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.
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.