24/10/ Dr Ray Adams

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1 Fermat s Conjecture 24/10/ Dr Ray Adams

2 Fermat s Conjecture 24/10/ Dr Ray Adams

3 Fermat s Conjecture 24/10/ Dr Ray Adams

4 Fermat s Conjecture 24/10/ Dr Ray Adams

5 Fermat s Conjecture 24/10/ Dr Ray Adams

6 Andrew Wiles & Fermat s Last Theorem aka FLT 24/10/2017 Dr Ray Adams 6

7 Do you want to read about Andrew Wiles & Fermat s Last Theorem? /10/2017 Dr Ray Adams 7

8 FLT Does x 3 + y 3 = z 3? What about x n + y n = z n? x, y and z are co-primes 24/10/2017 Dr Ray Adams 8

9 FLT: note that FLT is a negative statement! FLT says that x n + y n z n x n + y n IS NOT EQUAL TO z n 24/10/2017 Dr Ray Adams 9

10 Why FLT is correct Do you have a calculator on you? Try this calculation: = Did it work for you? If FLT is true, then this equation cannot hold!!!!! 24/10/2017 Dr Ray Adams 10

11 Why is Homer wrong? exact x x ,134,474,609,751,300,000,000,000,000,000,000,000,000, ,842,181,739,947,300,000,000,000,000,000,000,000,000, ,976,656,348,486,700,000,000,000,000,000,000,000,000,000 total 63,976,656,349,698,600,000,000,000,000,000,000,000,000,000 sums rounded to ten significant digits ,134,474,600,000,000,000,000,000,000,000,000,000,000, ,842,181,740,000,000,000,000,000,000,000,000,000,000, ,976,656,340,000,000,000,000,000,000,000,000,000,000,000 total 63,976,656,340,000,000,000,000,000,000,000,000,000,000,000 {Different! {The same? significant numbers only 6,397,665,634 6,397,665,634 {The same? 24/10/2017 Dr Ray Adams 11

12 FLT & TSW The Taniyama Shimura Weil Theorem is true for semistructured elliptic curves over Q Fermat s Last Theorem is true 24/10/2017 Dr Ray Adams 12

13 This is a proof by contradiction. The Crux of FLT Assume that FLT is false {i.e. a n + b n = c n, for n>2, a, b and c are integers. Ken Ribet showed that if FLT is false, then it is possible to produce a semistructured elliptic curve that is not modular (Frey curve) i.e. a n + b n = c n can be expressed as: E abc :y 2 = x(x a n )(x c n ) or y 2 = x(x a n )(x +b n )and the latter is not modular BUT Andrews Wiles has shown that all semistructured elliptic curve are modular This is a contradiction! Therefore the original assumption that FLT is false must be rejected. 24/10/2017 Dr Ray Adams 13

14 So, all we need to understand is: 1. Ken Ribet s relevant work {Professor of mathematics at the University of California, Berkeley.} 2. Ribet s work is based on the work of Gerhard Frey. {Frey is a German mathematician, known for his work in number theory. His Frey curve, a construction of an elliptic curve from a purported solution to the Fermat equation, was central to Wiles' proof of Fermat's Last Theorem.} 3. Andrew Wiles relevant work {Royal Society Research Professor at the University of Oxford} 24/10/2017 Dr Ray Adams 14

15 To do so, well need to understand the following concepts...(explanations follow) The Taniyama Shimura Weil Theorem (as we will see later, the proof of this theorem leads to a proof of FLT) Elliptic curves and the L functions Semistable elliptic curves Complex functions Modular forms 24/10/2017 Dr Ray Adams 15

16 We will start with the... Taniyama 24/10/2017 Dr Ray Adams 16

17 We will start with the... Taniyama Shimura 24/10/2017 Dr Ray Adams 17

18 We will start with the... Taniyama Shimura Weil 24/10/2017 Dr Ray Adams 18

19 We will start with the... Taniyama Shimura Weil Theorem Let E be an elliptic curve whose equation has integer coefficients, let N be the conductor of E and, for each n, let an be the number of the L function of E. Then there exists a corresponding modular elliptic curve. Underlined concepts will be defined, just not on this slide! 24/10/2017 Dr Ray Adams 19

