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The correspondence between binary quadratic forms and quadratic fields
| Content Provider | Semantic Scholar |
|---|---|
| Author | Nelson, Paul D. |
| Copyright Year | 2016 |
| Abstract | This document is the final report for a master semester project, whose goal was to study in detail the beautiful connection between integral binary quadratic forms and quadratic fields, along with its uses in these two settings. To do so, we begin by studying integral binary quadratic forms and the number-theoretic questions associated: which integers are represented by a given form/set of forms, how many representation does an integer admit by a given form/set of forms and so on. Doing so, we study in depth the equivalence of forms and the theories of reduction of definite and indefinite forms developed by Gauss. Class numbers of quadratic forms and their links with representation questions are introduced. Then, we recall some results about orders in the rings of integers of quadratic fields and we show how (classes) of forms can be associated to (classes) of ideals in such orders and vice-versa. Combining these results, we give the precise correspondence between classes of forms and narrow Picard groups of orders in quadratic fields. In the third chapter, we work on the correspondence obtained to transpose and answer questions from one setting to the other. The composition law on binary quadratic forms discovered by Gauss is derived from the group structure of Picard groups using the correspondence. We introduce how Manjul Bhargava recovered Gauss composition law in an elementary manner and how he generalized it to higher order spaces of forms. We show how to determine Picard groups of orders in quadratic fields (in particular ideal class groups and class numbers) very easily from the perspective of forms. These ideas are used to give tables summing up the correspondence for the first form discriminants. Working in the context of forms, we also present a proof of the class number one problem for even negative discriminants. Then, we study units in orders of rings of integers of quadratic fields, automorphisms of forms and show how they are closely related. We count the number of representations of an integer by the set of classes of forms of a given discriminant in two ways: working in quadratic fields thanks to the correspondence and without the latter, working in the point of view of forms. Doing so, we illustrate how insightful the correspondence is. After obtaining a closed formula for the number of representations of an integer by binary quadratic forms of given discriminant, we derive the Dirichlet class number formula from it, using an estimation by a L-series and a lattice point counting argument, again using the two settings of binary quadratic forms and quadratic fields. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://corentinperretgentil.gitlab.io/static/documents/correspondence-bqf-qf.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
