# Quod Erat Demonstrandum

## 2008/02/27

### To senior form mathematics students

Filed under: Information — johnmayhk @ 4:15 下午

Mathematics Curricula
http://www.edb.gov.hk/index.aspx?nodeid=2899&langno=1

Examination Syllabuses
http://www.hkeaa.edu.hk/en/hkale/Subject_and_Syllabuses/

Seminar and Workshop Handouts (For Mathematics Teachers)
http://www.edb.gov.hk/index.aspx?nodeid=4651&langno=1

In the link just above, it is quite interesting to find the NSS (New senior secondary 新高中) Enriching Knowledge for the Mathematics Curriculum (Advanced Topics in Mathematics) Rainbow in Mathematics, just click and read. If secondary students could come across and understand topics mentioned in the handout like

Homology
Pascal’s Theorem
Bézout’s Theorem
Cayley-Bacharach Theorem
Elliptic Curve
Mordell-Weil Theorem
Mazur’s Theorem
Birch and Swinnerton-Dyer conjecture

in NSS, then HK secondary mathematics education will take the lead over this lonely planet for sure! But there’s a long way to go, at least, I just heard about (not understand) some of the names mentioned in the list above. How poor I am.

## 2 則迴響 »

1. It’s good to know this excellent blog about mathematics. By the way, BSD conjecture = Birch and Swinnerton-Dyer Conjecture. I will probably tell you more about the conjecture next time. It’s one of the most important problems in modern number theory.

Basically, it connects two things that are quite different in nature.

1) Mordell-Weil Group: E(Q), which is a finitely generated abelian group. (thanks to the theorem of Mordell-Weil)
While the torsion part (i.e. the finite part of E(Q)) is completely understood due to a deep and difficult theorem of Mazur. The rank r, of E(Q), remains to be one of the most mysterious objects in number theory. Needless to say, the rank is E(Q) is defined in a totally algebraic way.

2) order of vanishing of the Hasse-Weil L-series, the construction of the L-series depends only on the information about the reduced curve E* (i.e. taking all the coefficients mod p), and then we will get an order of vanishing at 1, say the order of vanishing is r*.

Then the BSD-rank-conjecture says r = r*, this is quite an amazing result.
Partial results concerning the BSD conjecture has been known for curves of rank 0 or 1.

Let me remark one more thing, it is conjectured 100% of the elliptic curves have rank 0 or 1.
Note 100% does not mean all of them, for example, the curve E: y^2 + y = x^3 + x^2 – 2x is a rank 2 curve.

迴響 由 koopa — 2008/04/09 @ 10:14 上午 | 回應

2. Thank you koopa! Teach us more if you do have time! Thank you again!!

迴響 由 johnmayhk — 2008/04/09 @ 12:04 下午 | 回應