Overview on class groups Reducing the defining polynomial Reducing number field defining polynomials: An application to class group computations Alexandre Gélin Laboratoire d’Informatique de Paris 6, UPMC, Sorbonne Universités 09/11/2016 Alexandre Gélin Reducing number field defining polynomials
Overview on class groups Reducing the defining polynomial Number fields K number field ⇒ finite extension of Q ⇒ ∃ T ∈ Z [ X ] monic s.t. K = Q [ X ] / ( T ) . T is a defining polynomial of K . Alexandre Gélin Reducing number field defining polynomials
Overview on class groups Reducing the defining polynomial Number fields K number field ⇒ finite extension of Q ⇒ ∃ T ∈ Z [ X ] monic s.t. K = Q [ X ] / ( T ) . T is a defining polynomial of K . Two interesting structures: Group of ideals Group of units Alexandre Gélin Reducing number field defining polynomials
Overview on class groups Reducing the defining polynomial Number fields K number field ⇒ finite extension of Q ⇒ ∃ T ∈ Z [ X ] monic s.t. K = Q [ X ] / ( T ) . T is a defining polynomial of K . Two interesting structures: Group of ideals Quotient by principal ideals ⇒ class group Cl ( O K ) Group of units Finitely generated ⇒ fundamental units Alexandre Gélin Reducing number field defining polynomials
Overview on class groups Reducing the defining polynomial Number fields K number field ⇒ finite extension of Q ⇒ ∃ T ∈ Z [ X ] monic s.t. K = Q [ X ] / ( T ) . T is a defining polynomial of K . Two interesting structures: Group of ideals Quotient by principal ideals ⇒ class group Cl ( O K ) Group of units Finitely generated ⇒ fundamental units Aim: Compute the structure of the class group. Alexandre Gélin Reducing number field defining polynomials
Overview on class groups State of the art Reducing the defining polynomial General strategy for computation Outline Overview on class groups 1 State of the art General strategy for computation Reducing the defining polynomial 2 Theoretical results Our algorithm Alexandre Gélin Reducing number field defining polynomials
Overview on class groups State of the art Reducing the defining polynomial General strategy for computation Subexponential L -notation : L N (0 , c ) ≈ (log N ) c L N (1 , c ) ≈ N c � ( c + o (1))(log N ) α (log log N ) 1 − α � L N ( α, c ) = exp . 1 5 ) . 1969 Shanks: quadratic number fields in O ( | ∆ K | 1989 Hafner and McCurley: imaginary quadratic number fields in √ L | ∆ K | ( 1 2 , 2) . 1990 Buchmann: all number fields with fixed degree in L | ∆ K | ( 1 2 , 1 . 7) . 2014 Biasse and Fieker: all number fields in L | ∆ K | ( 2 3 + ε ) in general and L | ∆ K | ( 1 2 ) if n ≤ log( | ∆ K | ) 3 / 4 − ε . 2014 Biasse and Fieker: number fields defined by a good polynomial in L | ∆ K | ( a ) , 1 3 ≤ a < 1 2 . Alexandre Gélin Reducing number field defining polynomials
Overview on class groups State of the art Reducing the defining polynomial General strategy for computation Index calculus 1 Factor base Fix a factor base composed of small elements. 2 Relation collection Collect some relations between those small elements, corresponding to linear equations. 3 Linear algebra Deduce the sought result performing linear algebra on the system built. Alexandre Gélin Reducing number field defining polynomials
Overview on class groups State of the art Reducing the defining polynomial General strategy for computation The factor base B = { prime ideals in O K of norm below B } B determined so that B generates the whole class group . Minkowski’s bound: every class contains an ideal of norm smaller than � 4 � r 2 n ! � M K = | ∆ K | n n . π Bach’s bound: assuming GRH, classes of ideals of norm less than 12(log | ∆ K | ) 2 generate the class group. Alexandre Gélin Reducing number field defining polynomials
Overview on class groups State of the art Reducing the defining polynomial General strategy for computation The factor base B = { prime ideals in O K of norm below B } B determined so that B generates the whole class group . Minkowski’s bound: every class contains an ideal of norm smaller than � 4 � r 2 n ! � M K = | ∆ K | n n . π Bach’s bound: assuming GRH, classes of ideals of norm less than 12(log | ∆ K | ) 2 generate the class group. Practically B = L | ∆ K | ( β, c b ) . Alexandre Gélin Reducing number field defining polynomials
