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ELLIPSES AND ELLIPTIC CURVES M. Ram Murty Queen’s University Planetary orbits are elliptical x2 a2 What is an ellipse? y2 + b2 = 1 An ellipse has two foci From: gomath.com/geometry/ellipse.php Metric mishap causes loss of Mars orbiter (Sept. 30, 1999) How to calculate arc length Let f : [0; 1] ! R2 be given by t 7 ! (x(t); y(t)). p By Pyt hagoras, ¢ s = (¢ x) 2 + (¢ y) 2 r ds = Arc lengt h = ¡ R 1 0 dx dt r ³ ¢2 + ¡ ¢2 dx dt dy dt ´2 ³ + dy dt dt ´2 dt Circumference of an ellipse We can parametrize the points of an ellipse in the ¯rst quadrant by f : [0; ¼=2] ! (asint; bcost): Circumference = p R 4 ¼=2 a2 cos2 t + b2 sin2 tdt: 0 Observe t hat if a = b, we get 2¼a. We can simplify t his as follows. Let cos2 µ = 1 ¡ sin2 µ. T he int egral becomes p R 4a ¼=2 1 ¡ ¸ sin2 µdµ 0 Where ¸ = 1¡ b2 =a2 . We can expand t he square root using t heRbinomial t ³heorem: ´ P 1=2 (¡ 1) n ¸ n sin2n µdµ: 4a ¼=2 1 0 n= 0 n We R can use t he fact t hat 2 ¼=2 sin2n µdµ = ¼( 1=2) ( 1=2) + 1) ¢¢¢( ( 1=2) + ( n ¡ 0 n! 1) ) The final answer T he circumference is given by 2¼aF (1=2; ¡ 1=2; 1; ¸ ) where P 1 ( a) n ( b) n zn F (a; b; c; z) = n = 0 n !( c) n is t he hypergeomet ric series and (a) n = a(a + 1)(a + 2) ¢¢¢(a + n ¡ 1): We now use Landens t ransformat ion: F (a; b; 2b; 4x ) = (1 + x) 2a F (a; a ¡ b+ 1=2; b+ 1=2; x 2 ). (1+ x ) 2 We put a = ¡ 1=2; b = 1=2 and x = (a ¡ b)=(a + b). T he answer can now be writ t en as ¼(a + b)F (¡ 1=2; ¡ 1=2; 1; x 2 ). Approximations • In 1609, Kepler used the approximation (a+b). The above formula shows the perimeter is always greater than this amount. • In 1773, Euler gave the approximation 2√(a²+b²)/2. • In 1914, Ramanujan gave the approximation (3(a+b) √(a+3b)(3a+b)). What kind of number is this? For example, if a and b are rat ional, is t he circumeference irrat ional? In case a = b we have a circle and t he circumference 2¼a is irrat ional. In fact , ¼is t ranscendent al. What does this mean? T his means t hat ¼does not sat isfy an equat ion of t he type x n + an ¡ 1 x n ¡ 1 + ¢¢¢+ a1 x + a0 = 0 wit h ai rat ional numbers. In 1882, Ferdinand von Lindemann proved t hat ¼is t ranscendent al. 1852-1939 This means that you can’t square the circle! • There is an ancient problem of constructing a square with straightedge and compass whose area equals . • Theorem: if you can construct a line segment of length α then α is an algebraic number. • Since is not algebraic, neither is √. Some other interesting numbers • The number e is transcendental. • This was first proved by Charles Hermite (1822-1901) in 1873. Is ¼+ e t ranscendent al? Answer: unknown. Is ¼e t ranscendent al? Answer: unknown. Not both can be algebraic! • Here’s a proof. • (x If ¡both are algebraic, then 2 e)(x¡ ¼) = x ¡ (e + ¼)x + ¼e is a quadrat ic polynomial wit h algebraic coe± cient s. T his implies bot h e and ¼ are algebraic, a cont radict ion t o t he