20 We will start with the... Taniyama Shimura Weil Theorem Elliptic curve; see later Integer coefficients: Integers are positive & negative whole numbers, no fractions. For an elliptic curve E, the conductor f(e) is a parameter of the curve, so that: (The conductor summarises the L function of the elliptic curve). f(e) = Π p fp p Where f p is the field of the prime numbers that are the factors of E. Dirichlet s L function; see later Modular elliptic curve; see later 24/10/2017 Dr Ray Adams 20

21 Taniyama Shimura Weil Theorem We will see later that the proof of this theorem for a subset of elliptic curves provides a proof for FLT (i.e. semistable elliptic curves). This theorem relates elliptic curves to modular forms or modular elliptic curves. For every rational elliptic curve, there is a modular form or modular elliptic curve. For every elliptic curve with rational coefficients, there is a modular form with the same Dirichlet L series. For an elliptic curve of the form y 2 = Ax 3 + Bx 2 +Cx + D, there exists modular forms f(z) and g(z) such that: [f(z) 2 ] = A[g(z) 2 + Cg(z) + D 24/10/2017 Dr Ray Adams 21

22 Elliptic Curves Elliptic curves are not ellipses. Elliptic curves were given their name by G. C. Fagnano ( )who discovered them when calculating the arc lengths of an ellipse, producing elliptic integrals which are inverse functions of elliptic curves. An elliptic integral is any function f expressed as: where R is a rational function, P is a polynomial of degree 3 or 4 with no repeated roots. C is a constant that varies across functions. The above equation can be transformed eventually to y 2 = x(x a n )(x c n ) or y 2 = x(x a n )(x +b n ) 24/10/2017 Dr Ray Adams 22

23 Elliptic Curves An elliptic curve is of the general form y 2 = Ax 3 + Bx 2 + Cx +D 24/10/2017 Dr Ray Adams 23

24 Elliptic Curves An elliptic curve is a smooth, algebraic curve of genus one, for which there is a specific point O that defines it. By definition, it has no cusps or self intersections. When an elliptic curve is defined over complex numbers, it corresponds to a torus in the complex plane. Elliptic curves and modular forms are said to be complex as they cannot be simplified and invoke imaginary numbers. 24/10/2017 Dr Ray Adams 24

25 Elliptic Curves An elliptic curve is a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus i.e. a smooth, projective algebraic curve of genus one. If y 2 = P(x), where P is any polynomial of degree three in x with no repeated roots, then we obtain a non singular plane curve of genus one, which is thus an elliptic curve 24/10/2017 Dr Ray Adams 25

26 Elliptic Curves A semistable elliptic curve may be described as one that has bad reduction i.e. has singularities i.e. self intersections and cusps. 24/10/2017 Dr Ray Adams 26

27 Gerhard Frey s Elliptic Curve Starting with the familiar equation; a n + b n = c n, Frey was able to transform it to give the following equation: E a,b,c} y2 = x(x a n )(x c n ) or y 2 = x(x a n )(x +b n ) 24/10/2017 Dr Ray Adams 27

28 Gerhard Frey s Elliptic Curve Ken Ribet was able to demonstrate that, if it existed, Gerhard Frey s Elliptic Curve is not modular. Therefore to prove that all semistable elliptics were modular would be to reject the assumption that FLT was false, as this was the assumption on which Gerhard Frey s Elliptic Curve was built. 24/10/2017 Dr Ray Adams 28

29 Gerhard Frey s Elliptic Curve Starting with the familiar equation; a n + b n = c n, Frey was able to transform it to give the following equation: E a,b,c} y2 = x(x a n )(x c n ) or y 2 = x(x a n )(x +b n ) (n are odd primes and a, b and c are whole integers) VIP These two equations are built on the assumption that FLT is FALSE. If they exist, then FLT is FALSE. It is an example of a semistable elliptic curve. 24/10/2017 Dr Ray Adams 29

30 Elliptic Curves & Modular Forms The Taniyama Shimura Weil Theorem states that every elliptic curve has an equivalent modular form i.e. we say that the elliptic curve is modular. To compare elliptic curves with modular forms, he found that the Dirichlet L values of each elliptic form are equal to the parameters of the corresponding modular form. Two questions: What are Dirichlet s L numbers? What are modular forms? 24/10/2017 Dr Ray Adams 30

