The Arithmetic of Elliptic Curves

Transcription

1 The Arithmetic of Elliptic Curves Sungkon Chang The Anne and Sigmund Hudson Mathematics and Computing Luncheon Colloquium Series

2 OUTLINE Elliptic Curves as Diophantine Equations Group Laws and Mordell-Weil Theorem Torsion points and Rank Problems Solutions in different fields/rings. Applications: Integer Factoring The Birch & Swinnerton-Dyer Conjecture. BSD over a finite field Goldfeld s Conjecture and Results Taniyama-Shimura Conjecture (Theorem). Statement Proof of Fermat s Last Theorem.

3 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),...

4 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Arc length of an ellipse s = β α P(t) Q(t) dt. y 2 = Q(x) Abel sensed about the theory of complex multiplication x 2 a 2 + y2 b 2 = 1

5 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Example: E : y 2 = x 3 x. E(R)

6 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Example: E : y 2 = x 3 x. E(R)

7 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... E(C) is a Riemann surface of genus 1.

8 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... E(C)

9 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... E(R)

10 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... E(R)

11 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Diophantine Equations in two variables: e.g., X : x 2 + y 2 = 1 Q rat l X(Q) t ( ) 1 t 2 2t 1 +t 2, 1 +t 2

12 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Diophantine Equations in two variables: e.g., X : x 2 + y 2 = 1

13 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Diophantine Equations in two variables:

14 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Diophantine Equations in two variables: There are no non-constant rat l. maps: Q E(Q).

15 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),...

16 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... (P + Q) + R = P + (Q + R)

17 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... (P + Q) + R = P + (Q + R)

18 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... (P + Q) + R = P + (Q + R) =

19 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... E(Q) = E(Q) { } (P + Q) + R = P + (Q + R) =

20 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Group law: (a,b)+(c,d) := ( b2 + 2bd d 2 + a 3 a 2 c ac 2 + c 3 (a c) 2, 1 (a c) 3( b3 + 3b 2 d 3bd 2 + ba 3 + 2bc 3 +d 3 2da 3 + 3da 2 c dc 3 3bac 2 ) )

21 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Mordell-Weil Thm: There are points P 1,...,P r in E(Q) s.t. E(Q) = {m 1 P m r P r : m i are integers}.

22 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Mordell-Weil Thm: There are points P 1,...,P r in E(Q) s.t. E(Q) = {m 1 P m r P r : m i are integers}. example: E : y 2 = x E(Q) contains P = ( 1,4).

23 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Mordell-Weil Thm: There are points P 1,...,P r in E(Q) s.t. E(Q) = {m 1 P m r P r : m i are integers}. example: E : y 2 = x E(Q) contains P = ( 1,4). E(Q) contains 6P = ( , ).

24 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Mordell-Weil Thm: There are points P 1,...,P r in E(Q) s.t. E(Q) = {m 1 P m r P r : m i are integers}. example: E : y 2 = x has two generators P = ( 2, 3) and Q = (2, 5), i.e., every point in E(Q) is written as np + mq. E(Q) = Z Z

25 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Mordell-Weil Thm: The Mordell-Weil group E(Q) is a finitely generated abelian group, i.e., E(Q) = } Z {{ Z} E(Q) Tor. r ranke(q) := r where E(Q) Tor = {Q E(Q) : nq = for some n}.

26 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Mordell-Weil Thm: The Mordell-Weil group E(Q) is a finitely generated abelian group, i.e., E(Q) = } Z {{ Z} E(Q) Tor. r where E(Q) Tor = {Q E(Q) : nq = for some n}. Thm (B. Mazur, 1978) For elliptic curves E, there are finite possibilities for E(Q) Tor.

