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Gauss in Disquisitiones Arithmeticae has a long section on binary quadratic forms dealing with reduction and equivalence. This seems to do more than the other texts I have to hand, but (in Arthur A Clarke's translation, at least) seems to labour over things which would be done differently now.

JH Conway in "The Sensual (quadratic) Form" refers to the fact that he does not cover "Gauss's group of binary forms under composition" (preface p viii). Now this seems to be the core of the heavy lifting in the Disquisitiones (together with an theory of neighbouring forms, which Conway seems to have rewritten in a much more accessible way). Gauss also has an analysis of forms which are contained in others (where the determinant of a transformation of variables is not $\pm 1$), and these ideas seem to be absent from the texts I have to hand.

[On the other hand Gauss seems to deal almost in passing with Pell's equation as a necessary part of the discussion, and units in quadratic fields too]

Bits of Gauss seem to appear in many different places. But I haven't seen a source which does the whole job.

So this is a request for references for modern treatments of as much of Gauss as possible. Really I'm looking for works which replace what Gauss was doing in modern language. I don't mind if they are technical, but Gauss in the Disquisitiones had a preference for elementary methods where possible - maybe that is why it is the way to is. But elementary treatments would be very welcome.

[My other question on the Disquisitiones is whether anyone has ever tried to replicate the contents of the elusive chapter VIII].

So, references please for modern treatments of the Disquisitiones on quadratic forms.

Note that this is a reference request - there is much useful material, including references, in the comments, which would be valid in answers. Thanks to all who have commented.

Mark Bennet
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  • I don't know if these are exact matches, but here may be something. Will Jagy wrote a bit more for the now extinct MSE blog. – Jyrki Lahtonen Sep 01 '19 at 20:48
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    But, I would myself appreciate a reference to a text describing the connection between quadratic forms of a given discriminant and the ideal classes of the related quadratic field. The related questions/answers often go a step deeper. – Jyrki Lahtonen Sep 01 '19 at 20:50
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    Before delving into the classical theory it is helpful to be familiar with the translation into modern ideal-theoretic form, e.g. see here. The classical theory can be found in G. B. Mathews Theory of Numbers 1896, or in Buell's book on binary quadratic forms. There is also much of interest in many papers by Dan Shanks (esp. algorithmic matters). – Bill Dubuque Sep 01 '19 at 20:58
  • @BillDubuque Thank you for that. Buell looks pretty much along the lines I was looking for. I'm fairly familiar with the ideal/class group form from many years ago. Following the trail of references gives plenty to think about. – Mark Bennet Sep 01 '19 at 21:35
  • @JyrkiLahtonen That connection looks to be key. What seems to happen in modern treatments is that some of the questions Gauss was answering in explicit ways are no longer the main issues for the theory, and so get left out of modern treatments - so having taken the Disquisitiones on holiday with me, I started trying to translate, and found myself hoping that someone else had done the job. I should add that when I first studied class numbers and ideal class groups, the connection with quadratic forms was not mentioned (or so briefly that I missed it). – Mark Bennet Sep 01 '19 at 21:42
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    @Mark You can also find much more on the modern viewpoint in Lemmermeyer's Binary Quadratic Forms: An Elementary Approach to the Arithmetic of Elliptic and Hyperelliptic Curves – Bill Dubuque Sep 01 '19 at 21:45
  • Worth emphasis also is that Franz Lemmermeyer's book has many helpful references to related literature. Highly recommended. – Bill Dubuque Sep 01 '19 at 21:54
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    Mark and @Jyrki , there is a new book where the correspondence is, well, the whole book. https://bookstore.ams.org/dol-52/ Also inexpensive, don't know about shipping outside the U.S. – Will Jagy Sep 01 '19 at 22:13
  • As far as I. know, the classical development abandoned Gauss’s approach relatively early on in favor of other ways of thinking about these issues, such as ideals. However, it is also my understanding that somewhat recently Gauss’s approach was re-visited and expanded, and formed the basis of some of the work of Manjul Barghava. I remember him giving at talk at a Joint Meeting some years back in which he said he visualized the composition of forms as akin to a Rubik’s cube operations, and used it to generalize and obtain new results. You may want to look up his work. – Arturo Magidin Sep 01 '19 at 22:21
  • @Arturo The Barghava cubes composition is presented in Lemermeyer's book (see above), along with its relationship to the classical approaches. AFAIK, this is the best place to learn these topics nowadays. – Bill Dubuque Sep 02 '19 at 00:34
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    As for "Section VIII" of Disq. Arith., a good starting point is: G. Frei, The Unpublished Section Eight: On the Way to Function Fields over a Finite Field, pp. 159--198 in "The Shaping of Arithmetic after C. F. Gauss's Disquisitiones Arithmeticae" (C. Goldstein, N. Schappacher, J. Schwermer ed.), Springer, 2007. See also the MO question Origins of functional field arithmetic. – Bill Dubuque Sep 02 '19 at 16:02
  • @BillDubuque You could consolidate your comments as an answer. I have found them (and other well-informed comments) very helpful. – Mark Bennet Sep 02 '19 at 18:54
  • @Mark I'm glad to hear that my comments are helpful, but I think you deserve much better for an answer (alas, currently my time here is greatly constrained due to external factors so it is unlikely I'll have time to elaborate). – Bill Dubuque Sep 02 '19 at 22:53

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This is maybe not a reference in the sense you want, but my answer here might interest you (the second one down):

What's a BETTER way to see the Gauss's composition law for binary quadratic forms?

For 50 years after Gauss published his masterpiece, progress in developing the ideas on binary quadratic forms was slow. I think, initially, very few people knew of his work and understood it. One of those people was his protégé Dirichlet. There were very few papers touching on the subject until the 1850s. Then Dirichlet published papers in 1851 and 1854, which essentially laid out the modern way to compose forms that has been used every since. It has been streamlined a little, by Arndt in 1857 and by others. And Bhargava's cubes give a magnificent new way to do things, but at its core, the changes initiated by Dirichlet still exist.

In the process of streamlining Gauss's methods, Dirichlet drastically cut the generality of the methods. When you say Gauss "seems to do more than other texts," that is maybe an understatement. I think he did a lot more. What people now usually call "composition" of forms is really just composition of equivalence classes of forms of the same discriminant. Gauss showed how to compose individual pairs of forms, and how to compose forms even of different discriminants. I think there is no modern treatment that touches on this level of generality, even indirectly. And I do wonder from time to time if something was lost by removing that level of generality.

So as such, I'm not sure there's a treatment in "modern language" of the stuff in Gauss that essentially got cut out of subsequent treatments starting over 160 years ago. But at least the answer I linked talks about "SL_2(Z) actions."

The theory of infrastructure of binary quadratic forms involves composing forms, rather than just their classes. But this has been mostly used for computational purposes. I haven't seen a version of this that connects with Gauss's exposition.

I enjoy greatly Lemmemeyer's work, already mentioned in the comments, partly because for the reduction theory, he turns away from Gauss and uses the much less-known reduction theory devised by Zagier in Zetafunktionen und Quadratische Körper. I much prefer this version of reduction from a theoretical standpoint, although it is usually less efficient computationally than Gauss's theory. Zagier's book is really cool.

Barry Smith
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