Algebraic Number Theory
A.Y. 2025/2026
Learning objectives
The course provides standard results in algebraic number theory, hence introduce L-functions and their arithmetic relevance.
Expected learning outcomes
Learning the basic results in Algebraic Number Theory. Ability of computing the class groups and the group of units of a number field. Acquire familiarity with L-functions and other more advanced topics.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Course program (by week):
1. Number fields and global fields, rings of integers, ideal factorization (and failure of unique factorization of elements), decomposition of prime ideals, class groups, local fields, ramification theory.
2. Analysis of Galois groups of local fields, decomposition and inertia subgroups, description of abelian extensions of local fields (i.e. local class field theory). This latter subject will involve introducing the concept of formal groups and Lubin-Tate formal modules, which will carry over into the next week.
3. Continuation of formal groups and Lubin-Tate formal modules, statement and proof of the main theorem of local class field theory. This will entail discussion of group cohomology and the Brauer group.
4. We will then return to global fields and the problem of classifying their abelian extensions. Introduction to adèles and idèles, and the main statement of global class field theory. We will illustrate the problem of explicit class field theory, i.e. Kronecker's "Jungendtraum", and draw parallels between explicit local class field theory (provided by Lubin-Tate formal modules) and explicit class field theory over cyclotomic fields (Kronecker-Weber theorem), whose proof we will discuss.
5. We will then undertake the problem of explicit class field theory over imaginary quadratic fields. This will involve introducing elliptic curves (both as a projective planar equation equipped with group law, and as a group variety) and the notion of complex multiplication (CM). The parallel between CM elliptic curves and Lubin-Tate formal modules, and how they generate abelian extensions, will be discussed.
6. The proof of the main theorem of complex multiplication will be discussed, namely how the Galois action on torsion points of CM elliptic curves can be described in terms of the CM endomorphisms. Finally we will return to global class field theory, showing how the theory of CM makes a special case over imaginary quadratic fields explicit.
7. We will discuss aspects of the proof of global class theory and related results such as the Cebotarev density theorem. We will also introduce the notions of Hecke and Dedekind L-functions, their initial definitions as Euler product and discuss their analytic continuation.
8. We will discuss the proof of the class number formula for Dedekind L-functions. We will then return to elliptic curves and discuss the study of its integral and rational points. The group of rational points, the "Mordell-Weil group" will be introduced, as well as how to approximate this group via Galois cohomology (the Selmer group). We will discuss the proof of the Mordell-Weil theorem.
9. We then define the L-function of an elliptic curve, and analogies between its special value formulas and the class number formula. We will briefly discuss the prospect of analytically continuing this L-function and why this is a deep question. In the case of a CM elliptic curve, we will discuss the relation between the L-function and the Hecke L-function of a character arising from our previous discussion on explicit class field theory for the CM imaginary quadratic field. Discussion of these topics will likely continue into the next week.
10. We will continue the discussion of elliptic curves and their arithmetic, in particular a certain generating series called a "modular form" attached to an elliptic curve and its role in analytically continuing the L-function. We will discuss (in very broad terms) the connection of this question to the modularity theorem of Wiles and Taylor-Wiles, briefly touching on modern approaches to questions in arithmetic such as the Birch and Swinnerton-Dyer conjecture.
11. Continuation of the discussion on the Birch and Swinnerton-Dyer conjecture.
1. Number fields and global fields, rings of integers, ideal factorization (and failure of unique factorization of elements), decomposition of prime ideals, class groups, local fields, ramification theory.
2. Analysis of Galois groups of local fields, decomposition and inertia subgroups, description of abelian extensions of local fields (i.e. local class field theory). This latter subject will involve introducing the concept of formal groups and Lubin-Tate formal modules, which will carry over into the next week.
3. Continuation of formal groups and Lubin-Tate formal modules, statement and proof of the main theorem of local class field theory. This will entail discussion of group cohomology and the Brauer group.
4. We will then return to global fields and the problem of classifying their abelian extensions. Introduction to adèles and idèles, and the main statement of global class field theory. We will illustrate the problem of explicit class field theory, i.e. Kronecker's "Jungendtraum", and draw parallels between explicit local class field theory (provided by Lubin-Tate formal modules) and explicit class field theory over cyclotomic fields (Kronecker-Weber theorem), whose proof we will discuss.
