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Arithmetic of quadratic forms over $\mathbb{Z}$ (or lattices theory) has received much attention and there are many applications in broad area of mathematics (such as intersection forms on fourfolds). I now wonder whether or not a similar theory for cubic forms can be developed. I recently found a beautiful theorem about binary cubic forms:

Theorem (B. N. Delone and D. K. Faddeev,W.-T. Gan, B. H. Gross, and G. Savin) There is a canonical bijection between isomorphism classes of cubic rings and the set of $GL_{2}(\mathbb{Z})$-equivalence classes of integral binary cubic forms. Under this bijection, the discriminant of a cubic ring is equal to the discriminant of the corresponding binary cubic form.

I don't know how useful this theorem is because I don't know how difficult to classify cubic rings. My question is, are there any classification theory when the number of variable is small? I would appreciate it if anyone could give me a reference for recent development of cubic forms.

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Although this only indirectly related to your question, you may be interested in Bhargava's ICM proceedings article on higher composition laws: – Frank Thorne Jul 31 '12 at 5:19
Thanks! I will take a look at it. – user2013 Jul 31 '12 at 6:00
Long ago Manin wrote a famous book on Cubic Forms which deals with such issues as obstructions to the local-global principle for the existence of rational points on cubic surfaces. – Chandan Singh Dalawat Jul 31 '12 at 6:11
A cubic ring is one that is isomorphic to $\mathbf{Z}^3$ as a module. – Rob Harron Jul 31 '12 at 13:31
This may be of your interest. P.M.H.Wilson initiated the study of Calabi-Yau threefolds via Diophantine geometry of the cubic forms defined on second cohomologies (mod torsion). For example, he uses H.Davenport's work on arithmetic of cubic forms in his paper "Calabi-Yau manifolds with large Picard number" – Atsushi Kanazawa Jul 31 '12 at 19:49
up vote 7 down vote accepted

Let me begin with a historical comment. The correspondence between cubic rings and binary cubic forms that you mentioned is not due to Delone and Faddeev, but rather to F. Levi (Kubische Zahlkörper und binäre kubische Formenklassen, Leipz. Ber. 66, 26-37 (1914); this article presents the results of Levi's thesis, which was supervised by Weber in 1911). Actually Delone and Faddeev credit Levi in their book not in the chapter where the material is presented but in the preface.

BTW, Levi was Jewish and had to emigrate from Germany in 1936, when he went to the University of Calcutta and apparently was elected president of the Indian Mathematical Society for a few years. In 1949 he went to the Tata Institute, and he returned to Germany in 1952.

Now for your question: there is a theory of cubic forms analogous to that of binary quadratic forms, which was developed by Eisenstein but abandoned after the success of Dedekind's ideal theory. The theory deals not with arbitrary cubic forms but only those that can be written as products of three linear factors (decomposable forms), and more specifically norm forms. For a modern account of the arithmetic of binary cubic forms you may want to look at Hoffman and Morales, Arithmetic of binary cubic forms, Enseign. Math. 46 (2000) 61-94.

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I knew that a certain Levi was an early member of the Tata Institute but didn't know that he was fleeing Nazi Germany. Thanks for the information, Franz. – Chandan Singh Dalawat Oct 13 '12 at 3:28
Wonderful answer! – stankewicz Oct 13 '12 at 12:23

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