Primes of the form \( x^2 + ny^2 \) - Summer Semester 2017 - Tübingen

Contact

Office: C5A38.

Office hour: Thursdays 14:00-15:00, or by appointment.

Email: charlton(at)math(dot)uni-tuebingen(dot)de

Overview

A lecture course about finding conditions on which primes are represented by a given quaratic form. We want to investigate and generalise Fermat's conjecture that \( p = x^2 + y^2 \) if and only if \( p = 2 \) or \( p \equiv 1 \pmod{4} \).

Lecture notes and extra stuff

A preliminary version of the notes covering the lectures so far is availabe. Please let me know of any typos or mistakes in the notes!

Lecture outline

Lecture 1 (19/04/17): Introduction/motivation. We discuss Fermat's conjectures and the various generalisations we will study throughout the semester.
Lecture 2 (26/04/17): We give Euler's proof of Fermat's two-squares theorem. We discuss the two main steps: reciprocity and descent. We present the steps for Fermat's \( x^2 + 2y^2 \) and \( x^2 + 3y^2 \) theorems, via a sequence of exercises.
Lecture 3 (03/05/17): We begin to generalise the reciprocity step of Euler's proofs. We start by introducing the quadratic residue symbol \( \left( n / p \right) \) and prove some properties about it.
Lecture 4 (10/05/17): We state and prove the law of Quaratic Reciprocity, which solves the reciprocity step for every \( p \mid x^2 + ny^2 \).
Lecture 5 (17/05/17): We introduce quadratic forms, and some of their properties in preparation for generalising the Descent step.
Lecture 6 (24/05/17): We discuss the properties of \( \operatorname{GL}_n(R) \)-equivalence. We refine this to proper/\( \operatorname{SL}_n(\mathbb{Z}) \)-equivalence for integral forms, and connect this with proper representations. This gives an (implicit) solution to the descent step.
Lecture 7 (31/05/17): We give a procedure for listing the proper-equivalence classes of positive-definite and of indefinite binary quadratic forms. This gives an explicit solution to the descent step.

Lecture 8 (14/06/17): We investigate the class number 1 case, where our methods are already sufficient to give a solution. We begin to introduce genus theory to solve some further cases.
Lecture 9 (21/06/17): We continue the discussion of genus theory, with a larger example. We then begin to study the the composition of binary quadratic forms, and show that this composition gives a the equivalence classes the structure of an abelian group.
Lecture 10 (28/06/17): We finish the discussion of composition of binary quadratic forms, and start to use this to investigate some deeper properties of genus theory.
Lecture 11 (05/07/17): We establish further properties of the class group, and the consequences this has for genus theory.
Lecture 12 (12/07/17): Final exam 14:00 - 16:00.
Lecture 13 (19/07/17): We start to introduce some of the more advanced techniques for studying \( p = x^2 + ny^2 \). We look at using cubic reciprocity to solve \( p = x^2 + 27y^2 \). (NON-EXAMINABLE)
Lecture 14 (26/07/17): We discuss modular forms, and see how they can help study the number of representations of an integer by quadratic forms in \( n \)-variables. (NON-EXAMINABLE)

See Chapter 1, Section 4.3 (page 41) in The 1-2-3 of Modular Forms. for Zagier's example of modular forms giving conditions for binary quadratic forms of discriminant \( D = -23 \).

Problem sheets/exercises

Sheet 1:
- Some open ended questions to get a feeling for the type of results we study.
- Sheet 1 solutions.
Sheet 2:
- Proving Fermat's \( x^2 + 2y^2 \) and \( x^2 + 3y^2 \) theorems
- Sheet 2 solutions.
Sheet 3:
- Computations and applications involving quadratic reciprocity.
- Sheet 3 solutions.
Sheet 4:
- Properties of quadratic forms.
- Sheet 4 solutions.
Sheet 5:
- Class number, and reduction of quadratic forms.
- Sheet 5 solutions.
Sheet 6:
- Class number 1, and elementary genus theory.
- Sheet 6 solutions.
Sheet 7:
- Composition of quadratic forms.
- Sheet 7 solutions.
Sheet 8:
- Genus theory, revisted .
- Sheet 8 solutions.
Sheet 9:
- Cubic reciprocity, and modular forms.
- Sheet 9 solutions.

