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Integral Binary Quadratic Forms

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Integral Binary Quadratic Forms INTEGRAL BINARY QUADRATIC FORMS UTAH SUMMER REU JUNE JUNE 2 2 An integral binary quadratic form is a p olynomial ax bxy cy where a b and 2 2 c are integers 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 group 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 discriminant 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 Manjul Bhargavas Ph D thesis where the Gauss law is dened using integer 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 mathematics 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 ideal class group 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.
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