Binary quadratic forms

Annals of Mathematics In our first article [2] we developed a new view of Gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms. Moreover, we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number fields, while the spaces underlying those composition laws were closely related to certain exceptional Lie groups.

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Two centuries ago, in his celebrated work Disquisitiones Arithmeticae of 1801, Gauss laid down the beautiful law of composition of integral binary quadratic forms which would play such a critical role in number theory in the decades to follow. Even today, two centuries later, this law of composition still remains one of the primary tools for understanding and computing with the class groups of quadratic orders. It is hence only natural to ask whether higher analogues of this composition law exist that could shed light on the structure of other algebraic number rings and fields. ...

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