27 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Mordell-Weil Thm: The Mordell-Weil group E(Q) is a finitely generated abelian group, i.e., E(Q) = } Z {{ Z} E(Q) Tor. r ranke(q) := r

28 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Mordell-Weil Thm: E(Q) = Z Z E(Q) Tor. Open Questions Find generators of E(Q). Find an algorithm that computes ranke(q). Prove ly many E s with arbitrarily large ranke(q).

29 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Sets of solutions: E(Z), E(Q), E(R), E(C), E(Z/nZ),... Mordell-Weil Thm: E(Q) = Z Z E(Q) Tor. Open Questions Find generators of E(Q). Find an algorithm that computes ranke(q). Prove ly many E s with arbitrarily large ranke(q). Record: In 2006, N. Elkies found E for which ranke(q) 28.

30 y 2 + xy + y = x 3 x x A few independent points are P 1 = [ , ] P 2 = [ , ] P 3 = [ , ] P 4 = [ , ] P 5 = [ , ] P 6 = [ , ] P 7 = [ , ] P 8 = [ , ]

31 Modular Arithmetic An elliptic curve is an equation E : y 2 = x 3 + Ax + B where A,B are integers. The set of residue classes Z/nZ := {0,1,...,n 1}.

32 Modular Arithmetic An elliptic curve is an equation E : y 2 = x 3 + Ax + B where A,B are integers. The set of residue classes Z/nZ := {0,1,...,n 1}. Example: Z/5Z := {0,1,...,4}. 0 = {0,±5,±10,±15,...} = 5 1 = {..., 9, 4,1,6,11,16,21,...} = 11 2 = {..., 8, 3,2,7,12,17,22,...} = 9

33 Modular Arithmetic An elliptic curve is an equation E : y 2 = x 3 + Ax + B where A,B are integers. The set of residue classes Z/nZ := {0,1,...,n 1}. Example: Z/5Z := {0,1,...,4} = 6 = = 6 = 1 1/2 = 3. x y = 1 for all x 0.

34 Modular Arithmetic An elliptic curve is an equation E : y 2 = x 3 + Ax + B where A,B are integers. The set of residue classes Z/nZ := {0,1,...,n 1}. Example: Z/5Z := {0,1,...,4} = 6 = = 6 = 1 1/2 = 3. x y = 1 for all x 0. Z/5Z is the finite field with 5 elements, F 5.

35 Modular Arithmetic An elliptic curve is an equation E : y 2 = x 3 + Ax + B where A,B are integers. The set of residue classes Z/nZ := {0,1,...,n 1}. Example: Z/10Z := {0,1,...,9}. 2 y 1 for all y since 2y = 10q + 1 doesn t make sense. 1/2, not defined 3 7 = 1, i.e., 1/3 = 7.

36 Modular Arithmetic An elliptic curve is an equation E : y 2 = x 3 + Ax + B where A,B are integers. The set of residue classes Z/nZ := {0,1,...,n 1}. Prop. The class 1/k is defined if and only if k doesn t have a common factor with n.

37 Modular Arithmetic An elliptic curve is an equation E : y 2 = x 3 + Ax + B where A,B are integers. The set of residue classes Z/nZ := {0,1,...,n 1}. The set of solutions: E(Z/nZ) = {(x,y) : y 2 = x 3 + Ax + B}. Example: E : y 2 = x x 2 with coefficients in Z/4453Z E(Z/4453Z) contains P = (1,3).

38 Modular Arithmetic An elliptic curve is an equation E : y 2 = x 3 + Ax + B The set of residue classes Z/nZ := {0,1,...,n 1}. where A,B are integers. The set of solutions: E(Z/nZ) = {(x,y) : y 2 = x 3 + Ax + B}. Example: E : y 2 = x x 2 with coefficients in Z/4453Z E(Z/4453Z) contains P = (1,3). 2P = (97/6 2, 1441/6 3 ) = (4332,3230). 3P = ( ), , undefined since 61 is a common factor of 4331 and 4453.