31 Gustav Lejeune -Dirichlet s L series This concept makes use of clock arithmetic aka modular arithmetic. Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be = 15, but this is not the answer because clock time "wraps around" every 12 hours; in 12-hour time, there is no "15 o'clock". Likewise, if the clock starts at 12:00 (noon) and 21 hours elapse, then the time will be 9:00 the next day, rather than 33:00. Since the hour number starts over after it reaches 12, this is arithmetic modulo 12. We say that 12 is congruent not only to 12 itself, but also to 0, so the time called "12:00" could also be called "0:00", since 12 is congruent to 0 modulo 12 24/10/2017 Dr Ray Adams 31

32 Modular arithmetic On a 12 hour clock, =1 24/10/2017 Dr Ray Adams 32

33 What are Dirichlet s L values? For the elliptic curve: x 3 x 2 = y 2 + y For clock arithmetic 5 (modulo 5), the FOUR solutions are: x = 0, y = 0, x = 0, y = 4, x = 1, y = 0, x = 1, y = 4, We say that L 5 =4. For a 7 hour clock, there are nine solutions, so L 7 =9. (See Dirichlet 1837) 24/10/2017 Dr Ray Adams 33

34 What are Dirichlet s L values? The Dirichlet s L values for each elliptic are mirrored in the equivalent parameters of a modular form. 24/10/2017 Dr Ray Adams 34

35 What are Modular Elliptic Forms? A modular elliptic form is a complex function with extreme symmetry. 24/10/2017 c Dr Ray Adams 35

36 Degrees of Symmetry 24/10/ Dr Ray Adams

37 Measuring Symmetry 24/10/ Dr Ray Adams

38 Measuring Symmetry Irregular Shape Symmetry Analysis: Theory and Application to Quantitative Galaxy Classification This paper presents a set of imperfectly symmetric measures based on a series of geometric transformation operations for quantitatively measuring the amount of symmetry of arbitrary shapes. Pattern Analysis and Machine Intelligence, IEEE Transactions 2010 Qi Guo Strangeways Res. Lab., Univ. of Cambridge, Cambridge, UK Falei Guo ; Jiaqing Shao 24/10/ Dr Ray Adams

39 Measuring Symmetry Irregular Shape Symmetry Analysis: Theory and Application to Quantitative Galaxy Classification The asymmetry in the disk of NGC 4319 is the result of an interaction with another galaxy (not shown). NASA and The Hubble Heritage Team Like most galaxies, the Sombrero galaxy is highly symmetrical on all three axes 24/10/ Dr Ray Adams

40 Super Symmetric Particles 24/10/ Dr Ray Adams

41 Modular Elliptic Forms A modular elliptic form exists in four dimensional hyperbolic space, where the four dimensions are based on two dimensions, each based on complex numbers. y = y r +y i x = x r +x i 24/10/2017 c Dr Ray Adams 41

42 Modular Elliptic Forms A modular elliptic form exists in the upper half of the complex space. 24/10/2017 c Dr Ray Adams 42

43 Andrew Wiles Andrew Wiles took the next step. He proved that all semistable elliptic curves were modular and so each had an equivalent (same parameters) modular form. In so doing, he showed that the Frey semistable elliptic could not exist. Therefore, Fermat s Last Theorem was true i.e. X n + Y n Z n 24/10/2017 c Dr Ray Adams 43

44 Andrew Wiles Proof His proof followed the following steps: Identify the set of semistable elliptic curves Identify the L values of the semistable elliptic curves Demonstrate that the first L values of the semistable elliptic curves equal the first equivalent L values of the modular forms. Demonstrate that the n th L values of the semistable elliptic curves equal the n th equivalent L values of the modular forms. Demonstrate that the (n + 1) th L values of the semistable elliptic curves equal the (n+1) th equivalent L values of the modular forms. BY INDUCTION, as the equivalence applies to n and (n + 1) values, it applies to all such values. Therefore, the Taniyama, Shimura, Weil Conjecture is confirmed for all semistable elliptic curves. THUS FLT is confirmed. 24/10/2017 c Dr Ray Adams 44

45 Andrew Wiles How did Andrew Wiles take that final step? See: /10/2017 c Dr Ray Adams 45