Topics in Arithmetic Geometry 1-2, Fall 2009 and Spring 2010

Time: Mondays, 15:45--17:30. Place: Snellius 401.

Participation to the Fall semester is not a prerequisite for participation in the Spring semester!

Spring 2010

The (proposed) rules for obtaining 6 EC for this seminar are as follows:

• speak at least two times;
• not too much of the same;
• there must be some originality in the talk (e.g., at least work out an example of yourself, or state and prove your own new theorem!);
• students provide notes for their own lectures.

May 31.

• Ila Varma, Cubic rings and forms, II.

May 24. No meeting: 2nd day of Pentecost.

May 17.

• Joost Michielsen, Primes in arithmetic progressions in F_q[X].

May 10.

• Duong Hoang Dung, The probabilistic zeta function of a finitely generated profinite group.
• Igor Shparlinski, Sum-Product Problem: New Generalisations and Applications.

May 3. No meeting because of "course free week".

April 26.

• Ila Varma, Cubic rings and forms.
• Michiel Kosters, Uniserial rings (theory behind his master thesis).

April 19.

• Duong Hoang Dung, on the main result of this arxiv preprint; here are his notes.
• Michiel Kosters, Semi-simple modules, rings, and its application to representation theory.

April 12.

• Sunil Rana, Localization of Categories. Here is the relevant part from Schapira's book.
• Bas Edixhoven, Weighted projective spaces (preparation for Duong's talk on April 19).

April 5. No meeting because 2nd day of Easter.

March 29. No meeting because Peter and Bas are both abroad.

March 22. No meeting because this is a week without courses.

March 15.

• Hilko Chang, Linear Error Correcting Algebraic Geometry Codes (2 times 45 minutes).

March 8.

• Dung Duong Hoang, Finiteness of symmetric ideals, continued.
• Joost Michielsen, L-functions of the projective line (continuation). Here is his abstract.

March 1.
The theme today is vector bundles on P^1. See a paper by Winfried Scharlau for some historical details.

• Bas Edixhoven, Classification of vector bundles on P^1 over a field via geometry and cohomology.
• Ila Varma, Classification of vector bundles on P^1 over a field via elementary operations on matrices.

February 22.

• Michiel Kosters, Projective modules over Dedekind domains.
• Joost Michielsen, L-functions of the projective line (abstract) .

February 15.

• Sunil Rana, The mapping cylinder.
• Duong Hoang Dung, Finiteness of symmetric ideals.

February 8. No meeting because of anniversary of the university (`dies').

February 1. First meeting.

• Bas Edixhoven, Short introduction to simplistic algebraic topology and homological algebra.
• Sunil Rana, The origin of the mapping cone construction.

Fall 2009

December 7.

• Bas Edixhoven, Galois representations: maximal subgroups of PGL_2(Q_l), ramification.

November 30.

• Sunil Rana, The mapping cone construction.
• Bas Edixhoven, Some basics of Galois representations.

November 23.

• Ila Varma, Sums of 6 squares.
• Michiel Kosters, Injective modules and the injective hull of a module.

November 9.

• Ariyan Javanpeykar (2 times 45 minutes), Intersection theory hidden in K_0(X).

November 2.

• Bas Edixhoven, Adeles, product formula and ideles.
• Peter Stevenhagen, Class Field Theory, the theorem.

October 26.

• Bas Edixhoven, the projective line over Z. This will be very explicit, and does not require the contents of my previous lecture (which maybe could have seemed intimidating to some...). But it will give evidence for the point of view that I took there.
• Michiel Kosters, Classification of symmetric non-degenerate bilinear forms on finite Abelian groups. And the abstract.

October 19.

• Break.

October 12.

• Bas Edixhoven, Introduction to schemes.
• Alberto Vezzani, A functorial characterization of open immersions.

October 5.

• Ariyan Javanpeykar, Towards the Riemann-Roch Theorem (notes will be corrected).
• Peter Stevenhagen, Complex multiplication, introduction to class field theory (continued). Same reference.

September 28.

• Ariyan Javanpeykar, Grothendieck groups
• Peter Stevenhagen, Complex multiplication, introduction to class field theory. Here is a reference.

September 21.

• Joost Michielsen, Finite field isomorphisms.
• Bas Edixhoven, Modular forms, continuation.

September 14.

• Ila Varma, Calculating class numbers via special types of units.
• Bas Edixhoven, From counting sums of squares to modular forms, continued.

September 7.

• Organisation and information meeting.
• Bas Edixhoven, From counting sums of squares to modular forms.