Genus 1 Genus 2 Computing Igusa Class Polynomials Marco Streng Universiteit Leiden Explicit Methods in Number Theory Oberwolfach, July 2009 Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 The Hilbert class polynomial Definition The Hilbert class polynomial H K of an imaginary quadratic number field K is � � � H K = X − j ( E ) ∈ Z [ X ] . { E / C : End( E ) ∼ = O K } Applications: 1. K [ X ] / H K = Hilbert class field of K 2. Elliptic curves over F p : Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 The Hilbert class polynomial Definition The Hilbert class polynomial H K of an imaginary quadratic number field K is � � � H K = X − j ( E ) ∈ Z [ X ] . { E / C : End( E ) ∼ = O K } Applications: 1. K [ X ] / H K = Hilbert class field of K 2. Elliptic curves over F p : if π ∈ O K , ππ = p , then ( H K mod p ) is a product of linear factors and for any root j 0 ∈ F p , exists E with j ( E ) = j 0 and # E ( F p ) = p + 1 − tr( π ) Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Algorithm (sketch) 1. Bijection { E / C with CM by O K } / ∼ Cl K ↔ = [ a ] �→ C / a , a = z Z + Z with z in fund. domain: Im z > 0 , | Re z | ≤ 1 2 , | z | ≥ 1 Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Algorithm (sketch) 1. Bijection { E / C with CM by O K } / ∼ Cl K ↔ = [ a ] �→ C / a , a = z Z + Z with z in fund. domain: Im z > 0 , | Re z | ≤ 1 2 , | z | ≥ 1 2. j ( a ) = q − 1 + 744 + 196884 q + 21493760 q 2 + · · · ( q = e 2 π iz ) (or smarter approximation) 3. Compute H K = � z ( X − j ( z )) ∈ Z [ X ] Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Algorithms ◮ The Hilbert class polynomial is huge: the degree h K grows 1 2 , as do the logarithms of the coefficients. like | D | ◮ Three algorithms: ◮ Complex analytic method, ◮ p-adic, [Couveignes-Henocq 2002, Br¨ oker 2006] ◮ Chinese remainder theorem. [CNST 1998, ALV 2004] Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Algorithms ◮ The Hilbert class polynomial is huge: the degree h K grows 1 2 , as do the logarithms of the coefficients. like | D | ◮ Three algorithms: ◮ Complex analytic method, ◮ p-adic, [Couveignes-Henocq 2002, Br¨ oker 2006] ◮ Chinese remainder theorem. [CNST 1998, ALV 2004] ◮ Under GRH or heuristics, all O ( | D | 1+ ǫ ). ◮ [BBEL 2008, Sutherland 2009] turned CRT (the underdog) into the record holder: − D > 4 · 10 12 , h K = 5 , 000 , 000. Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Complex multiplication ◮ An elliptic curve has CM if End( E ) ∼ = O K with K imaginary quadratic. ◮ A curve of genus 2 has CM if End( J ( C )) ∼ = O K with K a CM field of degree 4. Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Complex multiplication ◮ An elliptic curve has CM if End( E ) ∼ = O K with K imaginary quadratic. ◮ A curve of genus 2 has CM if End( J ( C )) ∼ = O K with K a CM field of degree 4. ◮ A CM field is K 0 ( √ r ) with K 0 totally real and r ∈ K 0 , r << 0. ◮ K is primitive if it does not contain an imaginary quadratic subfield. Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Igusa invariants For 6 � C : y 2 = f ( x ) = a 6 ( x − α i ) , i =1 let ( ij ) = ( α i − α j ) and � a 2 (12) 2 (34) 2 (56) 2 , I 2 = 6 15 � a 4 (12) 2 (23) 2 (31) 2 (45) 2 (56) 2 (64) 2 , = I 4 6 10 � a 6 (12) 2 (23) 2 (31) 2 (45) 2 (56) 2 (64) 2 (14) 2 (25) 2 (36) 2 , I 6 = 6 60 � a 10 ( ij ) 2 = = discr . ( f ) � = 0 . I 10 6 i < j Bijection between set of genus-2 curves and points in a weighted projective 3-space. Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Igusa class polynomials Simplification: i 1 = I 5 i 2 = I 3 i 3 = I 2 2 I 4 2 I 6 I 10 , 2 and I 10 . I 10 Definition The Igusa class polynomials of a primitive quartic CM field K are the polynomials � � � H K , n ( X ) = X − i n ( C ) ∈ Q [ X ] , n ∈ { 1 , 2 , 3 } . { C / C : End( J ( C )) ∼ = O K } / ∼ = Applications: ◮ Class fields ◮ Curves over finite fields Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Algorithms 1. Complex analytic [Spallek 1994, Van Wamelen 1999] 2. 