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

Contact

Office: C5A38.

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

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

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.

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• 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

• Find the class number and reduced forms of discriminant \( D < 0 \).
• Find the class number and representatives of equivalence classes of indefinite forms of discriminan \( D > 0 \).
• Tables of ternary quadratic forms (odd).
• Tables of ternary quadratic forms (even).
• Tables of quaternary quadratic forms.
• Compute the composition of indefinite quadratic forms.
• Compute the composition of positive-definite quadratic forms.

• 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.