t heorems of Hermit e and Lindemann. Conject ure: ¼ and e are algebraically independent . But what about the case of the ellipse? Is t hep int egral R ¼=2 a2 cos2 t + b2 sin2 tdt 0 t ranscendent al if a and b are rat ional? Yes. T his is a t heorem of T heodor Schneider (1911-1988) Let s look at t he int egral again. Put t ing u = sin t t he int egral becomes R q 1 0 a 2 ¡ ( a 2 ¡ b2 ) u 2 1¡ u 2 du: Set k 2 = 1 ¡ b2 =a2 : T he qint egral becomes R a 1 1¡ k 2 u 2 du: 1¡ u 2 0 Put t = 1 ¡ k 2 u2 : 1 2 R 1 1¡ k 2 p t dt t ( t ¡ 1) ( t ¡ ( 1¡ k 2 ) ) Elliptic Integrals Int R egrals of t he form p dx x 3 + a2 x 2 + a3 x + a4 are called ellipt ic int egrals of t he ¯rst kind. Int R egrals of t he form p x dx x 3 + a2 x 2 + a3 x + a4 are called ellipt ic int egrals of t he second kind. Our int egral is of t he second kind. Elliptic Curves y 2 = x(x ¡ 1)(x ¡ (1 ¡ k 2 )) The equation is an example of an elliptic curve. 0 1 ¡ k2 1 One can writ e t he equat ion of such a curve as y2 = 4x 3 ¡ ax ¡ b. C Elliptic Curves over Let L be a lat t ice of rank 2 over R. T his means t hat L = Z! © Z! 1 2 We at t ach t he Weierst rass } -funct ion t o L : } (z) = 1 z2 + ³ P 06 = ! 2L ´ 1 ( z¡ ! ) 2 ¡ 1 ! 2 ! 2 ! 1 T he } funct ion is doubly periodic: T his means } (z) = } (z + ! 1 ) = } (z + ! 2 ). e2¼i = 1 T he exponent ial funct ion is periodic since ez = ez+ 2¼i : T he periods of t he exponent ial funct ion consist of mult iples of 2¼i . T hat is, t he period \ lat t ice; ; is of t he form Z(2¼i ). The Weierstrass function T he } -funct ion sat is¯es t he following di®erent ial equat ion: (} ; (z)) 2 = 4} (z) 3 ¡ g2 } (z)¡ g3 ; where P g2 = 60 !¡ 4 06 = ! 2L and P g3 = 140 ! ¡ 6: 06 = ! 2L T his means t hat (} (z); } ; (z)) is a point on t he curve y 2 = 4x 3 ¡ g2 x¡ g3 : The Uniformization Theorem Conversely, every complex point on t he curve y2 = 4x 3 ¡ g2 x ¡ g3 is of t he form (} (z); } ; (z)) for some z 2 C. Given any g2 ; g3 2 C, t here is a } funct ion such t hat (} ; (z)) 2 = 4} (z) 3 ¡ g2 } (z)¡ g3 : Even and Odd Functions Recall t hat a funct ion f is even if f (z) = f (¡ z). For example, z2 and cosz are even funct ions. P 1 } (z) is even since } (z) = + 06 = ! 2L z2 } ; (z) is odd since } ; (z) = ¡ 2 z3 ¡ 2 Not e t hat } ; (¡ ! =2) = ¡ } ; (! =2) T his means } ; (! =2) = 0: A funct ion is called odd if f (z) = ¡ f (¡ z). For example, z and sin z are odd funct ions. ³ ´ 1 ¡ 1 ( z¡ ! ) 2 ! 2 ³ P 06 = ! 2L ´ 1 ( z¡ ! ) 3 . In part icular } ; (! 1 =2) = 0 } ; (! 2 =2) = 0 and } ; ((! 1 + ! 2 )=2) = 0. T his means t he numbers } (! 1 =2); } (! 2 =2); } ((! 1 + ! 2 )=2) are root s of t he cubic 4x 3 ¡ g2 x¡ g3 = 0: One can show t hese root s are dist inct . In part icular, if g2 ; g3 are algebraic, t he root s are algebraic. Schneider’s Theorem If g2 ; g3 are algebraic and } is t he associat ed Weierst rass } -funct ion, t hen for ® algebraic, } (®) is t ranscendent al. Since } (! 1 =2), } (! 