Arithmetic Geometry
A.Y. 2023/2024
Learning objectives
Introduce the students to the arithmetic theory of elliptic modular forms and their arithmetic applications, in particular to explicit class field theory (Kronecker's Jugendtraum) and to the Birch and Swinnerton-Dyer conjecture.
Expected learning outcomes
The student will learn fundamental modularity results, at the centre of current research in arithmetic geometry.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
The course aims at introducing the audience to the arithmetic theory of elliptic modular forms.
Special emphasis will be put on the theory of complex multiplication of elliptic curves, and on the explicit
construction of generators of abelian extensions of imaginary quadratic fields via special values of
modular functions (cf. Hilbert's 12th Problem e Kronecker's Jugendtraum). The course will also discuss
Wiles's Modularity Theorem (which represents the key ingredient in the proof of Fermat's Last Theorem)
and its application to the arithmetic theory of elliptic curves (cf. Birch and Swinnerton-Dyer Conjecture).
Special emphasis will be put on the theory of complex multiplication of elliptic curves, and on the explicit
construction of generators of abelian extensions of imaginary quadratic fields via special values of
modular functions (cf. Hilbert's 12th Problem e Kronecker's Jugendtraum). The course will also discuss
Wiles's Modularity Theorem (which represents the key ingredient in the proof of Fermat's Last Theorem)
and its application to the arithmetic theory of elliptic curves (cf. Birch and Swinnerton-Dyer Conjecture).
Prerequisites for admission
Basics of Galois Theory and Algebraic Number Theory (cf. Algebra 3).
Basics of Complex Analysis (cf. Analisi Complessa, Prima Parte).
Basics of Complex Analysis (cf. Analisi Complessa, Prima Parte).
Teaching methods
Taught classes.
Teaching Resources
Lecture notes will be made available. In addition, the student is invited to consult the following references.
Shimura, Goro: Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press.
Lang, Serge: Elliptic functions, Springer.
Diamond, Fred and Im, John: Modular Forms and Modular Curves, in Seminar on Fermat's Last Theorem.
Silverman: The Arithmetic of Elliptic Curves, Springer.
Shimura, Goro: Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press.
Lang, Serge: Elliptic functions, Springer.
Diamond, Fred and Im, John: Modular Forms and Modular Curves, in Seminar on Fermat's Last Theorem.
Silverman: The Arithmetic of Elliptic Curves, Springer.
Assessment methods and Criteria
Homeworks and a seminar on a topic assigned by the lecturer.
Educational website(s)
Professor(s)