INTEGRAL BINARY QUADRATIC FORMS

UTAH SUMMER REU JUNE JUNE

2 2

An integral binary is a p olynomial ax bxy cy where a b and

2 2

c are One of the simplest quadratic forms that comes to mind is x y

A classical problem in numb er theory is to describ e which integers are sums of two

squares More generally one can ask what are integral values of any quadratic form

The aim of this workshop is to intro duce binary quadratic forms following a b o ok

The Sensual Quadratic Form by Conway and then go on to establish some deep

prop erties such as the Gauss law and a connection with continued fractions

More precisely we intend to cover the following topics

 Integral values In his b o ok Conway develops a metho d to visualize values of

a quadratic binary form by intro ducing certain top ographic ob jects rivers

lakes etc

 Gauss law In Gauss dened a comp osition law of binary quadratic forms

2

of a xed D b ac In a mo dern language this group law

corresp onds to multiplication of ideals in the quadratic eld of discriminant D

We shall take an approach to this topic from s Ph D

thesis where the Gauss law is dened using   cub es

 Reduction theory Two quadratic forms can dier only by a change of co ordi

nates Such quadratic forms are called equivalent We shall show that there

are only nitely many equivalence classes of binary quadratic forms of a xed

discriminant D If D the equivalence classes can b e interpreted as cycles

of purely p erio dic continued fractions This interpretation allows us to make

some unexp ected discoveries

Prerequisites for this course are rather minimal as Bhargava thesis is written in ele

mentary language early th century Our main to ol will b e rowcolumn

reduction In particular linear algebra and some familiarity with groups Zmo dules

rings ideals and elds are required Galois theory is not needed Topics to b e covered

I The rst part of this course will deal with some well known results in the theory

of quadratic elds orders ideals and the The relation with quadratic

binary forms and the Gauss law Reduction theory of quadratic forms as an eective

to ol to calculate the ideal class group A rough syllabus for this part is

Zmo dules

Quadratic elds over Q orders mo dules and class numb ers

UTAH SUMMER REU JUNE JUNE

Binary quadratic forms

Reduction theory and calculating class numb ers

 Upp er half plane in the case of complex elds

 Continued fractions in the case of real elds

I I The second part will b e based on Bhargavas thesis We intend to cover several

of his comp osition laws such as comp osition of   integer cub es and comp osition

of binary cubic forms Cubic rings and their parameterization by cubic binary forms

The law of comp osition of   integer b oxes and its relation to the ideal class

group of a cubic ring

I I I The third part will explore connections of Bhargavas comp osition laws with

exceptional Lie groups intro duction to ro ot systems through the example of sln

Quick construction of simple split Lie algebras and corresp onding Chevalley groups

In essence rowcolumn manipulations again