2-adic [GHKRW 2002] 3. Chinese remainder theorem [Eisentr¨ ager-Lauter 2005] Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Algorithms 1. Complex analytic [Spallek 1994, Van Wamelen 1999] 2. 2-adic [GHKRW 2002] 3. Chinese remainder theorem [Eisentr¨ ager-Lauter 2005] No bounds on the runtime: ◮ not explicit enough, ◮ no rounding error analysis for algorithm 1, ◮ no bound on denominator, ◮ no bound on absolute values of i n ( C ). Recently, bounds on the denominator were given [Goren-Lauter 2007], [Goren (unpublished)], [Yang (special cases 2007)]. Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Step 1: Enumerate ∼ = -classes K ⊗ R ∼ = R − alg. C 2 ◮ For Φ an isomorphism and a ⊂ O K , get lattice Λ = Φ( a ) ⊂ C 2 and End( C 2 / Λ) = O K Also need a principal polarization, so { (Φ , a , ξ ) } { C / C with CM by O K } ← → ∼ ∼ = Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Step 1: Enumerate ∼ = -classes K ⊗ R ∼ = R − alg. C 2 ◮ For Φ an isomorphism and a ⊂ O K , get lattice Λ = Φ( a ) ⊂ C 2 and End( C 2 / Λ) = O K Also need a principal polarization, so { (Φ , a , ξ ) } { C / C with CM by O K } ← → ∼ ∼ = ◮ symplectic basis gives Λ = Z Z 2 + Z 2 with Z = Z t , Im Z > 0 Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Step 2: Reduction ◮ Z is unique up to action of � � A � � A t D − C t B = 1 B Sp 4 ( Z ) = M = ∈ GL 4 ( Z ) : , A t C , D t B symmetric C D given by MZ = ( AZ + B )( CZ + D ) − 1 . ◮ Sp 4 ( Z )-reduce Z = ( z jk ), z jk = x jk + iy jk : 1. Im Z reduced: 0 ≤ 2 y 12 ≤ y 11 ≤ y 22 2. | x jk | ≤ 1 2 3. | det CZ + D | ≥ 1 for M ∈ Sp 4 ( Z ). Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Step 3: Igusa invariants ◮ Thomae’s formulae gives an equation for C , given Z , in terms of θ -constants. For c 1 , c 2 ∈ { 0 , 1 2 } 2 , let � exp( π i ( v + c 1 ) Z ( v + c 1 ) t +2 π i ( v + c 1 ) c 2t ) . θ [ c 1 , c 2 ]( Z ) = v ∈ Z 2 ◮ Write out, get pol. in θ ’s j n ( Z ) = ( � all θ ’s � = 0) ∗ ◮ Compute H K , n ∈ Q [ X ]. Have θ < 2 for reduced Z , so need lower bound on θ . Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Bounding θ Let Z = ( z jk ) be reduced and write z jk = x jk + iy jk . ◮ | θ [ c ]( Z ) | < 2. ◮ lower bounds on | θ [ c ]( Z ) | in terms of 1. upper bound on y 22 and 2. (weak) lower bound on | z 12 | . ◮ We know C 2 / ( Z Z 2 + Z 2 ) � = � 2 j =1 C / ( z jj Z + Z ), so z 3 � = 0, hence bound 2 follows from error analysis. Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Bounding the period matrix ◮ Genus 1: given positive upper and lower bounds on Im z ′ for z ′ ∈ C , get upper bound on Im z ′ Im Az ′ = | cz ′ + d | 2 � a � b independent of A = ∈ SL 2 ( Z ) . c d ◮ Similar results for genus 2, so look for good Z ′ , only in proof. Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Bounding the period matrix ◮ Genus 1: given positive upper and lower bounds on Im z ′ for z ′ ∈ C , get upper bound on Im z ′ Im Az ′ = | cz ′ + d | 2 � a � b independent of A = ∈ SL 2 ( Z ) . c d ◮ Similar results for genus 2, so look for good Z ′ , only in proof. ◮ We find Z ′ by taking a = z b + b − 1 with b ⊂ K 0 and z ∈ K . ◮ Bounds we need = upper and lower bounds on N K / Q ( b 2 ( z − z ) O K ) ◮ Lower bounds: pick z , b to maximize, use Minkowski’s convex body theorem. ◮ Upper bound from CM by K = K 0 ( √ r ). Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
Genus 1 Genus 2 Result Theorem Can compute the Igusa class polynomials of primitive quartic CM fields K in time O ( D 7 / 2 D 11 / 2 � ) , 1 0 where D 0 = D ( K 0 ), D = D ( K ) = D 1 D 2 0 and 2 , 3 � | D . The size of the output is between cst . ( D 1 D 0 ) 1 / 2 − ǫ O ( D 2 � 1 D 3 and 0 ) ◮ Ramification assumptions come from Goren’s unpublished work and it ‘should be’ possible to remove them. ◮ Preprint on Arxiv and on my web page http://www.math.leidenuniv.nl/ ∼ streng Marco Streng Universiteit Leiden Computing Igusa Class Polynomials
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