39 Modular Arithmetic An elliptic curve is an equation E : y 2 = x 3 + Ax + B where A,B are integers. The set of residue classes Z/nZ := {0,1,...,n 1}. Factoring Problem: It is given that n is a composite number. Find an integer factor. Lenstra s approach using elliptic curves: Play with random elliptic curves E and a point P in E(Z/nZ) to find a factor appearing in the denominators of the points mp.

40 Modular Arithmetic An elliptic curve is an equation E : y 2 = x 3 + Ax + B where A,B are integers. The set of residue classes Z/nZ := {0,1,...,n 1}. Factoring Problem: It is given that n is a composite number. Find an integer factor. Lenstra s approach using elliptic curves: Play with random elliptic curves E and a point P in E(Z/nZ) to find a factor appearing in the denominators of the points mp. Very effective for 60-digit numbers For larger numbers, effective for finding prime factors having around 20 to 30 digits

41 Local-to-Global Principle An elliptic curve is an equation E : y 2 = x 3 + Ax + B where A,B are integers. Question: Can we conclude something about E(Q) using the info {E(Z/nZ) : n Z + }? using the info {E(Z/p m Z) : p prime and m Z + }? using the info {E(Z/pZ) : p prime} = {E(F p ) : p}?

42 Local-to-Global Principle An elliptic curve is an equation E : y 2 = x 3 + Ax + B where A,B are integers. Question: Can we conclude something about E(Q) using the info {E(Z/nZ) : n Z + }? using the info {E(Z/p m Z) : p prime and m Z + }? using the info {E(Z/pZ) : p prime} = {E(F p ) : p}? Faltings proved in 1983 Tate s Isogeny Conjecture {E(F p ) : p} determines ranke(q). The Local-to-Global Principle Works!

43 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B Open Problems: Compute rank E(Q) = r limsup E ranke(q) =?, i.e., where A,B are integers. where E(Q) = Z } {{ Z} E(Q) Tor r Are there E s with arbitrarily large ranke(q)?

44 Elliptic Curves An elliptic curve is an equation E : y 2 = x 3 + Ax + B Open Problems: Compute rank E(Q) = r limsup E ranke(q) =?, i.e., where A,B are integers. where E(Q) = Z } {{ Z} E(Q) Tor r Are there E s with arbitrarily large ranke(q)? (Faltings) {E(F p ) : p} determines ranke(q).

45 Birch and Swinnerton-Dyer Conj. Award: $

46 Birch and Swinnerton-Dyer Conj. Award: $ Experiment: p<x # E(F p ) p c E (lnx) ranke(q)

47 Birch and Swinnerton-Dyer Conj. Award: $ The Hasse-Weil L-function L(E,s) is an analytic function on a domain of C L(E,s) := (1 a p p s + p 1 2s ) 1 p S (1 a p p s ) 1 p S made out of # E(F p ), p: where a p :=1 + p # E(F p ), and S is a finite set of primes.

48 Birch and Swinnerton-Dyer Conj. Award: $ Then, L(E,s) is analytically continued to C (proved by Wiles et al);

49 Birch and Swinnerton-Dyer Conj. Award: $ Then, L(E,s) is analytically continued to C (proved by Wiles et al);

50 Birch and Swinnerton-Dyer Conj. Award: $ Then, L(E,s) is analytically continued to C (proved by Wiles et al); L(E,s) = a(s 1) ranke(q) + b(s 1) ranke(q)+1 +

51 Birch and Swinnerton-Dyer Conj. Award: $ Then, L(E,s) is analytically continued to C (proved by Wiles et al); L(E,s) = a(s 1) ranke(q) + b(s 1) ranke(q)+1 + BSD implies: L(E,1) 0 ranke(q) = 0

52 Birch and Swinnerton-Dyer Conj. Award: $ Then, L(E,s) is analytically continued to C (proved by Wiles et al); L(E,s) = a(s 1) ranke(q) + b(s 1) ranke(q)+1 + BSD implies: L(E,1) 0 ranke(q) = 0 (proved by Kolyvagin + Gross-Zagier + Wiles).

