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# Teaching

**Winter term 2017-2018:**

**Algebraic Geometry 3**

This semester I am in charge of the Exercise sessions for the course Algebraic Geometry 3. We meet on **Fridays** from **10:00 to 12:00** at **WSC-S-U-3.02**. Lectures are given by Dr. Rodolfo Venerucci, on Tuesdays (12:00-14:00, WSC-S-U-3.01) and Wednesdays (12:00-14:00, WSC-S-U-3.01).

Exercises sheet 1 (Due by October 25th, 2017)

Exercises sheet 2 (Due by November 2nd, 2017)

Exercises sheet 3 (Due by November 16th, 2017)

Exercises sheet 4 (Due by November 23rd, 2017)

Exercises sheet 5 (Due by November 30th, 2017)

Exercises sheet 6 (Due by December 7th, 2017)

Exercises sheet 7 (Due by December 14th, 2017)

Exercises sheet 8 (Due by December 21st, 2017)

Exercises sheet 9 (Due by January 11th, 2018)

Exercises sheet 10 (Due by January 18th, 2018)

Exercises sheet 11 (Due by January 25th, 2018)

Exercises sheet 12 (Due by February 1st, 2018)

**Summer term 2017:**

**Master Seminar on Algebraic Geometry: Algebraic Curves and Elliptic Curves**

Together with Prof. Dr. Bertolini, we organize this seminar, which is devoted to the study of algebraic curves, also from an arithmetical point of view. A tentative program can be found here. On April 24th, we will have an organisational meeting to discuss the topics and set the calendar.

The seminar meets on Mondays from 12h to 14h, at **WSC-N-U-4.03**.

**Winter term 2016-2017:**

**Crash course on Commutative Algebra**

The aim of this intensive course is to provide the students with the basic notions of Commutative Algebra that will be required to attend the course Algebraic Geometry 1 by Prof. Dr. Bertolini. We will follow mainly the book *Introduction to Commutative Algebra*, by M. F. Atiyah and I. G. MacDonald, and the main topics to be covered will be the following:

1. Rings and ideals.

2. Modules.

3. Integral dependence.

4. Chain conditions, Noetherian and Artinian rings.

5. Noether's normalization lemma and Hilbert's Nullstellensatz.

6. Discrete valuation rings and Dedekind domains.

The course will take place from October 4th until October 14th, and there will be two lectures and one exercise session every weekday with the following schedule:

Lectures: from 10.15h to 11.45h and from 14.15h to 15.45h, at room **WSC-N-U-3.05.**

Exercise sessions (run by Andrea Agostini): from 16.15h to 17.45h, at room **WSC-N-U-3.05**.

You can find the program in pdf format here, and the details of the course on LSF here.

Update: we moved to **WSC-N-U-3.05**!

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**Summer term 2016:**

**Doctoral mini-course on Complex Multiplication for Abelian Varieties**

The goal of this mini-course will be to establish the foundations of the theory of complex multiplication for (elliptic curves and) abelian varieties. We will follow mostly the approach developed in Shimura's books "Introduction to Arithmetic Theory of Automorphic Functions" and "Abelian Varieties with Complex Multiplication and Modular Functions". The mini-course is aimed to PhD students, but also interested master students are welcome

We will meet on Wednesdays from 10h to 12h at WSC-O-3.46 (Teeraum). The course will start on April 13th with an introductory lecture.

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**Winter term 2015-2016:**

**Master Seminar: Introduction to p-adic Analysis and p-adic Zeta Functions**

This seminar will be run by Aprameyo Pal and myself. Please ask us in case of any question at:

aprameyo.pal (at) uni-due.de or carlos.de-vera-piquero (at) uni-due.de.

The aim of the seminar is to introduce students into the grounds of p-adic ananlysis with an eye towards applications in number theory. You can find the tentative program here (we will have an organisational meeting on October 20th to give more details and to discuss the distribution of talks).

The basic reference for the seminar will be the book "p-adic Numbers, p-adic Analysis and Zeta functions", by N. Koblitz.

**Summer term 2015: **

**Master Seminar: Introduction to Global Class Field Theory and Complex Multiplication**

This seminar was run by Aprameyo Pal and myself.

*Contents:*

Starting with a classical problem for elementary number theory about the representation of prime numbers by binary quadratic forms, in this seminar we will give an introduction to both global class field theory and complex multiplication from a classical point of view. The ultimate goal will be to relate these two remarkably rich areas in number theory through the theory of modular functions.

Depending on the background and interests of the participants, we will adapt the contents of the seminar and the distribution of the topics.

*Prerequisites:*

Students are expected to be familiar with Galois theory and to have a basic knowledge of algebraic number theory.

*References:*

- Primes of the form x^2+ny^2, by David Cox.

- Algebraic Number Theory, by Serge Lang.