5. We will then undertake the problem of explicit class field theory over imaginary quadratic fields. This will involve introducing elliptic curves (both as a projective planar equation equipped with group law, and as a group variety) and the notion of complex multiplication (CM). The parallel between CM elliptic curves and Lubin-Tate formal modules, and how they generate abelian extensions, will be discussed.
6. The proof of the main theorem of complex multiplication will be discussed, namely how the Galois action on torsion points of CM elliptic curves can be described in terms of the CM endomorphisms. Finally we will return to global class field theory, showing how the theory of CM makes a special case over imaginary quadratic fields explicit.
7. We will discuss aspects of the proof of global class theory and related results such as the Cebotarev density theorem. We will also introduce the notions of Hecke and Dedekind L-functions, their initial definitions as Euler product and discuss their analytic continuation.
8. We will discuss the proof of the class number formula for Dedekind L-functions. We will then return to elliptic curves and discuss the study of its integral and rational points. The group of rational points, the "Mordell-Weil group" will be introduced, as well as how to approximate this group via Galois cohomology (the Selmer group). We will discuss the proof of the Mordell-Weil theorem.
9. We then define the L-function of an elliptic curve, and analogies between its special value formulas and the class number formula. We will briefly discuss the prospect of analytically continuing this L-function and why this is a deep question. In the case of a CM elliptic curve, we will discuss the relation between the L-function and the Hecke L-function of a character arising from our previous discussion on explicit class field theory for the CM imaginary quadratic field. Discussion of these topics will likely continue into the next week.
10. We will continue the discussion of elliptic curves and their arithmetic, in particular a certain generating series called a "modular form" attached to an elliptic curve and its role in analytically continuing the L-function. We will discuss (in very broad terms) the connection of this question to the modularity theorem of Wiles and Taylor-Wiles, briefly touching on modern approaches to questions in arithmetic such as the Birch and Swinnerton-Dyer conjecture.
11. Continuation of the discussion on the Birch and Swinnerton-Dyer conjecture.
Prerequisites for admission
It will be assumed that you have a good knowledge of the following subjects.
1. Group theory, particularly the theory of normal and p-Sylow subgroups, the structure theory of finitely generated abelian groups and character theory will be assumed. It is also helpful to be familiar with the concept of representations of groups.
2. The theory of commutative rings and modules, field and ring extensions as well as prime ideals. Knowing the concept and existence of integral and algebraic closures will also be assumed, as well as Familiarity with valuation rings and Dedekind domains will also be helpful, although these concepts will be reviewed in the first part of the course.
3. The notion of the p-adic numbers and completions of rings (particularly at primes ideals) will be crucially used throughout the course. It is good to have a familiarity with the p-adic valuation and the notion of a nonarchimedean metric. In this vein, having a familiarity with complete local rings and semilocal rings is highly recommended.
4. Galois theory, including the Galois correspondence and the explicit theory of cyclotomic extensions. It is recommended to be very familiar with the theory of polynomials, their splitting fields, the concept of minimal and characteristic polynomials, the Cayley-Hamilton theorem, and criteria for irreducibility such as Eisenstein's criterion. In particular, knowing the irreducibility of the p^mth cyclotomic polynomial (and its consequences in Galois theory, e.g. the structure of Gal(Q(\mu_N)/Q)) is highly recommended.
5. Kummer theory and basic results on Galois cohomology such as Hilbert's Theorem 90. Knowing the explicit descriptions of first group cohomology will also be helpful, as well as some familiarity with the description of second group cohomology.
6. A good understanding of complex analysis, particularly the concepts of meromorphic functions, Cauchy's residue theorem, contour integration and analytic continuation. Basic results from Fourier and harmonic analysis such as the existence of Fourier expansions, Fourier duality and Poisson summation, will also be helpful.
7. Basic algebraic geometry, such as the notion of a ring spectrum and an algebraic variety, will be assumed. It is recommended to understand Spec(Z[x]), for example. Knowing what a regular morphism of algebraic varieties is also highly recommended. It will also be useful to be familiar with rings of power series.