2 =2) are algebraic It follows t hat t he periods ! 1 ; ! 2 must be t ranscendent al when g2 ; g3 are algebraic. T his is t he ellipt ic analog of t he Hermit e-Lindemann t heorem t hat says if ® is a non-zero algebraic number, t hen e® is t ranscendent al. Not e t hat we get ¼ t ranscendent al by set t ing ® = 2¼i Why should this interest us? Let y = sin x. T hen ( dy ) 2 + y 2 = 1. dx Let y = } (x). T ³ hen ´2 dy = 4y3 ¡ g2 y ¡ g3 . dx T hus p dy 1¡ y 2 = dx. T hus p = dx: dy 4y 3 ¡ g2 y ¡ g3 Int egrat ing bot h sides, we get Int egrat ing bot h sides, we get R R si n b p dy } ( ( ! 1 + ! 2 ) =2) p = b. dy = ! 0 1¡ y 2 } ( ! 1 =2) 4y 3 ¡ g2 y ¡ g3 2 2 R Let’s look at our formula for the circumference of an ellipse again. 1 1¡ k 2 p where k 2 = 1¡ t dt t ( t ¡ 1) ( t ¡ ( 1¡ k 2 ) ) b2 a2 : T he cubic in t he int egrand is not in Weierst rass form. It can be put in t his form. p But let us look at t he case k = 1= 2. Put t ing t = s + 1=2, t he int egral becomes R 1=2 2s+ 1 ds: p 0 4s 3 ¡ s T he int egral R 1=2 p ds 0 4s 3 ¡ s is a period of t he ellipt ic curve y2 = 4x 3 ¡ x: But what about R 1=2 p sds ? 0 4s 3 ¡ s Let us look at t he Weierst rass ³ -funct ion: ³ ´ P 1 1 + 1 + z . ³ (z) = + z 06 = ! 2L z¡ ! ! ! 2 Observe t hat ³ ; (z) = ¡ } (z). is not periodic! • It is “quasi-periodic’’. • What does this mean? ³ (z + ! ) = ³ (z) + ´ (! ): Put z = ¡ ! =2 t o get ³ (! =2) = ³ (¡ ! =2) + ´ (! ). Since ³ is an odd funct ion, we get ´ (! ) = 2³ (! =2). T his is called a quasi-period. What are these quasi-periods? Observe t hat ³ (! 1 =2) = ³ (¡ ! 1 =2 + ! 1 ) = ³ (¡ ! 1 =2) + ´ (! 1 ) But ³ is odd, so ³ (¡ ! 1 =2) = ¡ ³ (! 1 =2) so t hat 2³ (! 1 =2) = ´ (! 1 ): Similarly, 2³ (! 2 =2) = ´ (! 2 ) Since ´ is a linear funct ion on t he period lat t ice, we get 2³ ((! 1 + ! 2 )=2) = ´ (! 1 ) + ´ (! 2 ). T hus d³ = ¡ } (z)dz. Recallpt hat d} = 4} (z) 3 ¡ g2 } (z) ¡ g3 dz: Hence, d³ = ¡ p We can int egrat e bot h sides from z = ! 1 =2 to z = R (! 1 + ! 2 )=2 t o get ´ = 2 e2 p x dx 2 e1 x 3 ¡ g2 x ¡ g3 where e1 = } (! 1 ) and e2 = } ((! 1 + ! 2 )=2): Hence, our original int egral is a sum of a period and a quasi-period. } ( z) d} 4} ( z) 3 ¡ g2 } ( z ) ¡ g3 Are there triply periodic functions? • In 1835, Jacobi proved that such functions of a single variable do not exist. • Abel and Jacobi constructed a function of two variables with four periods giving the first example of an abelian variety of dimension 2. 1804-1851 1802 - 1829 What exactly is a period? • These are the values of absolutely convergent integrals of algebraic functions with algebraic coefficients defined by domains in Rn given by polynomial inequalities with algebraic coefficients. • For example is a period. RR ¼= dxdy x2 + y2 · 1 Some unanswered questions • • • • • Is e a period? Probably not. Is 1/ a period? Probably not. The set of periods P is countable but no one has yet given an explicit example of a number not in P. NµZµQµAµP