Elliptic curves over F q Formulas for Addition Computer assited proof of associativity Proof of associativity via combinatorial incidence Geometry Bezout Theorem Cayley-Bacharah Theorem Pappus Theorem Pascal’s Theorem Associativity E LLIPTIC CURVES II ( THE ASSOCIATIVITY ) The algebraic proof of associativity the ring of functions on the elliptic curve F RANCESCO P APPALARDI from points to maximal ideal #3 - T HIRD L ECTURE . A UGUST 9 TH 2016 Saigon University Vietnam August 1, 2016
Elliptic curves over F q Formulas for Addition on E (Summary for special equation) Formulas for Addition Computer assited proof of associativity E : y 2 = x 3 + Ax + B Proof of associativity via combinatorial incidence Geometry P 1 = ( x 1 , y 1 ) , P 2 = ( x 2 , y 2 ) ∈ E ( k ) \ {O} , Bezout Theorem Cayley-Bacharah Theorem Addition Laws for the sum of affine points Pappus Theorem Pascal’s Theorem • If P 1 � = P 2 Associativity The algebraic proof of P 1 + E P 2 = O associativity x 1 = x 2 ⇒ the ring of functions on the • elliptic curve x 1 � = x 2 • from points to maximal ideal y 2 − y 1 y 1 x 2 − y 2 x 1 λ = ν = x 2 − x 1 x 2 − x 1 • If P 1 = P 2 P 1 + E P 2 = 2 P 1 = O y 1 = 0 ⇒ • y 1 � = 0 • 3 x 2 x 3 1 + A 1 − Ax 1 − 2 B λ = , ν = − 2 y 1 2 y 1 Then P 1 + E P 2 = ( λ 2 − x 1 − x 2 , − λ 3 + λ ( x 1 + x 2 ) − ν )
Elliptic curves over F q Properties of the operation “ + E ” Formulas for Addition Computer assited proof of Theorem associativity Proof of associativity via The addition law on E ( k ) has the following properties: combinatorial incidence Geometry (a) P + E Q ∈ E ( k ) ∀ P , Q ∈ E ( k ) Bezout Theorem Cayley-Bacharah Theorem (b) P + E O = O + E P = P ∀ P ∈ E ( k ) Pappus Theorem Pascal’s Theorem (c) P + E ( − P ) = O ∀ P ∈ E ( k ) Associativity (d) P + E ( Q + E R ) = ( P + E Q ) + E R ∀ P , Q , R ∈ E ( k ) The algebraic proof of associativity the ring of functions on the (e) P + E Q = Q + E P ∀ P , Q ∈ E ( k ) elliptic curve from points to maximal ideal • ( E ( k ) , + E , O ) commutative group • − P = − ( x 1 , y 1 ) = ( x 1 , − y 1 ) • All group properties are easy except associative law (d) • Today we shall discuss three proofs: Computer assisted proof 1 Combinatorial incidence Geometry proof 2 Algebraic proof via the Picard group 3 • If L / k is a field extension, we can E ( L ) also if E is defined over k ; Theorem holds for ( E ( L ) , + E ) • In particular, if E / k , can consider the groups E ( k ) .
Elliptic curves over F q Computer assited proof of the associativity We need to explain to the computer how to check that: Formulas for Addition Computer assited proof of P + E ( Q + E R ) = ( P + E Q ) + E R ∀ P , Q , R ∈ E associativity Proof of associativity via combinatorial incidence In the case when either one of P , Q , R , P + E Q or Q + E R equals O the above identity is clearly satisfied. Here Geometry we deal with the generic case . i.e. All the points ± P , ± R , ± Q , ± ( Q + E R ) , ± ( P + E Q ) all different.We have Bezout Theorem Cayley-Bacharah Theorem the following Pappus Theorem Pascal’s Theorem Lemma Associativity Let P 1 = ( x 1 , y 1 ) , P 2 = ( x 2 , y 2 ) , P 3 = ( x 3 , y 3 ) ∈ k 2 distinct. Suppose there exists an elliptic curve E such that The algebraic proof of associativity P 1 , P 2 , P 2 ∈ E ( k ) \ {O} and P 1 + P 2 + P 3 = O the ring of functions on the � � elliptic curve x 3 1 − y 2 1 x 1 � 1 � from points to maximal ideal x 3 2 − y 2 = ⇒ det = 0 . � 1 x 2 � 2 � � x 3 3 − y 2 1 x 3 � � 3 Mathematica code L[x_,y_,r_,s_]:=(s-y)/(r-x); M[x_,y_,r_,s_]:=(yr-sx)/(r-x); A[{x_,y_},{r_,s_}]:={(L[x,y,r,s]) 2 -(x+r), -(L[x,y,r,s]) 3 +L[x,y,r,s](x+r)-M[x,y,r,s]} Together[A[A[{x,y},{u,v}],{h,k}]-A[{x,y},A[{u,v},{h,k}]]] det = Det[({{1,x 1 ,x 3 1 -y 2 1 },{1,x 2 ,x 3 2 -y 2 2 },{1,x 3 ,x 3 3 -y 2 3 }})] PolynomialQ[Together[Numerator[Factor[res[[1]]]]/det], {x 1 ,x 2 ,x 3 ,y 1 ,y 2 ,y 3 }] PolynomialQ[Together[Numerator[Factor[res[[2]]]]/det], {x 1 ,x 2 ,x 3 ,y 1 ,y 2 ,y 3 }] One more case: P + E 2 Q = ( P + E Q )+ E Q
