CM-Points on Straight Lines A joint work with Amalia - PowerPoint PPT Presentation

Complex Multiplication ◮ End (Λ) = { α ∈ C : α Λ ⊆ Λ } ; End Λ ⊇ Z ◮ Λ has Complex Multiplication if End Λ � Z ◮ Λ = � τ, 1 � has CM ⇐ ⇒ [ Q ( τ ) : Q ] = 2 ◮ In this case: ◮ O = End Λ is an order in K = Q ( τ ) of discriminant ∆ = Df 2 ; ◮ D = D K < 0 the fundamental discriminant ; ◮ f = [ O K : O ] the conductor ; √ � � ◮ O = Z + f O K = Z ∆+ ∆ =: O ∆ ; 2 ◮ if τ is root of at 2 + bt + c ∈ Z [ t ] , ( a , b , c ) = 1 then ∆ = b 2 − 4 ac and τ = − b + √ ∆ . 2 a

Class Field Theory ◮ j ( τ ) algebraic number (even algebraic integer)

Class Field Theory ◮ j ( τ ) algebraic number (even algebraic integer) ◮ K ( j ( τ )) is abelian extension of K = Q ( τ ) (the “Ring Class Field”)

Class Field Theory ◮ j ( τ ) algebraic number (even algebraic integer) ◮ K ( j ( τ )) is abelian extension of K = Q ( τ ) (the “Ring Class Field”) ◮ [ K ( j ( τ )) : K ] = [ Q ( j ( τ )) : Q ] = h (∆)

Class Field Theory ◮ j ( τ ) algebraic number (even algebraic integer) ◮ K ( j ( τ )) is abelian extension of K = Q ( τ ) (the “Ring Class Field”) ◮ [ K ( j ( τ )) : K ] = [ Q ( j ( τ )) : Q ] = h (∆) ◮ h (∆) the class number of the order O ∆

Class Field Theory ◮ j ( τ ) algebraic number (even algebraic integer) ◮ K ( j ( τ )) is abelian extension of K = Q ( τ ) (the “Ring Class Field”) ◮ [ K ( j ( τ )) : K ] = [ Q ( j ( τ )) : Q ] = h (∆) ◮ h (∆) the class number of the order O ∆ ◮ moreover: Gal ( K ( j ( τ )) / K ) = Cl (∆)

The Class Number ◮ h (∆) → ∞ as | ∆ | → ∞ (Siegel)

The Class Number ◮ h (∆) → ∞ as | ∆ | → ∞ (Siegel) ◮ In other words, for h ∈ Z > 0 there exist finitely many ∆ with h (∆) = h .

The Class Number ◮ h (∆) → ∞ as | ∆ | → ∞ (Siegel) ◮ In other words, for h ∈ Z > 0 there exist finitely many ∆ with h (∆) = h . ◮ In particular (Heegner-Stark) there exist thirteen ∆ with h (∆) = 1 (the corresponding j belong to Z ): − 3 · 2 2 − 3 · 3 3 − 4 · 2 2 − 7 · 2 2 ∆ − 3 − 4 − 7 − 8 2 4 3 3 5 3 − 2 15 3 · 5 3 2 6 3 3 2 3 3 3 11 3 − 3 3 5 3 3 3 5 3 17 3 2 6 5 3 j 0 ∆ − 11 − 19 − 43 − 67 − 163 − 2 15 − 2 15 3 3 − 2 18 3 3 5 3 − 2 15 3 3 5 3 11 3 − 2 18 3 3 5 3 23 3 29 3 j

The Class Number ◮ h (∆) → ∞ as | ∆ | → ∞ (Siegel) ◮ In other words, for h ∈ Z > 0 there exist finitely many ∆ with h (∆) = h . ◮ In particular (Heegner-Stark) there exist thirteen ∆ with h (∆) = 1 (the corresponding j belong to Z ): − 3 · 2 2 − 3 · 3 3 − 4 · 2 2 − 7 · 2 2 ∆ − 3 − 4 − 7 − 8 2 4 3 3 5 3 − 2 15 3 · 5 3 2 6 3 3 2 3 3 3 11 3 − 3 3 5 3 3 3 5 3 17 3 2 6 5 3 j 0 ∆ − 11 − 19 − 43 − 67 − 163 − 2 15 − 2 15 3 3 − 2 18 3 3 5 3 − 2 15 3 3 5 3 11 3 − 2 18 3 3 5 3 23 3 29 3 j ◮ A funny example (Hermite): √ 163 = 262537412640768743 . 99999999999925007 . . . e π