53 Birch and Swinnerton-Dyer Conj. Searching for Evidence Let E be given by y 2 = x 3 + Ax + B. A quadratic twist of E is an elliptic curve given by E D : Dy 2 = x 3 + Ax + B where D is a square-free integer. ranke D (Q) varies as D varies.

54 Birch and Swinnerton-Dyer Conj. Searching for Evidence Let E be given by y 2 = x 3 + Ax + B. A quadratic twist of E is an elliptic curve given by E D : Dy 2 = x 3 + Ax + B where D is a square-free integer. ranke D (Q) varies as D varies. BSD implies the uniform distribution of the parities of ranke D (Q).

55 Birch and Swinnerton-Dyer Conj. Searching for Evidence Let E be given by y 2 = x 3 + Ax + B. A quadratic twist of E is an elliptic curve given by E D : Dy 2 = x 3 + Ax + B where D is a square-free integer. ranke D (Q) varies as D varies. BSD implies the uniform distribution of the parities of ranke D (Q). Are the parities of ranke D (Q) uniformly distributed?

56 Birch and Swinnerton-Dyer Conj. Searching for Evidence Let E be given by y 2 = x 3 + Ax + B. A quadratic twist of E is an elliptic curve given by E D : Dy 2 = x 3 + Ax + B D T (X) ranke D (Q) Goldfeld s Conj: lim = 1/2 X #T (X) T (X) = {0 < D < X : D square-free}.

57 Birch and Swinnerton-Dyer Conj. Searching for Evidence Let E be given by y 2 = x 3 + Ax + B. A quadratic twist of E is an elliptic curve given by E D : Dy 2 = x 3 + Ax + B D T (X) ranke D (Q) Goldfeld s Conj: lim = 1/2. X #T (X) Heath-Brown (Invent. 94): Let E be y 2 = x 3 x. D T (X) odd ranke D (Q) lim sup X #T (X) odd 1.26.

58 Birch and Swinnerton-Dyer Conj. Searching for Evidence Let E be given by y 2 = x 3 + Ax + B. A quadratic twist of E is an elliptic curve given by E D : Dy 2 = x 3 + Ax + B D T (X) ranke D (Q) Goldfeld s Conj: lim = 1/2. X #T (X) C. (JNT, 06): Let E be y 2 = x 3 A lim sup X where A 1,25 mod 36, and sq. fr. D N(X) ranke D (Q) 1, A > 0 #N(X) 4/3, A < 0. where N(X) = {D T (X) : D > 0, D 1 mod 12A}.

59 Theoretical Results Birch and Swinnerton-Dyer Conj. J. Coates & A. Wiles (1978): L(E,1) = 0 and E has CM ranke(q) 1. A. Wiles at et al: L(E,s), analytically continued as a corollary of the proof of the Taniyama-Shimura Conj. Kolyvagin + Gross-Zagier: L(E,1) 0 ranke(q) = 0. BSD: L(E,s) = a(s 1) ranke(q) + b(s 1) ranke(q)+1 +

60 Taniyama-Shimura Conjecture Y. Taniyama G. Shimura

61 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). Z/5Z : 1 = 16 since 16 =

62 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). C/(Zv + Zw) : z 1 = z 2 if z 2 = z 1 + (mv + nw) where m,n Z.

63 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). C/(Zv + Zw) : z 1 = z 2 if z 2 = z 1 + (mv + nw) where m,n Z. Example: C/(Zi + Z(1 + i)). 1 = 1 + i = 0 since 1 = (1 + i) + (( 1) i + 0 (1 + i)) i = 14i = 0 a + bi = 0 if a,b Z.

