97926 - Geometric Number Theory

Academic Year 2022/2023

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Mathematics (cod. 5827)

Learning outcomes

At the end of the course, the students have an advanced knowledge in number theory and arithmetic geometry. They are able to use this knowledge both in algebraic and geometric settings.

Course contents

Integral extensions and integral closure.

Number fields and rings of integers.

Norms, traces, bilinear forms and discriminants.

Dedekind domains: Unique factorization of ideals; The ideal class group; Integral extensions and closures; Ramification theory of primes in extensions.

Class numbers of number fields. Finiteness of the class number.

Group of units in a number field: Finite generatedness; rank and torsion subgroup of the group of units. CM fields and real quadratic fields.

Cyclotomic fields, their class numbers and units. Fermat's last theorem for regular primes.

Absolute values, local fields, completions. Hensel's lemma.

Algebraic curves. Rational points on curves over number fields. The Hasse principle.

Readings/Bibliography

JAMES S. MILNE. Algebraic Number Theory.

JAMES S. MILNE. Elliptic Curves, 2006 electronic version.

Both books can be downloaded for free from author's personal webpage.

Teaching methods

Class lectures and exercise classes (48 hours).

Assessment methods

Oral exam. There is also one mandatory assignment (pass/not pass), which the student need to pass in order to be eligible for the oral exam. The grade is decided only on the basis of the oral exam.

Teaching tools

Professor's lecture notes.

Office hours

See the website of Lars Halvard Halle