The Class Number ◮ h (∆) → ∞ as | ∆ | → ∞ (Siegel) ◮ In other words, for h ∈ Z > 0 there exist finitely many ∆ with h (∆) = h . ◮ In particular (Heegner-Stark) there exist thirteen ∆ with h (∆) = 1 (the corresponding j belong to Z ): − 3 · 2 2 − 3 · 3 3 − 4 · 2 2 − 7 · 2 2 ∆ − 3 − 4 − 7 − 8 2 4 3 3 5 3 − 2 15 3 · 5 3 2 6 3 3 2 3 3 3 11 3 − 3 3 5 3 3 3 5 3 17 3 2 6 5 3 j 0 ∆ − 11 − 19 − 43 − 67 − 163 − 2 15 − 2 15 3 3 − 2 18 3 3 5 3 − 2 15 3 3 5 3 11 3 − 2 18 3 3 5 3 23 3 29 3 j ◮ A funny example (Hermite): √ 163 = 262537412640768743 . 99999999999925007 . . . e π

The Class Number ◮ h (∆) → ∞ as | ∆ | → ∞ (Siegel) ◮ In other words, for h ∈ Z > 0 there exist finitely many ∆ with h (∆) = h . ◮ In particular (Heegner-Stark) there exist thirteen ∆ with h (∆) = 1 (the corresponding j belong to Z ): − 3 · 2 2 − 3 · 3 3 − 4 · 2 2 − 7 · 2 2 ∆ − 3 − 4 − 7 − 8 2 4 3 3 5 3 − 2 15 3 · 5 3 2 6 3 3 2 3 3 3 11 3 − 3 3 5 3 3 3 5 3 17 3 2 6 5 3 j 0 ∆ − 11 − 19 − 43 − 67 − 163 − 2 15 − 2 15 3 3 − 2 18 3 3 5 3 − 2 15 3 3 5 3 11 3 − 2 18 3 3 5 3 23 3 29 3 j ◮ A funny example (Hermite): √ 163 = 262537412640768743 . 99999999999925007 . . . e π √ ◮ τ = 1 + − 163 2

The Class Number ◮ h (∆) → ∞ as | ∆ | → ∞ (Siegel) ◮ In other words, for h ∈ Z > 0 there exist finitely many ∆ with h (∆) = h . ◮ In particular (Heegner-Stark) there exist thirteen ∆ with h (∆) = 1 (the corresponding j belong to Z ): − 3 · 2 2 − 3 · 3 3 − 4 · 2 2 − 7 · 2 2 ∆ − 3 − 4 − 7 − 8 2 4 3 3 5 3 − 2 15 3 · 5 3 2 6 3 3 2 3 3 3 11 3 − 3 3 5 3 3 3 5 3 17 3 2 6 5 3 j 0 ∆ − 11 − 19 − 43 − 67 − 163 − 2 15 − 2 15 3 3 − 2 18 3 3 5 3 − 2 15 3 3 5 3 11 3 − 2 18 3 3 5 3 23 3 29 3 j ◮ A funny example (Hermite): √ 163 = 262537412640768743 . 99999999999925007 . . . e π √ ◮ τ = 1 + − 163 2 √ 163 = − e 2 π i τ ≈ − j ( τ ) + 744 ∈ Z ◮ e π

The Class Number ◮ h (∆) → ∞ as | ∆ | → ∞ (Siegel) ◮ In other words, for h ∈ Z > 0 there exist finitely many ∆ with h (∆) = h . ◮ In particular (Heegner-Stark) there exist thirteen ∆ with h (∆) = 1 (the corresponding j belong to Z ): − 3 · 2 2 − 3 · 3 3 − 4 · 2 2 − 7 · 2 2 ∆ − 3 − 4 − 7 − 8 2 4 3 3 5 3 − 2 15 3 · 5 3 2 6 3 3 2 3 3 3 11 3 − 3 3 5 3 3 3 5 3 17 3 2 6 5 3 j 0 ∆ − 11 − 19 − 43 − 67 − 163 − 2 15 − 2 15 3 3 − 2 18 3 3 5 3 − 2 15 3 3 5 3 11 3 − 2 18 3 3 5 3 23 3 29 3 j ◮ A funny example (Hermite): √ 163 = 262537412640768743 . 99999999999925007 . . . e π √ ◮ τ = 1 + − 163 2 √ 163 = − e 2 π i τ ≈ − j ( τ ) + 744 ∈ Z ◮ e π ◮ Currently all ∆ with h ∆ ≤ 100 are known (Watkins 2006).