The results we will use will be briefly reviewed during the course, but elaborations on their proofs will not be given unless explicitly needed.
1. Group theory, particularly the theory of normal and p-Sylow subgroups, the structure theory of finitely generated abelian groups and character theory will be assumed. It is also helpful to be familiar with the concept of representations of groups.
2. The theory of commutative rings and modules, field and ring extensions as well as prime ideals. Knowing the concept and existence of integral and algebraic closures will also be assumed, as well as Familiarity with valuation rings and Dedekind domains will also be helpful, although these concepts will be reviewed in the first part of the course.
3. The notion of the p-adic numbers and completions of rings (particularly at primes ideals) will be crucially used throughout the course. It is good to have a familiarity with the p-adic valuation and the notion of a nonarchimedean metric. In this vein, having a familiarity with complete local rings and semilocal rings is highly recommended.
4. Galois theory, including the Galois correspondence and the explicit theory of cyclotomic extensions. It is recommended to be very familiar with the theory of polynomials, their splitting fields, the concept of minimal and characteristic polynomials, the Cayley-Hamilton theorem, and criteria for irreducibility such as Eisenstein's criterion. In particular, knowing the irreducibility of the p^mth cyclotomic polynomial (and its consequences in Galois theory, e.g. the structure of Gal(Q(\mu_N)/Q)) is highly recommended.
5. Kummer theory and basic results on Galois cohomology such as Hilbert's Theorem 90. Knowing the explicit descriptions of first group cohomology will also be helpful, as well as some familiarity with the description of second group cohomology.
6. A good understanding of complex analysis, particularly the concepts of meromorphic functions, Cauchy's residue theorem, contour integration and analytic continuation. Basic results from Fourier and harmonic analysis such as the existence of Fourier expansions, Fourier duality and Poisson summation, will also be helpful.
7. Basic algebraic geometry, such as the notion of a ring spectrum and an algebraic variety, will be assumed. It is recommended to understand Spec(Z[x]), for example. Knowing what a regular morphism of algebraic varieties is also highly recommended. It will also be useful to be familiar with rings of power series.
The results we will use will be briefly reviewed during the course, but elaborations on their proofs will not be given unless explicitly needed.
Teaching methods
Course material will be primarily taught in regular (roughly biweekly) lectures. Suggested exercises will be assigned during the course (particularly for working out concrete examples to supplement examples given in class, and in working out details of proofs given in class). Collaboration and group work will be highly encouraged in solving these suggested exercises as well as in reviewing the course material and in preparing for the final seminar presentation. Attending regular weekly office hours is also highly encouraged.
Teaching Resources
Algebraic Number Theory, edited by J.W.S. Cassels and A. Fröhlich
Published by the London Mathematical Society
ISBN-10: 0950273422, ISBN-13: 978-0950273426
Algebraic Number Theory, by J. Neukirch, Springer, Berlin, Heidelberg. Springer (1999).
Class Field Theory, notes by J. Milne, https://www.jmilne.org/math/CourseNotes/CFT.pdf.
"The Arithmetic of Elliptic Curves", J.H. Silverman. Graduate Texts in Mathematics, Vol. 106. Springer, New York. (2009).
Published by the London Mathematical Society
ISBN-10: 0950273422, ISBN-13: 978-0950273426
Algebraic Number Theory, by J. Neukirch, Springer, Berlin, Heidelberg. Springer (1999).
Class Field Theory, notes by J. Milne, https://www.jmilne.org/math/CourseNotes/CFT.pdf.
"The Arithmetic of Elliptic Curves", J.H. Silverman. Graduate Texts in Mathematics, Vol. 106. Springer, New York. (2009).
Assessment methods and Criteria
There will be a final seminar-style presentation on an advanced topic of your choice (from a given list of advanced topics in number theory) at the end of the semester which will determine the grade for the course. However, during this seminar talk you will be asked questions pertaining to the subject and to various aspects of the course, including recalling aspects of the course (such as outlines of proofs of important theorems and solutions to suggested exercises).
Professor(s)
Reception:
Fridays from 10:30 to 12:30, please fix appointment beforehand via email.
Department of Mathematics - Studio 2092