Rational isogenies Computing rational isogenies from the equations - PowerPoint PPT Presentation

Rational isogenies Computing rational isogenies from the equations of the kernel David Lubicz, Damien Robert

Damien Robert – Rational isogenies 2 1 Theta functions

Complex abelian varieties Quasi-periodicity: Damien Robert – Rational isogenies 3 upper half space. • Abelian variety over � : A = � g / ( � g + Ω � g ) , where Ω ∊ � g ( � ) the Siegel • The theta functions with characteristic are analytic (quasi periodic) functions on � g . � e π i t ( n + a )Ω( n + a )+ 2 π i t ( n + a )( z + b ) ϑ [ a a , b ∊ � g b ]( z , Ω) = n ∊ � g b ]( z + m 1 Ω + m 2 , Ω) = e 2 π i ( t a · m 2 − t b · m 1 ) − π i t m 1 Ω m 1 − 2 π i t m 1 · z ϑ [ a ϑ [ a b ]( z , Ω) . • Projective coordinates: � n g − 1 A −→ � z �−→ ( ϑ i ( z )) i ∊ Z ( n ) � 0 � ( ., Ω where Z ( n ) = � g / n � g and ϑ i = ϑ n ) . i n

4 Damien Robert – Rational isogenies Theta functions of level n • Translation by a point of n -torsion: ϑ i ( z + m 1 n Ω + m 2 n ) = e − 2 π i t i · m 1 ϑ i + m 2 ( z ) . n • ( ϑ i ) i ∊ Z ( n ) : basis of the theta functions of level n ⇔ A [ n ] = A 1 [ n ] ⊕ A 2 [ n ] : symplectic decomposition. � coordinates system n � 3 • ( ϑ i ) i ∊ Z ( n ) = coordinates on the Kummer variety A / ± 1 n = 2 • Theta null point: ϑ i ( 0 ) i ∊ Z ( n ) = modular invariant.

Riemann Relations Theorem (Koizumi–Kempf) Damien Robert – Rational isogenies 5 Let F be a matrix of rank r such that t F F = ℓ Id r . Let X ∊ ( � g ) r and Y = F ( X ) ∊ ( � g ) r . Let j ∊ ( � g ) r and i = F ( j ) . Then we have � 0 � � 0 � ( Y 1 , Ω ( Y r , Ω ϑ n ) ... ϑ n ) = i 1 i r � � 0 � � 0 � ( X 1 + t 1 , Ω ( X r + t r , Ω ϑ ℓ n ) ... ϑ ℓ n ) , j 1 j r t 1 ,..., t r ∊ 1 ℓ � g / � g F ( t 1 ,..., t r )=( 0,...,0 ) (This is the isogeny theorem applied to F A : A r → A r .) � � • If ℓ = a 2 + b 2 , we take F = a b , so r = 2 . − b a • In general, ℓ = a 2 + b 2 + c 2 + d 2 , we take F to be the matrix of multiplication by a + bi + c j + dk in the quaternions, so r = 4 .

6 where Damien Robert – Rational isogenies The differential addition law ( k = � ) � � χ ( t ) ϑ i + t ( x + y ) ϑ j + t ( x − y ) � . � � χ ( t ) ϑ k + t ( 0 ) ϑ l + t ( 0 ) � = t ∊ Z ( 2 ) t ∊ Z ( 2 ) � � χ ( t ) ϑ − i ′ + t ( y ) ϑ j ′ + t ( y ) � . � � χ ( t ) ϑ k ′ + t ( x ) ϑ l ′ + t ( x ) � . t ∊ Z ( 2 ) t ∊ Z ( 2 ) χ ∊ ˆ Z ( 2 ) , i , j , k , l ∊ Z ( n ) ( i ′ , j ′ , k ′ , l ′ ) = F ( i , j , k , l )   1 1 1 1   F = 1  1 1 − 1 − 1     1 − 1 1 − 1  2 1 − 1 − 1 1

7 Differential Addition Algorithm: Damien Robert – Rational isogenies Example: addition in genus 1 and in level 2 Input: P = ( x 1 : z 1 ) , Q = ( x 2 : z 2 ) and R = P − Q = ( x 3 : z 3 ) with x 3 z 3 ̸ = 0 . Output: P + Q = ( x ′ : z ′ ) . 1. x 0 = ( x 2 1 + z 2 1 )( x 2 2 + z 2 2 ) ; 2. z 0 = A 2 B 2 ( x 2 1 − z 2 1 )( x 2 2 − z 2 2 ) ; 3. x ′ = ( x 0 + z 0 ) / x 3 ; 4. z ′ = ( x 0 − z 0 ) / z 3 ; 5. Return ( x ′ : z ′ ) .

