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Prof M.Bhargava's work on "Higher Composition Law" which solved some outstanding conjectures on number theory seems to be very interesting topic. I have seen his papers but, in spite of the titles, it is not easy to understand (Of course in my point of view, for sure there are many people who can understand it easily).

Do you know any lecture note or expository paper which explains more details and some explicit example? especially his work on composition law for binary quadratic form.

Thanks

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  • $\begingroup$ Have you looked at his ICM notes or notes in this Algorithmic Number Theory Volume? His composition law for binary quadratic forms is of course the same as Gauss's. However, one new thing was a composition law on triples of binary quadratic forms. $\endgroup$
    – Kimball
    Commented Nov 8, 2010 at 14:08

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I have lecture notes that I'd like to turn into a book one day. I have not yet had time to adapt to my new TeX system, and the drawings done via ps-tricks do not yet come out as planned. In addition, there are gaps and mistakes that I have not yet had time to fill and correct (the chapter on composition should be essentially correct and complete, however). In any case, the present set of notes, for the time being, can be found here. Comments and corrections are welcome.

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  • $\begingroup$ What you mean by "Give me a day or so and I will post a link to the relevant sections."? Is it possible to have your notes? $\endgroup$
    – M.B
    Commented Nov 8, 2010 at 14:04
  • $\begingroup$ @ Franz: Thanks for your note. It is great. It seems that in chapter "Bhargava Cube" you have developed only "Gauss Composition", of course with respect of "Bhargava Cube". $\endgroup$
    – M.B
    Commented Nov 9, 2010 at 4:26
  • $\begingroup$ M.B, what do you mean by "only Gauss composition"? $\endgroup$
    – Franz Lemmermeyer
    Commented Nov 9, 2010 at 5:59
  • $\begingroup$ @Franz: I think according to Theorem 1 of Bhargava's paper "Higher composition laws I" if $Q_{id,D}$ be any primitive binary quadratic form of discriminant $D$ such that there is a cube $A_0$ with $Q^{A0}_1= Q^{A0}_2=Q^{A0}_3=Q_{id,D}$ then there is a unique group law. For an specific $Q_{id,D}$ one can get usual Guass Composition Law. It seems that you have picked this specific case. I might be confused. $\endgroup$
    – M.B
    Commented Nov 9, 2010 at 7:03
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    $\begingroup$ Given any group G and an element g in G, you can define a group law on G with neutral element g by demanding that ab = a+b-g. You can do the same on an elliptic curve, where you choose a flex point as your neutral element in order to make the group law as simple as possible. But this is just playing around with *the group law, and you gain nothing by trying to be as general as possible. $\endgroup$
    – Franz Lemmermeyer
    Commented Nov 9, 2010 at 11:33
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Try the Bourbaki talk given by Karim Belabas a couple of years ago :

Paramétrisation de structures algébriques et densité de discriminants [d'après Bhargava] Astérisque, Vol. 299 (2005), Exp. No. 935, pp. 267-299, Séminaire Bourbaki. Vol. 2003/2004,MR 2167210.

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