## Math 274 - Rational points on varieties

Course notes (.pdf) (.dvi.gz) (these are continually under construction - last revision 4/28/03)

Homework 1 and solutions (.pdf) (.dvi.gz)
Homework 2 and solutions (.pdf) (.dvi.gz)
Homework 3 and solutions (.pdf) (.dvi.gz)
Homework 4 and solutions (.pdf) (.dvi.gz)
Homework 5 and solutions (.pdf) (.dvi.gz)
Homework 6 and solutions (.pdf) (.dvi.gz)
Homework 7 and solutions (.pdf) (.dvi.gz)
Homework 8 and solutions (.pdf) (.dvi.gz)
Homework 9 and solutions (.pdf) (.dvi.gz)
Homework 10 and solutions (.pdf) (.dvi.gz)
Homework 11 and solutions (.pdf) (.dvi.gz)

Instructor: Bjorn Poonen

Lectures: MWF 10-11am in 51 Evans

Course Control Number: 55146

Office: 703 Evans, e-mail: poonen@math

Office Hours: Wed, Fri 11:10-12 or by appointment.

Prerequisites: Algebraic number theory (254A is enough), algebraic geometry (256AB - e.g. Hartshorne's book, the more the better), and some group cohomology (e.g., Chapter 2 of Milne's course notes on class field theory, available from his website, or Chapters IV (sections 1-8) and V in Cassels and Frohlich, Algebraic number theory).

Syllabus: The course will use the language of schemes to study varieties over fields that are not algebraically closed, and in particular to understand their rational points. Topics will include some subset of the following:

• basic facts about local and global fields
• the theory of C_r fields,
• cohomological dimension of fields,
• Brauer groups of fields (with the structure of the Brauer groups of local and global fields stated without proof)
• properties of varieties over arbitrary fields (geometrically integral, etc.),
• smooth and etale morphisms
• faithfully flat descent (with Galois descent as an important special case),
• twists (e.g. Severi-Brauer varieties),
• Weil restriction of scalars,
• the classification of curves of low genus over arbitrary fields,
• computational methods for determining rational points on curves over number fields
• etale cohomology (the goal here would be a brief survey with precisely stated definitions and theorems, with a view towards applications needed later in the course, but little in the way of proofs)
• torsors,
• cohomological Brauer groups of schemes,
• the Hasse principle and its failure,
• the Brauer-Manin obstruction, and
• applications of all of the above to rational points on geometrically rational surfaces, including the Iskovskih-Manin classification involving conic bundles and Del Pezzo surfaces (including cubic surfaces) - this has connections to the combinatorics of exceptional root systems.
Instead of working toward the proof of one or two big theorems, the goal will be to develop techniques that have wide application in arithmetic geometry.

Required Text: None

Recommended Reading: Notes will be available online. Also, the following books have been put on reserve at the Berkeley math library:

• QA241.A42 Cassels, Frohlich: Algebraic number theory
• QA201.G76 Greenberg: Lectures on forms in many variables
• QA1.A665 no.67 Shatz: Profinite groups, arithmetic, and geometry
• QA247.S4613 1979 Serre: Local fields
• QA247.S45813 2002 Serre: Galois cohomology
• QA242.5 .S47 1989 Serre: Lectures on the Mordell-Weil theorem
• QA1.E7 ser.3:21 Bosch, Lutkebohmert, Raynaud: Neron Models
• QA1.E7 ser.3:13 Freitag, Kiehl: Etale cohomology and the Weil conjecture
• QA564.M52 Milne: Etale cohomology
• QA573.M25131 Manin: Cubic forms
• QA251.3 .S62 2001 Skorobogatov: Torsors and rational points
We're not going to cover everything in these books! There are also Milne's course notes on various subjects: the notes on class field theory and etale cohomology are the most relevant to this course.

Grading: There will be no exams. Grades will be based on weekly homework.

Homework: Assignments will be due in class in Mondays. The first one will be due February 3. Late homeworks will not be accepted. You are free to consult any sources (animate or inanimate) while doing your homework, but if you use anything (or anyone) other than your class notes or the texts listed above, you should say so on your homework. Staple loose sheets!!!