Damien Robert – Rational isogenies 8 2 Computing isogenies (geometrically)

The isogeny formula Proposition Damien Robert – Rational isogenies Corollary 9 A = � g / ( � g + Ω � g ) , B = � g / ( � g + ℓ Ω � n ) ℓ ∧ n = 1, �� � �� � � 0 � 0 · , Ω · , ℓ Ω ϑ A ϑ B b : = ϑ , b : = ϑ b b n n n n Let F be a matrix of rank r such that t F F = ℓ Id r . Let X in ( � g ) r and Y = X F − 1 ∊ ( � g ) r . Let i ∊ ( Z ( n )) r and j = iF − 1 . Then we have � ϑ B i 1 ( Y 1 ) ... ϑ B ϑ A j 1 ( X 1 + t 1 ) ... ϑ A i r ( Y r ) = j r ( X r + t r ) , t 1 ,..., t r ∊ 1 ℓ � g / � g ( t 1 ,..., t r ) F =( 0,...,0 ) � ϑ B k ( 0 ) ϑ B 0 ( 0 ) ... ϑ B ϑ A j 1 ( t 1 ) ... ϑ A ( j = ( k ,0,...,0 ) F − 1 ∊ Z ( n )) 0 ( 0 ) = j r ( t r ) , t 1 ,..., t r ∊ K ( t 1 ,..., t r ) F =( 0,...,0 )

Normalizing points a good lift (coming from the affine theta functions); Damien Robert – Rational isogenies also good lifts. Theorem ([LR12]) 10 normalize the coordinates; • The isogeny formula assume that the points are in affine coordinates. In practice, given A / � q we only have projective coordinates ⇒ we need to • Let P be a projective point on A , and � P be any lift. Note � P = λ � P 0 where � P 0 is • If ℓ = 2 m + 1 , we have that ϑ i (( m + 1 ) � P 0 ) = ϑ − i ( m � P 0 ) ; • By computing m � P , ( m + 1 ) � P using differential additions, we recover an equation λ ℓ = µ . We call P a potential good lift if µ = 1 . Let e 1 ,... e g be a basis of the maximal isotropic kernel K . Assume that we have e i , � chosen a potential good lift for � e i + e j . Then e i , � • Up to an action of Sp 2 g ( � ) , the � e i + e j are good lifts; • If all points in K are then computed using the Riemann relations, they are

Damien Robert – Rational isogenies points of the kernel. 11 Working in the algebra { λ ℓ i = µ i } • Taking potential good lift involve computing ℓ -roots; • But by [CR11], each choice give the same final results; • More precisely, the isogeny formulas only involves the λ ℓ i ; ⇒ We can compute the isogeny over the field defined by the geometric

The algorithm additions. Damien Robert – Rational isogenies 8. Distinguish between the isogeneous curve and its twist. 12 the zeta function known): H hyperelliptic curve of genus 2 over k = � q , J = Jac ( H ) , ℓ odd prime, 2 ℓ ∧ char k = 1 . Compute all rational ( ℓ , ℓ ) -isogenies J �→ Jac ( H ′ ) (we suppose 1. Compute the extension � q n where the geometric points of the maximal isotropic rational kernels of J [ ℓ ] lives. 2. Compute a “symplectic” basis of J [ ℓ ]( � q n ) . 3. Find all rational maximal isotropic kernels K . 4. For each such kernel K , convert its basis from Mumford to theta coordinates of level 2 (Rosenhain then Thomae). 5. Compute the other points in K in theta coordinates using differential 6. Apply the change level formula to recover the theta null point of J / K . 7. Compute the Igusa invariants of J / K (“Inverse Thomae”).

The complexity is much worse over a number field because we need to work Remark with extensions of much higher degree. Damien Robert – Rational isogenies 13 Complexity over � q • The geometric points of the kernel live in a extension k ′ of degree at most ℓ g − 1 over k = � q ; • Computing the normalization factor takes O ( log ℓ ) operations in k ′ ; • Computing the points of the kernel via differential additions take O ( ℓ g ) operations in k ′ ; • If ℓ ≡ 1 ( mod 4 ) , applying the isogeny formula take O ( ℓ g ) operations in k ′ ; • If ℓ ≡ 3 ( mod 4 ) , applying the isogeny formula take O ( ℓ 2 g ) operations in k ′ ; ⇒ The total cost is � O ( ℓ 2 g ) or � O ( ℓ 3 g ) operations in � q .

Damien Robert – Rational isogenies 14 3 Computing isogenies (rationally)

Equations of the Kernel normalization factor. dimension (yet). Damien Robert – Rational isogenies 15 • We suppose that we have (projective) equations of K in diagonal form over the base field k : P 1 ( X 0 , X 1 ) = 0 ... X n X d 0 = P n ( X 0 , X 1 ) • By setting X 0 = 1 we can work with affine coordinates. The projective solutions can be written ( x 0 , x 0 x 1 ,..., x 0 x n ) so X 0 can be seen as the • Note: I don’t know how to obtain equations of K without computing the geometric points of K as we don’t have modular polynomials in higher

Operations on generic points zero, otherwise computing generic differential additions get tricky; irreducible points”. Damien Robert – Rational isogenies 16 • We can work in the algebra A = k [ X 1 ] / ( P 1 ( X 1 )) , each operation takes � O ( ℓ g ) operations in k (this is also “true” for number fields). • A generic point is η = ( X 0 , X 0 X 1 , X 0 P 2 ( X 1 ) ,..., X 0 P n ( X 1 )) ; • By computing differential additions over the algebra A , one can recover a generic normalization X ℓ 0 = µ ∊ A ; • We assume here that none of the coordinates of the geometric points are • If we suppose P 1 irreducible, the Galois action on η give “linearly free

Remark The generic algorithm (first version) This look nice, but in fact this is just a fancy way of working over the splitting Damien Robert – Rational isogenies 17 • Use the Galois action to compute g “linearly independent generic points” η 1 ,..., η g ; • Compute the η i + η j over A ; • Normalize each of these points; • Use differential additions to formally compute each points of the kernel; • Apply the isogeny formula. The result is computed in A but will actually be in k . field of P 1 . In this case we can as well work directly with the geometric points of K so we gain nothing!