The Arithmetic of Elliptic Curves

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

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.

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),...

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

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)

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.

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)

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)

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

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),...

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)

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) =

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) =

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 ) )

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 1 + + m r P r : m i are integers}. example: E : y 2 = x 3 + 17. E(Q) contains P = ( 1,4). E(Q) contains 6P = ( 3703710745675909854372790801 699522170992056989781928164, 260615534370184701706424971733686333999801 18501299163096718454375010188444627517288 ).

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 1 + + m r P r : m i are integers}. example: E : y 2 = x 3 + 17 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

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}.

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.

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).

y 2 + xy + y = x 3 x 2 20067762415575526585033208209338542750930230312178956502x +34481611795030556467032985690390720374855944359319180361266 008296291939448732243429 A few independent points are P 1 = [ 2124150091254381073292137463,259854492051899599030515511070780628911531] P 2 = [2334509866034701756884754537,18872004195494469180868316552803627931531] P 3 = [ 1671736054062369063879038663,251709377261144287808506947241319126049131] P 4 = [2139130260139156666492982137,36639509171439729202421459692941297527531] P 5 = [1534706764467120723885477337,85429585346017694289021032862781072799531] P 6 = [ 2731079487875677033341575063,262521815484332191641284072623902143387531] P 7 = [2775726266844571649705458537,12845755474014060248869487699082640369931] P 8 = [1494385729327188957541833817,88486605527733405986116494514049233411451]

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}.

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

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}. 2 + 4 = 6 = 1 2 3 = 6 = 1 1/2 = 3. x y = 1 for all x 0.

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.

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.

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 3 + 10x 2 with coefficients in Z/4453Z E(Z/4453Z) contains P = (1,3).

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 3 + 10x 2 with coefficients in Z/4453Z E(Z/4453Z) contains P = (1,3). 2P = (97/6 2, 1441/6 3 ) = (4332,3230). 3P = ( 81266098284 ), 262062512821742, undefined 4331 2 4331 3 since 61 is a common factor of 4331 and 4453.

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.

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}?

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)?

Birch and Swinnerton-Dyer Conj. Award: $ 2 6 5 6

Birch and Swinnerton-Dyer Conj. Award: $ 2 6 5 6 Experiment: p<x # E(F p ) p c E (lnx) ranke(q)

Birch and Swinnerton-Dyer Conj. Award: $ 2 6 5 6 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.

Birch and Swinnerton-Dyer Conj. Award: $ 2 6 5 6 Then, L(E,s) is analytically continued to C (proved by Wiles et al);

Birch and Swinnerton-Dyer Conj. Award: $ 2 6 5 6 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 +

Birch and Swinnerton-Dyer Conj. Award: $ 2 6 5 6 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

Birch and Swinnerton-Dyer Conj. Award: $ 2 6 5 6 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).

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).

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.

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}.

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 +

Taniyama-Shimura Conjecture Y. Taniyama G. Shimura

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 = 1 + 5 3

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.

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)). 55 + 69i = 14i = 0 a + bi = 0 if a,b Z.

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)

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).

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?

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

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

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.

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.

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

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

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

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]

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]

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

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

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

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

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

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)

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

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!

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.

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).

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: [ ] [ 0 1 1 0 and 0 1 ] 1 1.

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

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

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.

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

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).

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.

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),

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

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 3 + + a p q p + + and a p = p + 1 # E(F p ).

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.

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.

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.

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).

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) =?

Serre s Conjecture

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

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.