64 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). C/(Zv + Zw) : z 1 = z 2 if z 2 = z 1 + (mv + nw)

65 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). C/(Zv + Zw) : z 1 = z 2 if z 2 = z 1 + (mv + nw)

66 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots).

67 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B (where x 3 + Ax + B = 0 has distinct roots). What is that E given v and w?

68 Taniyama-Shimura Conjecture The additive group Λ = (Zv + Zw) is called a lattice. The elliptic curve is E : y 2 = 4x 3 + Ax + B where 1 A = 60 λ 4 B = 140 λ Λ\{0} λ Λ\{0} 1 λ 6. The map : C/Λ E(C) is given by z ( (z), (z) ) where (z) = 1 z 2 + λ Λ\{0} 1 (z λ) 2 1 λ 2

69 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B. lattices elliptic curves

70 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B. E 1 (C) = E 2 (C) if there is a 1-to-1 rat l map.

71 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B. [E 1 ] = [E 2 ] if there is a 1-to-1 rat l map.

72 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B. Z1 + Zτ Zv + Zw [E]

73 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B. {Z1 + Zτ : τ H} {[E]}

74 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B. {Z1 + Zτ : τ H} {[E]} Further reduction Z1 + Zτ = Zv + Zw for some ( ) v and( w. )( ) 1 = av + bw τ = cv + dw 1 a b v = τ c d w

75 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B. {Z1 + Zτ : τ H} {[E]} Further reduction Z1 + Zτ = Zv + Zw for some ( ) v and( w. )( ) 1 = av + bw τ = cv + dw 1 a b v = τ c d w ( ) a b is invertible! c d ( ) a b is in SL 2 (Z) c d

76 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B. {Z1 + Zτ : τ H} {[E]} Further reduction Z1 + ( Zτ is ) a lattice. a b SL 2 (Z) c d Z(aτ + b) + Z(cτ + d) = Z1 + Zτ If Z(aτ + b) + Z(cτ + d) E, then Z1 + Z aτ + b cτ + d [E]

77 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B. Further reduction {Z1 + Zτ : τ H} {[E]} Z1 + ( Zτ is ) a lattice. a b SL 2 (Z) c d Z(aτ + b) + Z(cτ + d) = Z1 + Zτ Z1 + Zτ Z1 + Z aτ + b cτ + d [E]

78 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B. {Z1 + Zτ : τ H} {[E]} H {[E]} The quotient group SL 2 (Z)\H is the equivalence classes [τ] s.t. [τ] = [ ] aτ + b cτ + d ( ) a b where SL 2 (Z) c d

79 Taniyama-Shimura Conjecture An elliptic curve is an equation E : y 2 = x 3 + Ax + B. [τ] = [ ] aτ + b cτ + d

80 Taniyama-Shimura Conjecture Modular Forms of weight k The map: H {E} is given by τ E : y 2 = 4x 3 + A(τ)x + B(τ). Thus, A(τ) and B(τ) are functions: H C. It turns out that ( ) aτ + b A = (cτ + d) 4 A(τ) cτ + d ( ) aτ + b B = (cτ + d) 6 B(τ) cτ + d

81 Taniyama-Shimura Conjecture Modular Forms of weight k The map: H {E} is given by τ E : y 2 = 4x 3 + A(τ)x + B(τ). Thus, A(τ) and B(τ) are functions: H C. It turns out that ( ) aτ + b A = (cτ + d) 4 A(τ) cτ + d ( ) aτ + b B = (cτ + d) 6 B(τ) cτ + d The function A(τ) is called a modular form of weight 4, and B(τ), a modular form of weight 6.

