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Shimura-Taniyama-Weil Conjecture (Modularity Theorem)

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Shimura and Taniyama are two Japanese mathematicians first put up the conjecture in 1955, later the French mathematician André Weil re-discovered it in 1967.

The British Andrew Wiles proved the conjecture and used this theorem to prove the 380-year-old Fermat’s Last Theorem (FLT) in 1994.

It is concerning the study of these strange curves called Elliptic Curve with 2 variables cubic equation:

Example:
\boxed {y^{2} + y = x^{3} - x^{2} } — (I)

There are many solutions in integers N, real R or complex C numbers, but solutions in modulo N hide the most beautiful gem in Mathematics.

For modulo 5, the above equation has 4 solutions:
(x, y) = (0, 0)
(x, y) = (0, 4)
(x, y) = (1, 0)
(x, y) = (1, 4)

Note 1: the last solution when y=4,
Left-side = 16 + 4 = 20 = 4×5 = 0 (mod 5).
Right-side = 1-1= 0 (mod 5).

Note 2: We call the equation (I) a “Curve over a finite field” since {0, 1,2,.. p-1} is a Field with finite p elements.

Mathematicians for some time have known that if N is a prime number (p), there will be roughly p solutions.

However, the most interesting number is a_{p} = the difference between p and the actual number of solutions.

For N = p = 5, the above equation has actually 4 solutions,
\boxed {a_{5} = 5 - 4 = 1}

Note: a_{p} can be positive or negative.

There is a ‘general rule’ (generating function) to predict a_{p}, and it is inspired from the ubiquitous Fibonacci numbers.

Recall:
Definition of the Fibonacci sequence as a recurrence relation:
\boxed{ F_{n}= \begin{cases} 0, & \text{for }n=0\\ 1, & \text{for }n=1\\ F_{n-2} + F_{n-1} , & \text{for } n \geq { 2} \end{cases} }

Alternatively there is also a generating function for Fibonacci numbers:
q + q(q+q^{2})+ q(q+q^{2})^{2} + q(q+q^{2})^{3} + q(q+q^{2})^{4} + ...

Let’s expand it we get the infinite series:
q + q^{2} + 2q^{3} + 3q^{4 } + 5q^{5} + 8q^{6} + 13q^{7} +...

The above coefficients coincide with
Fibonacci sequence: {0, 1, 1, 2, 3, 5, 8, 13…}

In 1954, the genius German mathematician Martin Eichler took the cue from the above, discovered another generating function:

\boxed {q(1-q^{1})^{2} (1-q^{11})^{2} (1-q^{2})^{2} (1-q^{22})^{2}(1-q^{3})^{2} (1-q^{33})^{2}(1-q^{4})^{2} (1-q^{44})^{2} ... } — (II)

Let’s expand it, we get:
q-2q^{2} -q^{3}+ 2q^{4} + q^{5}+2 q^{6}-2q^{7} -2q^{9} -2q^{10}+ q^{11} -2q^{12}+ 4q^{13}
Let b_{m} denotes the coefficient of the term q^{m}:
b_{1} = 1, b_{2} = -2, b_{3} = -1, b_{4} = 2, b_{5} = 1, ...

Eichler discovered that for any prime p,
\boxed { b_{p} = a_{p}}

Check: b_{5} = 1 = a_{5}

The random numbers of solutions in the elliptic curve equation (I) lies on the generating function (II).

If we view q as a point inside a unit disc on the complex plane, there is a group of symmetries and the function (II) is invariant under this group. The function (II) is called a modular form.

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The advanced generalisation of the Shimura-Taniyama-Weil Conjecture : we replace each cubic equation by a Representation of the Galois Group; and the modular form generalised by the generating function the “automorphic” function.:

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Remarks:
1. The Shimura-Taniyama-Weil Conjecture is a special case of Langlands Program.

2. Weil’s “Rosetta stone”:
Number Theory -> Curves over Finite Fields -> Riemann Surfaces

References:

http://en.m.wikipedia.org/wiki/Modularity_theorem

Love and Math by Edward Frenkel http://www.amazon.co.uk/dp/0465050743/ref=cm_sw_r_udp_awd_53swtb16779PY

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Click below for more free loan at the Singapore National Library Branches: http://www.nlb.gov.sg/mobile/searches/view_availability/200154975

Shimura Memoire on André Weil

Education, French, Modern Math 2 Comments

Goro Shimura (志村 五郎 born 23 February 1930) is a Japanese mathematician, and currently a professor emeritus of mathematics (former Michael Henry Strater Chair) at Princeton University

Shimura is known to a wider public through the important Modularity Theorem (previously known as the Taniyama-Shimura conjecture before being proven in the 1990s); Kenneth Ribet has shown that the famous Fermat’s Last Theorem (FLT) follows from a special case of this theorem. Shimura dryly commented that his first reaction on hearing of 1994 Andrew Wiles’s proof of the semi-stable case of the FLT theorem was ‘I told you so’.

Shimura’s mémoire on the 20th century great French mathematician André Weil (Fields Medal, Founder of Bourbaki):

1. Weil advised us not to stick to a wrong idea too long. “At some point you must be able to tell whether your idea is right or wrong; then you must have the guts to throw away your wrong idea.”

2. According to him, one of the best way to learn French or any foreign language was to see the same movie in that language again and again, staying in the same seat in the same movie theatre.

3. A French gentleman’s ideal is to have three concurrent loves: the first one, whom he cares about at present; the second, a potential one, whom he has his eye on with the hope that she will eventually be his principal love; the third, the past one, with whom he hasn’t completely cut off his relations. Then he observed: “It’s a good idea for a mathematician to have three mathematical loves in the same sense.”

4. As to Fields medals, he said: “It’s a kind of lottery. There are so many eligible candidates, and the whole selection process is a matter of chance. Therefore the prize could be given to any of them as in a lottery.”

5. He used to say that a good mathematician must have two good ideas. “It is possible for someone to have a really good idea, but it may be just a fluke. Once the person has a second good idea, then there is a good chance for him to develop into a better mathematician.”

6. In the summer of 1970 after the Nice Congress, I was talking with him somewhere in the Institute about French mathematicians. He observed that there were three young mathematicians in Paris who started brilliantly, and so there were high expectations for them. He mentioned three well-known names and said, “What happened to them? They utterly failed to produce anything great.” After around 1975 he expressed, more than once, his pessimistic view that French mathematics had been declining for some time.

{Note: This recorded memoire of Shimura made the French very unhappy, for which Shimura refused to delete it from the book }

7. Weil told me several anecdotes about Hardy. “Hardy’s opinion that mathematics is a young man’s game is nonsense,” Weil said.

8. When I prodded the guests to tell their ambitions in their next lives… “I want to be a Chinese scholar studying Chinese poems,” said Weil. After visiting China twice, he had been reading English translations of Chinese standard literature like ＂The Dream of the Red Chamber＂(红楼梦).

{Note: Weil met Hua LuoGeng 华罗庚 and remarked if every Chinese is like Hua, very soon in future the Westerners will have to learn Math in Chinese}

9. Weil said, “I would like to see the Riemann hypothesis settled before I die, but that is unlikely.” Weil died at 92 in 1998.