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. 



Diophantine theory of homogeneous cubic polynomials
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.