Handouts and online resources

Handout 1: An explanation of Euler's proof of reciprocity for \( p \mid x^2 + y^2 \), using finite differences .

Handout 2: The connection between binary quadratic forms, and ideals in quadratic number fields .

Handout 3: The Jacobi symbol, and lemma relating to \( p \mid x^2 + ny^2 \).

Exam/assessment information

For those talking the course for credit, a ~2 hour final exam will be available on 12 July. Exam questions will be (largely) taken from the Problem Sheet 2--6, or slight variations. Do look at the posted solutions!

English-German/English-Spanish/... dictionaries ARE permitted.

Basic scientific calculators (i.e. not programmable) ARE permitted.

Exam focus:

Chapter 2. Look at solutions to sheet 2
Section 3.3 and 3.4 - no proofs. Look at solutions to sheet 3
Section 4.4
Chapter 6. Look at oslutions to sheet 6

Exam guidelines

Material from Chapter 2 - Chapter 6 only
Chapter 2:
- Proofs with guidance
Chapter 3:
- definition and properties of the Legendre symbol \( (a/p) \).
- statement of quadratic reciprocity, and supplements (proofs not required)
- Solving \( (a/p) = 1 \) as a congruence \( p \equiv n_1, \ldots, n_k \pmod{N} \).
- Solution to the Reciprocity step.
Chapter 4:
- Definitions about quadratic forms (only versions for integral binary), including primitive, integral, positive-definite/indefinite, disciminant, (proper) equivalence, (proper) representatoin, ...
- Properties of equivalence of quadratic forms, including relation between discrminants, representing same integers, ...
- Relationship between proper equivalence, and proper representation
- Condition sone when an integer is represented by some BQF
- Soution to the Descent step.
Chapter 5: Reduction in the positive-definite case only!
- Definition of a reduced form
- Statement that every positive-definite form is equivalence to a unique reduced form (including idea of the proof of existence, but not uniqueness)
- Bounds on \( a \) in terms of discriminant \( D \), and finiteness of the class number
- Reduction, up to discriminant \( -D < 108 \), say. (With optimisation, the table has < 20 rows)
- SKIP SECTION 5.2 - INDEFINITE REDUCTION
- SKIP SECTION 5.3 - FINITENESS OF THE CLASS NUMBER IN GENERAL
Chapter 6

Definition of `principal forms'.
Principal form represents a subgroup \( H \subset \ker \chi \subset (\mathbb{Z}/D\mathbb{Z})^\ast \). (No proof.)
Other forms represent cosets of \( H \) in \( \ker \chi \). (No proof.)
Definition of a genus.
Theorem of genus theory (Thm 16.3) to find congruence conditions, when \( n < 27 \). (In the cases where genus theory does work, meaning when each genus contains 1 form. Say for \( p = x^2 + 21y^2 \).)
Idea of proof of why genus theory works. (Idea is in Example 6.6, for \( p = x^2 + 5y^2 \). Important idea is that there is 1 form in each genus.)

Literature

Main textbook for the course. Covers all the essential topics in detail.

David A. Cox. "Primes of the form \( x^2 + ny^2 \): Fermat, class field theory, and complex multiplication". Vol. 34. John Wiley & Sons, 2011.

Other books which present some of the material in different ways (more general, alternative viewpoins, ...). I will try to indicate which sections are appropriate, when we cover the relevant material.

John Horton Conway, and Francis YC Fung. "The sensual (quadratic) form". No. 26. MAA, 1997.

John William Scott Cassels. "Rational quadratic forms". Courier Dover Publications, 2008.

Don Bernard Zagier. "Zetafunktionen und quadratische Körper: eine Einführung in die höhere Zahlentheorie". Springer-Verlag, 2013.