Elliptic curves over F q Combinatorial incidence Geometry Formulas for Addition Computer assited proof of associativity We specialize to the case k = C If P ∈ C [ x , y ] has degree d , we consider the affine curve V P = { ( x 0 , y 0 ) ∈ C 2 : P ( x 0 , y 0 ) = 0 } and the Proof of associativity via combinatorial incidence Geometry associated projective curve Bezout Theorem Cayley-Bacharah Theorem P V F P = { [ x 0 , y 0 , z o ] ∈ P 2 ( C ) : F P ( x 0 , y 0 , z 0 ) = 0 } Pappus Theorem Pascal’s Theorem where F P ( X , Y , Z ) := Z d P ( X / Z , Y / Z ) is the corresponding homogenized polynomial. Associativity The algebraic proof of associativity A curve (affine or projective) of degree one is a line P V F P : aX + bY + cZ = 0 1 the ring of functions on the elliptic curve A curve (affine or projective) of degree two is called a quadric 2 from points to maximal ideal P V F P : aX 2 + bXY + cXZ + dY 2 + eYZ + fZ 2 = 0 A curve (affine or projective) of degree three is called a cubic 3 P V F P : aX 3 + bX 2 Y + cX 2 Y + dXY 2 + eXYZ + fXZ 2 + gY 3 + hY 2 Z + jYZ 2 + kZ 3 = 0 A curve my have multiple components when P (or F P ) is not irreducible. When P is irreducible (so is F P ), 4 V P (and P V F P ) are called irreducible Examples: Q : X 2 − XY = 0 is a reducible quadric; C : X ( X 2 + Y 2 + Z 2 ) = 0 is a reducible cubic. In 5 this case we write Q ∩ C = ℓ . Where ℓ : { X = 0 } is a common component. An irreducible quadric is called a conic 6 A cubic which is irreducible, smooth and is also called elliptic curve 7
Elliptic curves over F q Bézout Theorem Formulas for Addition Computer assited proof of associativity We shall use the fundamental: Proof of associativity via combinatorial incidence Geometry Theorem (Bézout Theorem) Bezout Theorem Cayley-Bacharah Theorem Any two (projective) curves with degrees d and d ′ without common components, meet in exactly dd ′ points Pappus Theorem counted with moltiplicity. Pascal’s Theorem Associativity The algebraic proof of For example if there are no common components, a line meets a curve of degree d in d points and a quadric associativity the ring of functions on the curve meets it in 2 d points. Two cubic (irreducible or not) meet in 9 points and so on. elliptic curve from points to maximal ideal Note (Consequences of Linear Algebra) A line depends on 3 parameters; A quadric depends on 6 parameters; A cubic depends on 1 0,...A curve of degree d , depends on ( d + 1 )( d + 2 ) / 2 parameters. Hence, applying linear algebra: Through any 2 given points in P 2 ( C ) it passes a line 1 Through any 5 given points in P 2 ( C ) it passes a quadric 2 Through any 9 given points in P 2 ( C ) it passes a cubic 3 Through any d ( d + 3 ) / 2 given in P 2 ( C ) points it passes a curve of degree d 4
Elliptic curves over F q Note (Example) Formulas for Addition Given [ X j , Y j , Z j ] ∈ P 2 ( C ) , j = 1 , 2 , 3 , 4 , 5, solve for a , b , c , d , e , f the linear system: Computer assited proof of associativity Proof of associativity via combinatorial incidence aX 2 1 + bX 1 Y 1 + cX 1 Z 1 + dY 2 1 + eY 1 Z 1 + fZ 2 Geometry 1 = 0 Bezout Theorem aX 2 2 + bX 2 Y 2 + cX 2 Z 2 + dY 2 1 + eY 2 Z 2 + fZ 2 2 = 0 Cayley-Bacharah Theorem aX 2 3 + bX 3 Y 3 + cX 3 Z 3 + dY 2 3 + eY 3 Z 3 + fZ 2 3 = 0 Pappus Theorem Pascal’s Theorem aX 2 4 + bX 4 Y 4 + cX 4 Z 4 + dY 2 4 + eY 4 Z 1 + fZ 2 4 = 0 Associativity aX 2 5 + bX 5 Y 5 + cX 5 Z 5 + dY 2 5 + eY 5 Z 5 + fZ 2 5 = 0 The algebraic proof of associativity the ring of functions on the elliptic curve from points to maximal ideal For degree 1 , if the points are distinct, the line is unique 1 For degree 2 2 if 5 points are collinear, then there are infinitely many quadric (all reducible) through the 5 points • if 3 points are collinear, then there exists no conic through the 5 points (Bezout Theorem) but only union of lines • For degree 3 3 if 8 points are in a quadric, then there are infinitely many cubic (all reducible) through the 9 points • if 7 points are in a quadric, then there exists no irreducible cubic through the 9 points (Bezout Theorem) but only union • of a quadric and a line if 4 points are collinear, then there exists no irreducible cubic through the 9 points (Bezout Theorem) but only union of a • quadric and a line The notion of General Position may be introduced to recover uniqueness? For example: If five point of P 2 ( C ) are such that no three of them are collinear, then the quadric is unique and it is a conic.
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