Complex Multiplication Lattices j -invariant Complex Multiplication Class Field Theory The Class Number Theorem of André Special Points and Special Curves Theorem of André CM-Points on Straight Lines Kühne’s “uniformity observation” CM-Points on Straight Lines The Proof Equality of CM-fields The Proof Proof of Theorem ECMF Discriminants with Class Group Annihilated by 2 Proof of Theorem ECMF

Special Points and Special Curves τ imaginary quadratic ⇒ j ( τ ) ∈ ¯ Q

Special Points and Special Curves τ imaginary quadratic ⇒ j ( τ ) ∈ ¯ Q CM-point or special point on C 2 : � � j ( τ 1 ) , j ( τ 2 ) .

Special Points and Special Curves τ imaginary quadratic ⇒ j ( τ ) ∈ ¯ Q CM-point or special point on C 2 : � � j ( τ 1 ) , j ( τ 2 ) . Question: can an irreducible plane curve F ( x 1 , x 2 ) = 0 contain infinitely many CM-points?

Special Points and Special Curves τ imaginary quadratic ⇒ j ( τ ) ∈ ¯ Q CM-point or special point on C 2 : � � j ( τ 1 ) , j ( τ 2 ) . Question: can an irreducible plane curve F ( x 1 , x 2 ) = 0 contain infinitely many CM-points? Special curves: ◮ vertical line x 1 = j ( τ 1 )

Special Points and Special Curves τ imaginary quadratic ⇒ j ( τ ) ∈ ¯ Q CM-point or special point on C 2 : � � j ( τ 1 ) , j ( τ 2 ) . Question: can an irreducible plane curve F ( x 1 , x 2 ) = 0 contain infinitely many CM-points? Special curves: ◮ vertical line x 1 = j ( τ 1 ) ◮ horizontal line x 2 = j ( τ 2 )

Special Points and Special Curves τ imaginary quadratic ⇒ j ( τ ) ∈ ¯ Q CM-point or special point on C 2 : � � j ( τ 1 ) , j ( τ 2 ) . Question: can an irreducible plane curve F ( x 1 , x 2 ) = 0 contain infinitely many CM-points? Special curves: ◮ vertical line x 1 = j ( τ 1 ) ◮ horizontal line x 2 = j ( τ 2 ) ◮ Y 0 ( N ) realized as Φ N ( x 1 , x 2 ) = 0

Modular Curves and Modular Polynomials ◮ Φ N ( x 1 , x 2 ) N th “modular polynomial”: Φ N � � j ( z ) , j ( Nz ) = 0

Modular Curves and Modular Polynomials ◮ Φ N ( x 1 , x 2 ) N th “modular polynomial”: Φ N � � j ( z ) , j ( Nz ) = 0 ◮ � � j ( τ ) , j ( N τ ) ∈ Y 0 ( N ) for every τ .

Modular Curves and Modular Polynomials ◮ Φ N ( x 1 , x 2 ) N th “modular polynomial”: Φ N � � j ( z ) , j ( Nz ) = 0 ◮ � � j ( τ ) , j ( N τ ) ∈ Y 0 ( N ) for every τ . ◮ More generally: for γ ∈ GL 2 ( Q ) there exists N such that � � j ( τ ) , j ( γτ ) ∈ Y 0 ( N ) for every τ .

Modular Curves and Modular Polynomials ◮ Φ N ( x 1 , x 2 ) N th “modular polynomial”: Φ N � � j ( z ) , j ( Nz ) = 0 ◮ � � j ( τ ) , j ( N τ ) ∈ Y 0 ( N ) for every τ . ◮ More generally: for γ ∈ GL 2 ( Q ) there exists N such that � � j ( τ ) , j ( γτ ) ∈ Y 0 ( N ) for every τ .

Modular Curves and Modular Polynomials ◮ Φ N ( x 1 , x 2 ) N th “modular polynomial”: Φ N � � j ( z ) , j ( Nz ) = 0 ◮ � � j ( τ ) , j ( N τ ) ∈ Y 0 ( N ) for every τ . ◮ More generally: for γ ∈ GL 2 ( Q ) there exists N such that � � j ( τ ) , j ( γτ ) ∈ Y 0 ( N ) for every τ . Polynomials Φ N , N ≤ 3 Φ 1 ( x , y ) = x − y Φ 2 ( x , y ) = − x 2 y 2 + x 3 + y 3 + 1488 x 2 y + 1488 xy 2 + 40773375 xy − 162000 x 2 − 162000 y 2 + 8748000000 x + 8748000000 y − 157464000000000 x 4 + y 4 − x 3 y 3 + 2232 x 3 y 2 + 2232 x 2 y 3 − 1069956 x 3 y − 1069956 xy 3 Φ 3 ( x , y ) = + 36864000 x 3 + 36864000 y 3 + 2587918086 x 2 y 2 + 8900222976000 x 2 y + 8900222976000 xy 2 + 452984832000000 x 2 + 452984832000000 y 2 − 770845966336000000 xy + 1855425871872000000000 x + 1855425871872000000000 y

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points.

