TIME and LOCATION
- Tuesday 3 July to Saturday 7 July 2018
Recommended arrival/departure date: Monday 2/Sunday 8
- Campus Belval, University of Luxembourg
COURSES
The lecturers are kindly helped by assistants for the exercise sessions. Many thanks to all involved!
Sara Arias-de-Reyna (with Samuele Anni, Pietro Mercuri and Maria Sabitova),
ELLIPTIC CURVES
- Group law, torsion points
- Isogenies
- Mod p and p-adic Galois representation, compatible systems
- Néron-Ogg-Shafarevich criterion
- p-adic uniformization of elliptic curves: Tate’s Curve
- Image of inertia at a prime of multiplicative reduction
- Conductor
- Tate’s algorithm
- LMFDB - information about elliptic curves
A video of the first lecture can be found here: In
mp4 format (639 MB), and in
wmv format (114 MB).
Henri Cohen (with Samuele Anni, Jean-Stefan Koskivirta and Tomomi Ozawa),
MODULAR FORMS
- Elliptic functions
- Modular forms and functions
- Hecke operators
- Euler products, functional equations
- Modular forms on subgroups
- More general modular forms
A video of the first lecture can be found here: In
mp4 format (602 MB), and in
wmv format (108 MB).
Bas Edixhoven (with Samuele Anni, Jean-Stefan Koskivirta and Tomomi Ozawa),
GALOIS REPRESENTATIONS ATTACHED TO MODULAR FORMS
- Galois representations
- Modular curves over ℂ
- Elliptic curves over ℂ
- Modular forms geometrically
- Arithmetic moduli of elliptic curves
- Construction of the Galois representation attached to a Hecke eigenform
- Local study of the Galois representation
- Computational aspects
A video of the first lecture can be found here: In
mp4 format (545 MB), and in
wmv format (143 MB).
Davide Lombardo (with Samuele Anni, Pietro Mercuri and Maria Sabitova),
ABELIAN VARIETIES
- The Néron-Ogg-Shafarevich criterion for bad reduction and the semistable reduction theorem
- Point-counting (especially on Jacobians) via characteristic polynomials of Frobenius
- The classification of endomorphism algebras
- CM abelian varieties and their corresponding representations
- Polarizations and the Weil pairing
A video of the first lecture can be found here: In
mp4 format (545 MB), and in
wmv format (143 MB).
MATHEMATICAL SOFTWARE
Please bring your laptop: in case, you can work together with the other participants,
or access a computer in the room MSA 2.200.
Before the conference,
please check out the programs PARI/GP and
SAGE
(if applicable, please set up an account to access the online version of the program, and install the program).
ROOMS
The activities take place in the MSA (Maison du Savoir) which is a building on the north side of the campus
(number 1 on the map).
You recognize the MSA because it is covered by metal squares and it consists of two parts connected by a bridge
(note, the auditoriums are in the lower part of the building).
- The main lecture room is MSA 3.110, on the third floor.
- We also have reserved another lecture room, MSA 3.040.
- For the exercise sessions we have the rooms MSA 2.240, 2.230 and 2.200.
Notice that the room MSA 2.200 is equipped with computers.
- Coffee breaks take place next to the main lecture room MSA 3.110.
- On the first floor of the MSA, in the main hall, there are several tables (and there is eduroam access).
SCHEDULE
|
Tuesday |
Wednesday |
Thursday |
Friday |
Saturday |
| 09.00 - 10.30 | Elliptic Curves 1 | Modular Forms 1 | Elliptic Curves 2 | Elliptic Curves 3 | Abelian Varieties 3 |
|---|
| 10.30 - 11.00 | Coffee break | Coffee break | Coffee break | Coffee break | Coffee break |
|---|
| 11.00 - 12.30 | Abelian Varieties 1 | Galois Reps. 1 | Abelian Varieties 2 | Modular Forms 2 | Modular Forms 3 |
|---|
| 12.30 - 13.30/14:00 | Lunch break | Lunch break | Lunch break | Lunch break | Lunch break |
|---|
| 14.00 - 15.30 | Short Talks 1 | Short Talks 2 (at 13.30!) | Galois Reps. 2 | Galois Reps. 3 | Project presentation |
|---|
| 15.30 - 15.45 | Coffee break | Coffee break | Excursion | Coffee break | Free work |
|---|
| 15.45 - 18.00 | Exercises/Projects 1 | Exercises/Projects 2 | Excursion | Exercises/Projects 3 | Free work |
|---|
| 18.00 - 21.00 | PIZZA and Free/project work | Free/project work | Excursion | Free/project work | |
|---|
PROJECT WORK AND MATERIAL
Course notes and
Projects by Sara Arias de Reyna on
Elliptic curves.
Course notes, projects
(
1,
2,
3,
4,
5)
and project answers
(
1,
2,
3,
4,
5)
by Henri Cohen on
Modular Forms.
Course notes and
Projects by Davide Lombardo on
Abelian Varieties.
Course notes and
Projects by Bas Edixhoven on
Galois Representations.
Samuele Anni's
SAGEMATH presentation.
Please read up on complex abelian varieties before the start of the school, here are some references:
- Milne. Abelian varieties. Available online.
- Mumford. Abelian varieties. Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, 1970.
- Moonen and van der Geer, Abelian Varieties. Available online.
CONTRIBUTED TALKS
On *Tuesday, July 3* there will be two parallel sessions of contributed talks: 13:30-15:30, Room MSA 3.110 and Room MSA 3.040.
The second session includes an introduction to Sage and Pari/GP: 14:00-15:30, Room MSA 3.040.
On *Wednesday, July 4* there will be two parallel sessions of contributed talks: 13:30-15:30, Room MSA 3.110 and Room MSA 3.040.
The schedule and abstracts of the contributed talks are here: pdf.