82 Taniyama-Shimura Conjecture Modular Forms f ( aτ+b cτ+d) = (cτ + d) k f (τ)

83 Taniyama-Shimura Conjecture Modular Forms Modular Functions f ( aτ+b cτ+d) = (cτ + d) k f (τ) f ( aτ+b cτ+d) = f (τ)

84 Taniyama-Shimura Conjecture Modular Forms Modular Functions Example: f ( aτ+b cτ+d) = (cτ + d) k f (τ) f ( aτ+b cτ+d) = f (τ) j(τ) = 1728A(τ)3 27B(τ) 2 A(τ) 3 where A ( aτ+b cτ+d ) = (cτ + d) 4 A(τ) B ( aτ+b cτ+d ) = (cτ + d) 6 B(τ)

85 Taniyama-Shimura Conjecture Modular Forms/Functions for SL 2 (Z). f ( ) aτ+b cτ+d = (cτ + d) k f (τ) and g ( aτ+b cτ+d) = g(τ) j : SL 2 (Z)\H C is bijective! j(τ) = 1728A(τ)3 27B(τ) 2 A(τ) 3

86 Taniyama-Shimura Conjecture Modular Forms/Functions for SL 2 (Z). f ( ) aτ+b cτ+d = (cτ + d) k f (τ) and g ( aτ+b cτ+d) = g(τ) j : SL 2 (Z)\H C is bijective! j(τ) = 1728A(τ)3 27B(τ) 2 A(τ) 3

87 Taniyama-Shimura Conjecture Modular Forms/Functions for SL 2 (Z). f ( ) aτ+b cτ+d = (cτ + d) k f (τ) and g ( aτ+b cτ+d) = g(τ) j : SL 2 (Z)\H C is bijective!

88 Taniyama-Shimura Conjecture Modular Forms/Functions for SL 2 (Z). f ( ) aτ+b cτ+d = (cτ + d) k f (τ) and g ( aτ+b cτ+d) = g(τ) j : SL 2 (Z)\H C is bijective! SL 2 (Z)\H compactified is a Riemann surface of genus 0, i.e., a Riemann sphere.

89 Taniyama-Shimura Conjecture Modular Forms/Functions for SL 2 (Z). f ( ) aτ+b cτ+d = (cτ + d) k f (τ) and g ( aτ+b cτ+d) = g(τ) j : SL 2 (Z)\H C is bijective! The manifold SL 2 (Z)\H as the solutions of an equation: (SL 2 (Z)\H) = Y (C) given by [τ] ( j(τ), j(τ) ) where Y : x y = 0. (SL 2 (Z)\H) = X(C) where X(C) is the compactification of Y (C).

90 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ ] a b c d SL2 (Z) : c 0 mod N } The manifold Γ 0 (N)\H compactified is called the modular curve for Γ 0 (N). Example: Γ 0 (2)\H. Matrices: [ ] [ and 0 1 ] 1 1.

91 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ ] a b c d SL2 (Z) : c 0 mod N } The manifold Γ 0 (N)\H compactified is called the modular curve for Γ 0 (N). Example: Γ 0 (2)\H. Matrices: [ ] [ and 0 1 ] 1 1.

92 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ ] a b c d SL2 (Z) : c 0 mod N } The manifold Γ 0 (N)\H compactified is called the modular curve for Γ 0 (N). Example: Γ 0 (2)\H. Matrices: [ ] [ and 0 1 ] 1 1.

93 Taniyama-Shimura Conjecture Copies of the shaded region ( SL 2 (Z)\H under SL 2 (Z)):

94 Taniyama-Shimura Conjecture Example: Γ 0 (16)\H has genus 0.

95 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ ] a b c d SL2 (Z) : c 0 mod N } The manifold Γ 0 (N)\H compactified is called Example: the modular curve for Γ 0 (N). Γ 0 (11)\H has genus 1, i.e., an elliptic curve. Γ 0 (22)\H has genus 2.

