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The Weil Conjectures for Elliptic Curves The Weil Conjectures for Elliptic Curves Gaurish Korpal Elliptic curves Definition Group structure Weil conjectures The Weil Conjectures for Elliptic Curves Zeta function Statement Counting points over finite fields Main lemma Gaurish Korpal An example The University of Arizona, Tucson December 10, 2020 Outline The Weil Conjectures for Elliptic Curves Gaurish Korpal 1 Elliptic curves Elliptic curves Definition Definition Group structure Group structure Weil conjectures Zeta function Statement Counting points 2 Weil conjectures over finite fields Zeta function Main lemma An example Statement 3 Counting points over finite fields Main lemma An example The Weil Conjectures for Elliptic Curves Gaurish Korpal Elliptic curves Definition Group structure Weil conjectures Zeta function Statement Counting points over finite fields Elliptic curves Main lemma An example Definition The Weil Conjectures for Elliptic Curves Since we won't be discussing any proofs in this seminar, it will be Gaurish Korpal sufficient to restrict ourselves to the following definition of elliptic Elliptic curves curves: Definition Group structure Elliptic curve Weil conjectures Zeta function An elliptic curve over a field K is a nonsingular projective plane curve Statement over K given by an affine equation of the form Counting points over finite fields 2 3 2 Main lemma y + a1xy + a3y = x + a2x + a4x + a6 An example with an extra point O = [0 : 1 : 0] at infinity. Therefore, in homegenous coordinates the equation will be: 2 2 3 2 2 3 Y Z + a1XYZ + a3YZ = X + a2X Z + a4XZ + a6Z where x = X =Z and y = Y =Z. Examples and non-examples over R The Weil Conjectures for Elliptic Curves 2 3 Gaurish Korpal y = x y 2 = x3 + x2 Elliptic curves y Definition 2 3 2 Group structure y = x + x + x 2 3 2 Weil conjectures y = x + x + x + 1 2 Zeta function 2 3 Statement y = x x Counting points − over finite fields Main lemma An example x 3 2 1 1 2 3 − − − 2 − Group structure The Weil Conjectures for Elliptic Curves Gaurish Korpal 2 Elliptic curves Since the equation of an elliptic curve E P has degree three, any 2 ⊂ Definition line L P intersects E at exactly three points, counted with Group structure multiplicities⊂ (B´ezout'stheorem). Keeping this fact in mind, we Weil conjectures Zeta function define the following addition law on the points of E: Statement Counting points Addition law over finite fields Main lemma Let P; Q E be two points. Suppose L is the line through P and Q An example (if P = Q2, then let L be the tangent line to E at P), and R is the third point of intersection of L with E. Then, let L0 be the line through R and O which will intersect E at a third point. We denote that third point by P + Q. Illustration over R The Weil Conjectures for Elliptic Curves Gaurish Korpal 12 R 3 4 Q P Elliptic curves Q' Definition P P Group structure P Weil conjectures P' Q Zeta function Q Statement R' Counting points P + Q = Q' over finite fields P + Q = R' 2Q = P' P + Q = O 2P = O Main lemma An example Figure: Addition of points on elliptic curve [SuperManu, CC BY-SA 3.0, via Wikimedia Commons, https://commons.wikimedia.org/wiki/File:ECClines-2.svg] Key observation The addition law makes E into an abelian group with identity element O. The Weil Conjectures for Elliptic Curves Gaurish Korpal Elliptic curves Definition Group structure Weil conjectures Zeta function Statement Counting points over finite fields Weil conjectures Main lemma An example Zeta function The Weil Conjectures for Fq is a finite field with q elements, where q is a power of prime p. Elliptic Curves Fqn for each integer n 1, is the extension of Fq of degree n, so Gaurish Korpal n ≥ #Fqn = q . Elliptic curves V = q is a projective variety, such that V is the set of solutions to Definition F Group structure f1(x0;:::; xN ) = = fm(x0;:::; xN ) = 0 Weil conjectures ··· Zeta function Statement where f1;:::; fm are homogeneous polynomials with coefficients Counting points in Fq. over