# Diophantine theory of homogeneous cubic polynomials

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: icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_13.pdf –  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" springerlink.com/content/g0583k343q720095 –  Atsushi Kanazawa Jul 31 '12 at 19:49