96 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ ] a b c d SL2 (Z) : c 0 mod N } Modular Forms/Functions for Γ 0 (N): f ( ) aτ+b cτ+d = (cτ + d) k f (τ) and g ( aτ+b cτ+d) = g(τ) where [ ] a b c d Γ0 (N). Γ 0 (N)\H g : Γ 0 (N)\H C

97 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ a b c d Eichler-Shimura Theory Thm. Let N be a positive integer. ] SL2 (Z) : c 0 mod N } (a) There is Y 0 (N), the system of polynomial equations with integer coefficients such that there is a complex analytic isomorphism j N : (Γ 0 (N)\H) X 0 (N)(C) where X 0 (N)(C) is a compactification of Y 0 (N)(C); (Γ 0 (N)\H) = X 0 (N)(C). (b) Sometimes, there is an elliptic curve E : y 2 = x 3 +Ax+B with A,B Z such that there is a surjective rat l map with rational coefficients φ : X 0 (N)(C) E(C).

98 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ a b c d Eichler-Shimura Theory ] SL2 (Z) : c 0 mod N } Given N, sometimes a nice way to construct an elliptic curve E using X 0 (N).

99 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ a b c d Eichler-Shimura Theory ] SL2 (Z) : c 0 mod N } Given N, sometimes a nice way to construct an elliptic curve E using X 0 (N). Sometimes, a special modular form f of weight 2 for Γ 0 (N), called a new form of level N. Consider the map Ψ f : Γ 0 (N) C given by [ a b ] c d Then, Ψ f (Γ 0 (N)) is... aτ 0 +b cτ 0 +d τ 0 f (z) dz.

100 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ a b c d Eichler-Shimura Theory ] SL2 (Z) : c 0 mod N } Given N, sometimes a nice way to construct an elliptic curve E using X 0 (N). Sometimes, a special modular form f of weight 2 for Γ 0 (N), called a new form of level N. Consider the map Ψ f : Γ 0 (N) C given by [ a b ] c d Then, Ψ f (Γ 0 (N)) is... aτ 0 +b cτ 0 +d τ 0 f (z) dz. a lattice Zv + Zw, an elliptic curve.

101 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ a b c d Eichler-Shimura Theory Thm. Let N be a positive integer. ] SL2 (Z) : c 0 mod N } (a) (Γ 0 (N)\H) = X 0 (N)(C) where X 0 (N)(C) is the compactification of the solutions of equations with interger coefficients. (b) Each new form f (τ) of level N generates an elliptic curve E s.t. there is a rat l map: X 0 (N)(C) E(C). Moreover, if the Fourier expansion of f (τ) (around ) has integer coefficients, i.e., f (τ) = a 1 q + a 2 q 2 + a 3 q 3 + with a i Z then the elliptic curve E has integer coefficients. where q = exp(2πiτ),

102 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ a b c d Eichler-Shimura Theory Thm. Let N be a positive integer. (a) (Γ 0 (N)\H) = X 0 (N)(C) ] SL2 (Z) : c 0 mod N } where X 0 (N)(C) is the compactification of the solutions of equations with interger coefficients. (b) {new forms} {E}, and {new forms with Z-coeff.} {E with Z-coeff.} X 0 (N) E

103 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ a b c d Taniyama-Shimura Conjecture ] SL2 (Z) : c 0 mod N } Given an elliptic curve E with integer coefficients, (a) There is a rat l map with rational coefficients φ : X 0 (N)(C) E(C) for some N; all E s are modular. (b) The image Ψ f (Γ 0 (N)) for some new form f of level N is a lattice for E. (c) f (τ) = a 1 q + a 2 q 2 + a 3 q a p q p + + and a p = p + 1 # E(F p ).

104 Taniyama-Shimura Conjecture Modular Congruence Subgroups Γ 0 (N) = {[ a b c d Taniyama-Shimura Conjecture There is a correspondence between ] SL2 (Z) : c 0 mod N } new forms of all levels with integer coefficients and elliptic curves with integer coefficients. The corresponding level N for each elliptic curve E is called the conductor of E, and given E, we know how to compute N.

105 Proof of Fermat s Last Theorem Fermat asserted: If n is an integer > 2, then the following equation has no positive integer solutions x n + y n = z n.

