is "MF2".

References:

  1. Goldfeld, Hundley, Automorphic Representations and L-functions for the General Linear Group, vol. 1 (online access through university library here)
  2. Bump, Automorphic Forms and Representations (online access through university library here)
  3. Miyake, Modular Forms

 

SS2025: Master Seminar on Number Theory (joint with Dr. Gautier Ponsinet)

In this seminar, we will study modular curves from an algebraic point of view, Eichler-Shimura relations, Galois representations and formulate the Modularity Theorem for elliptic curves over the rational numbers Q. We will follow Chapters 6-9 of Diamond-Shurman.

Prerequistes: the course Modular Forms-1 (chapters 1-5 of Diamond-Shurman, A first course in modular forms; in particular the definitions of modular forms and Hecke operators).

Time and Room: Tuesday 12h15-13h45, WSC-S-U-3.01

 

The name of the course on Moodle page is "Master Seminar on Number Theory" (here) and the enrolling key is "Modularity".

Tentative program:

  1. [DS]6.1-6.2, Jacobians of Riemann surfaces
  2. [DS]6.3-6.5, Hecke algebras
  3. [DS]6.6, Modular abelian varieties and modularity theorem over complex numbers C
  4. [DS]7.1-7.2, Elliptic curves as algebraic curves
  5. [DS]7.3-7.4, Weil pairing on elliptic curves
  6. [DS]7.5-7.6, Function fields of modular curves over C and over Q
  7. [DS]7.7-7.9, Modular curves and algebraic curves and modularity theorem over Q
  8. [DS]8.1-8.2, Elliptic curves and algebraic curves in arbitrary characteristic
  9. [DS]8.3-8.4, Reductions mod p of elliptic curves
  10. [DS]8.5-8.6, Reductions mod p of modular curves
  11. [DS]8.7-8.8, Eichler-Shimura relation
  12. *[DS]9.1-9.2, Galois number fields
  13. [DS]9.3-9.4, Galois representations and elliptic curves
  14. [DS]9.5-9.6, Galois representations and modular forms

*: might be skipped depending on the backgrounds of the participants

References: 

  1. [DS] Diamond and Shurman, A first course in modular forms (online access through university library here)

 

 

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".

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

References:

  1. Borel, Chowla, Herz and Serre, Seminar on complex multiplication
  2. Diamond and Shurman, A first course in modular forms
  3. Lang, Introduction to modular forms
  4. 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".

Time and 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

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

References:

  1. [S]Serre, Abelian l-adic representations and elliptic curves, McGill University lecture notes, (1968).
  2. 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

  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

 

References:

  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;

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.