Overview on class groups State of the art Reducing the defining polynomial General strategy for computation Relation collection B = ( p 1 , · · · , p N ) Surjective morphism: φ π Z N − → I − → Cl ( O K ) � � i p e i i [ p i ] e i ( e 1 , · · · , e N ) �− → �− → i � ( e 1 , · · · , e N ) ∈ Z N | � � Cl ( O K ) ≃ Z N / i p e i i = ( α ) O K Alexandre Gélin Reducing number field defining polynomials
Overview on class groups State of the art Reducing the defining polynomial General strategy for computation Relation collection B = ( p 1 , · · · , p N ) Surjective morphism: φ π Z N − → I − → Cl ( O K ) � � i p e i i [ p i ] e i ( e 1 , · · · , e N ) �− → �− → i � ( e 1 , · · · , e N ) ∈ Z N | � � Cl ( O K ) ≃ Z N / i p e i i = ( α ) O K Idea: 1 Pick at random A = � i p v i i . 2 Find a reduced ideal A ′ in the same class. 3 If A ′ splits on B ( ⇔ A ′ = � i p v ′ i ) then i A ( A ′ ) − 1 = � i p v i − v ′ is principal . i i Alexandre Gélin Reducing number field defining polynomials
Overview on class groups State of the art Reducing the defining polynomial General strategy for computation Linear algebra Relations stored in a matrix of size about N × N . Structure of the class group given by the Smith Normal Form of the matrix. First compute Hermite Normal Form with a premultiplier because we need kernel vectors. Storjohann and Labahn algorithm, runtime in N ω +1 ( 2 ≤ ω ≤ 3 exponent of matrix multiplication) Alexandre Gélin Reducing number field defining polynomials
Overview on class groups State of the art Reducing the defining polynomial General strategy for computation Verification We find a tentative class group H , but the class group Cl ( O K ) may be only a quotient of H . ⇒ Need an approximation of the class number h K = | Cl ( O K ) | . Alexandre Gélin Reducing number field defining polynomials
Overview on class groups State of the art Reducing the defining polynomial General strategy for computation Verification We find a tentative class group H , but the class group Cl ( O K ) may be only a quotient of H . ⇒ Need an approximation of the class number h K = | Cl ( O K ) | . Class number formula + Euler Product: � h K Reg K = EP · w K · | ∆ K | 2 r 1 · (2 π ) r 2 . Alexandre Gélin Reducing number field defining polynomials
Overview on class groups State of the art Reducing the defining polynomial General strategy for computation Verification We find a tentative class group H , but the class group Cl ( O K ) may be only a quotient of H . ⇒ Need an approximation of the class number h K = | Cl ( O K ) | . Class number formula + Euler Product: � h K Reg K = EP · w K · | ∆ K | 2 r 1 · (2 π ) r 2 . From the relations, we can also deduce a candidate for an approximation of Reg K and perform the verification step. Alexandre Gélin Reducing number field defining polynomials
Overview on class groups Theoretical results Reducing the defining polynomial Our algorithm Outline Overview on class groups 1 State of the art General strategy for computation Reducing the defining polynomial 2 Theoretical results Our algorithm Alexandre Gélin Reducing number field defining polynomials
Overview on class groups Theoretical results Reducing the defining polynomial Our algorithm What is a good polynomial ? We want a polynomial that defines a fixed number field: The degree is fixed, We want the coefficients as small as possible. Alexandre Gélin Reducing number field defining polynomials
Overview on class groups Theoretical results Reducing the defining polynomial Our algorithm What is a good polynomial ? We want a polynomial that defines a fixed number field: The degree is fixed, We want the coefficients as small as possible. Definition Let T = � a k X k ∈ Z [ X ] . The height of T is defined as the maximal norm of its coefficients, namely H ( T ) = max | a k | . k Alexandre Gélin Reducing number field defining polynomials
Overview on class groups Theoretical results Reducing the defining polynomial Our algorithm What is a good polynomial ? We want a polynomial that defines a fixed number field: The degree is fixed, We want the coefficients as small as possible. Definition Let T = � a k X k ∈ Z [ X ] . The height of T is defined as the maximal norm of its coefficients, namely H ( T ) = max | a k | . k Proposition For every defining polynomial T of a degree- n number field K , the discriminants satisfy | ∆ K | ≤ | ∆( T ) | ≤ n 2 n H ( T ) 2 n − 2 . Alexandre Gélin Reducing number field defining polynomials
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