106 Proof of Fermat s Last Theorem Fermat asserted: x n + y n = z n... Gerhard Frey suggested that if a l +b l = c l where a,b,c > 0 and l is an odd prime > 5, then the elliptic curve y 2 = x(x a l )(x + b l ) is not modular. Serre reformulated this problem, and almost proved it; hence, called the epsilon conjecture.

107 Proof of Fermat s Last Theorem Fermat asserted: x n + y n = z n... Serre s Epsilon Conjecture proved by Ken Ribet in 1990: Let E : y 2 = x(x a l )(x + b l ) where a l + b l = c l. The conductor N of E is an even square-free number. Serre s Epsilon Conjecture X 0 (N) E X 0 (N ) E where N = N/p for any odd prime p.

108 Proof of Fermat s Last Theorem Fermat asserted: x n + y n = z n... Serre s Epsilon Conjecture proved by Ken Ribet in 1990: Let E : y 2 = x(x a l )(x + b l ) where a l + b l = c l. The conductor N of E is an even square-free number. Serre s Epsilon Conjecture X 0 (N) X 0 (N 1 ). X 0 (N s ) E E 1. E s X 0 (2) E where N k+1 = N k /p k+1.

109 Proof of Fermat s Last Theorem Fermat asserted: x n + y n = z n... Serre s Epsilon Conjecture proved by Ken Ribet in 1990: Let E : y 2 = x(x a l )(x + b l ) where a l + b l = c l. The conductor N of E is an even square-free number. Taniyama-Shimura new form f for Γ 0 (N). Serre s ε new form g for Γ 0 (m) where m divides N/p for any odd prime factor p of N; he used a connection with a representation. Induction with Serre s ε new form h for Γ 0 (m ) where m divides 2. Eichler-Shimura X 0 (2)(C) E (C).

110 Proof of Fermat s Last Theorem Fermat asserted: x n + y n = z n... Proof: If a l + b l = c l, then E : y 2 = x(x a l )(x + b l ) has conductor N = 2p 1 p s where p i s are odd primes. If we assume the Taniyama-Shimura conjecture and use the epsilon conjecture, then X 0 (2)(C) E (C) where is E is an elliptic curve. X 0 (2) =?

111 Proof of Fermat s Last Theorem Fermat asserted: x n + y n = z n... Proof: If a l + b l = c l, then E : y 2 = x(x a l )(x + b l ) has conductor N = 2p 1 p s where p i s are odd primes. If we assume the Taniyama-Shimura conjecture and use the epsilon conjecture, then X 0 (2)(C) E (C) X 0 (2) =? ( Γ0 (2)\H ) = X0 (2) where is E is an elliptic curve.

112 Proof of Fermat s Last Theorem Fermat asserted: x n + y n = z n... Proof: If a l + b l = c l, then E : y 2 = x(x a l )(x + b l ) has conductor N = 2p 1 p s where p i s are odd primes. If we assume the Taniyama-Shimura conjecture and use the epsilon conjecture, then

113 Proof of Fermat s Last Theorem Fermat asserted: x n + y n = z n... Proof: If a l + b l = c l, then E : y 2 = x(x a l )(x + b l ) has conductor N = 2p 1 p s where p i s are odd primes. If we assume the Taniyama-Shimura conjecture and use the epsilon conjecture, then Q.E.D.

114 Serre s Conjecture

115 Serre s Conjecture

116 Serre s Conjecture

117 Serre s Conjecture

118 Serre s Conjecture

119 Serre s Conjecture Khare and Wittenberger (2005) proved Serre s Conjecture for odd conductor.

120 The Connections {a p : p prime} Elliptic Curves: a p := p + 1 #E(F p ). New Forms: a p := the p-th Fourier coefficient. Galois representations: a p := the trace of an field automorphism called a Frobenius automorphism at p.