finite fields Main lemma V (Fqn ) is the set of points of V with coordinates in Fqn An example We encode the number of points in V (Fqn ) for all n 1 into the following generating function: ≥ Zeta function The zeta function of V =Fq is the power series 1 ! X tn n ZV = q (t) = exp (#V (Fq )) F n n=1 Statement for elliptic curves The Weil Conjectures for When the variety V is an elliptic curve E, the zeta function exhibits Elliptic Curves the following properties referred to as Weil conjectures: Gaurish Korpal Weil conjectures for elliptic curves Elliptic curves Definition Let E=Fq be an elliptic curve. Then there is an a Z such that Group structure 2 Weil conjectures 2 Zeta function 1 at + qt Statement ZE= q (t) = − Q(t) F (1 t)(1 qt) 2 Counting points over finite fields − − Main lemma Further, over C we have An example 1 at + qt2 = (1 αt)(1 βt) − − − with α = β = pq. j j j j The zeta function also satisfies the following functional equation: 1 Z = Z (t) E=Fq qt E=Fq The Weil Conjectures for Elliptic Curves Gaurish Korpal Elliptic curves Definition Group structure Weil conjectures Zeta function Statement Counting points over finite fields Counting points over finite fields Main lemma An example Main lemma The Weil Conjectures for Elliptic Curves Gaurish Korpal Following is the main lemma which enables us to simplify the zeta Elliptic curves function and get the rational function stated above. Definition Group structure Weil conjectures Lemma Zeta function Statement Let E=Fq be an elliptic curve and a = q + 1 #E(Fq). If α; β C 2 − 2 Counting points are the roots of the polynomial p(t) = t at + q. Then α and β are over finite fields − Main lemma complex conjugates satisfying α = β = pq and for every n 1, j j j j ≥ An example we have n n n #E(Fqn ) = q + 1 α β − − The quantity a is called the trace of Frobenius, because it is equal to the trace of the q-power Frobenius map considered as a linear transformation of the `-adic Tate module of E. An example over F2 The Weil Conjectures for Elliptic Curves Consider the following elliptic curve over F2: Gaurish Korpal E : y 2 y = x 3 x 2 Elliptic curves − − Definition 2 Group structure Hence, E is a variety in P given by the homogeneous equation: Weil conjectures Zeta function E¯ : Y 2Z YZ 2 = X 3 X 2Z Statement − − Counting points over finite fields Since #F2 = 2 is a small number, we can easily find that Main lemma An example [0 : 0 : 1]; [1 : 0 : 1]; [1 : 1 : 1]; [0 : 1 : 1] and O = [0 : 1 : 0] are the only points lying on E over F2. That is, #E(F2) = 5 . Therefore, a = 2 + 1 5 = 2 and we can get α; β by finding the roots of p(t) = t2 + 2−t + 2.− Hence, we have n n n #E(F2n ) = 2 + 1 ( 1 + i) ( 1 i) − − − − − where i = p 1. − An example over F2 (contd.) The Weil Conjectures for Elliptic Curves We can simplify the above formula to get: Gaurish Korpal Elliptic curves n n +1 3n n 2 π Definition #E(F2 ) = 2 + 1 2 cos Group structure − 4 8 n n +1 Weil conjectures 2 + 1 2 2 if n 0 (mod 8) Zeta function > > n − n+1 ≡ Statement >2 + 1 + 2 2 if n 1 (mod 8) <> Counting points n ≡ ± over finite fields = 2 + 1 if n 2 (mod 8) Main lemma > n n+1 ≡ ± > 2 An example >2 + 1 2 if n 3 (mod 8) > n − n +1 ≡ ± :2 + 1 + 2 2 if n 4 (mod 8) ≡ Observe that, #E(F2) = #E(F4) = #E(F8) = 5, hence there are no new solutions even though we are going to a bigger field. Using this formula, we can easily find that 10 #E(F1024) = 2 + 1 = 1025 . General estimate The Weil Conjectures for Hasse's inequality Elliptic Curves Gaurish Korpal q + 1 #E(Fq) 2pq j − j ≤ Elliptic curves Definition For example, for q = 1021, the prime nearest to 210 = 1024, Group structure p Weil conjectures #E(F1021) = 1000 satisfying 1021 + 1 1000 = 22 2 1021 64. Zeta function j − j ≤ ≈ Statement 1000 Counting points over finite fields 800 Main lemma An example 600 400 200 200 400 600 800 1000 2 3 Figure: Plot of 999 affine points on y − y = x − x over F1021. SageMathCell code: E=EllipticCurve(GF(1021),[0,-1,-1,0,0]); A=E.plot(); A.save('v1.pgf').
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