Abelian Varieties

Lecturer Dr. Andreas Mihatsch
Email mihatsch (add ''at''math.uni-bonn.de)
Office Room 4.024

Lecture Mon 10:15 -- 11:45, Thu 14:15 -- 15:45
via Zoom (ID 991 9507 0584, Pwd Abelian)
Exercises Wed 14:15 -- 15:45 w/ Lucas Mann (via Zoom)
Registration If you wish, you may send me your email for a mailing list, so you will
recieve announcements (if any).


Examination

Oral exams will take place during the three weeks July 26 -- August 13 and in the week September, 27 -- 30.


Lecture Notes and References

Algebraic Geometry: Course notes for AG 1 and AG 2 as well as Stacks Project
Cohmology and Base Change: Madapusi-Pera's notes
Group Schemes: First chapter of Milne's notes on algebraic groups
Elliptic Curves: Silverman's Arithmetic of Elliptic Curves
Abelian varieties: Milne's notes, the book draft of van der Geer--Moonen and the book of Mumford.
Mordell Conjecture/Faltings' Theorem: Kleine AG program by Johannes Anschütz and Gregor Pohl

1 - Elliptic Curves
2 - Smoothness
3 - ECs over C, j-invariant
4 - Modular Curve
5 - ECs are Cubics
6 - Cubics are ECs - Part 1
7 - Cohomology and Base change
8 - Cubics are ECs - Part 2
9 - Complements on Flatness, Relative Curves
10 - Torsion and Tate module
11 - Endomorphisms
12 - Pic and Rosati Involution
13 - Quotients I
14 - Quotients II
15 - Isogenies I
16 - Isogenies II
17 - Isogenies III and Discussion
18 - Abelian Varieties
19 - Pic^0
20 - Theorem of the Cube
21 - Ample Line Bundles
22 - Divisibility and deg [n]
23 - Dual AV I
24 - Dual AV II
25 - Dual AV III
26 - Outlook


Exercise Sheets

Sheets will be discussed during exercise sessions. It is very much recommended to give these problems a try yourself before that! Students of Bonn may hand in their solutions to Lucas (mannluca w/ added @math.uni-bonn.de) to obtain personal feedback. However, this is optional.

1st Session - Group Schemes - Sheet and Discussion notes
2nd Session - Smoothness - Sheet and Discussion notes
3rd Session - Curves - Sheet and Discussion notes
4th Session - Curves II - Sheet and Discussion notes
5th Session - Relative Curves - Sheet and Discussion notes
6th Session - Endomorphisms of ECs - Sheet and Discussion notes
7th Session - Endomorphisms of ECs II - Sheet and Discussion notes
8th Session - (Noetherian) approximation, Endomorphisms III - Sheet and Discussion notes
9th Session - Endomorphisms IV - Sheet and Discussion notes
10th Session - Sheet and Discussion notes
11th Session - Sheet (version with hints) and Discussion notes
12th Session - Sheet (version with hints) and Discussion notes
13th Session - Sheet (version with hints) and Discussion notes


Abelian Varieties

An abelian variety is a connected, smooth, proper group variey. Any such is necessarily commutative, hence the name. AVs are of interest for two reasons. First, they form themselves a not-easy-but-accessible class of varieties that may be studied from a number-theoretic point of view. For example, the famous Mordell--Weil Theorem states that if A/Q is an AV over the rationals, A(Q) is a finitely generated abelian group. Specializing this to one-dimensional AVs over Q, which are the so-called elliptic curves, shows how these fall into the sweet spot of complexity between the projective line and higher genus curves. Namely the latter have only finitely many rational points by Faltings' Theorem (Mordell Conjecture).

The second reason is that moduli spaces of AVs provide algebraic models for the so-called Shimura varieties. For example, modular curves are first defined as Riemann surfaces, namely as the quotients of the complex upper half plane by congruence subgroups of SL_2(Z). Interpreting these as moduli spaces of elliptic curves with level structure instead allows to define them as curves over number fields and to study their reduction behaviour mod p.

This Course

An introduction to elliptic curves and abelian varieties with a view towards arithmetic geometry. Contents will be a subset of
  • Elliptic Curves
  • Analytic theory of abelian varieties
  • Torsion subgroups
  • Duality Theory, Polarizations
  • Endomorphism rings
  • Mordell--Weil Theorem (see above)
I intend to follow up with a course on moduli spaces of elliptic curves and abelian varieties. My aim is to eventually explain some results from arithmetic intersection theory (Gross--Zagier Formula, Arithmetic Fundamental Lemma, Kudla Program), details being not yet decided.