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points. Different proofs:

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points. Different proofs: ◮ André (1998)

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points. Different proofs: ◮ André (1998) ◮ Edixhoven (1998, GRH)

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points. Different proofs: ◮ André (1998) ◮ Edixhoven (1998, GRH) ◮ Pila (2009, extends to higher dimension)

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points. Different proofs: ◮ André (1998) ◮ Edixhoven (1998, GRH) ◮ Pila (2009, extends to higher dimension) All non-effective, use Siegel-Brauer

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points. Different proofs: ◮ André (1998) ◮ Edixhoven (1998, GRH) ◮ Pila (2009, extends to higher dimension) All non-effective, use Siegel-Brauer ◮ Breuer (2001, GRH, effective)

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points. Different proofs: ◮ André (1998) ◮ Edixhoven (1998, GRH) ◮ Pila (2009, extends to higher dimension) All non-effective, use Siegel-Brauer ◮ Breuer (2001, GRH, effective) ◮ B., Masser, Zannier (2013, effective)

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points. Different proofs: ◮ André (1998) ◮ Edixhoven (1998, GRH) ◮ Pila (2009, extends to higher dimension) All non-effective, use Siegel-Brauer ◮ Breuer (2001, GRH, effective) ◮ B., Masser, Zannier (2013, effective) ◮ Kühne (2012, 2013, effective)

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points. Different proofs: ◮ André (1998) ◮ Edixhoven (1998, GRH) ◮ Pila (2009, extends to higher dimension) All non-effective, use Siegel-Brauer ◮ Breuer (2001, GRH, effective) ◮ B., Masser, Zannier (2013, effective) ◮ Kühne (2012, 2013, effective)

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points. Different proofs: ◮ André (1998) ◮ Edixhoven (1998, GRH) ◮ Pila (2009, extends to higher dimension) All non-effective, use Siegel-Brauer ◮ Breuer (2001, GRH, effective) ◮ B., Masser, Zannier (2013, effective) ◮ Kühne (2012, 2013, effective) Particular cases :

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points. Different proofs: ◮ André (1998) ◮ Edixhoven (1998, GRH) ◮ Pila (2009, extends to higher dimension) All non-effective, use Siegel-Brauer ◮ Breuer (2001, GRH, effective) ◮ B., Masser, Zannier (2013, effective) ◮ Kühne (2012, 2013, effective) Particular cases : ◮ no CM-points on x 1 + x 2 = 1 (Kühne 2013)

Theorem of André Theorem (André, 1998) A non-special irreducible plane curve can have only finitely many special points. Different proofs: ◮ André (1998) ◮ Edixhoven (1998, GRH) ◮ Pila (2009, extends to higher dimension) All non-effective, use Siegel-Brauer ◮ Breuer (2001, GRH, effective) ◮ B., Masser, Zannier (2013, effective) ◮ Kühne (2012, 2013, effective) Particular cases : ◮ no CM-points on x 1 + x 2 = 1 (Kühne 2013) ◮ no CM-points on x 1 x 2 = 1 (B., Masser, Zannier 2013)

Complex Multiplication Lattices j -invariant Complex Multiplication Class Field Theory The Class Number Theorem of André Special Points and Special Curves Theorem of André CM-Points on Straight Lines Kühne’s “uniformity observation” CM-Points on Straight Lines The Proof Equality of CM-fields The Proof Proof of Theorem ECMF Discriminants with Class Group Annihilated by 2 Proof of Theorem ECMF

Kühne’s “uniformity observation” Kühne (2013): ◮ If ( j ( τ 1 ) , j ( τ 2 )) belongs to a non-special straight line over a n.f. L , then | ∆ 1 | , | ∆ 2 | ≤ c eff ([ L : Q ]) .

Kühne’s “uniformity observation” Kühne (2013): ◮ If ( j ( τ 1 ) , j ( τ 2 )) belongs to a non-special straight line over a n.f. L , then | ∆ 1 | , | ∆ 2 | ≤ c eff ([ L : Q ]) . ( ∆ i discriminant of the CM-order End � τ i , 1 � )

Kühne’s “uniformity observation” Kühne (2013): ◮ If ( j ( τ 1 ) , j ( τ 2 )) belongs to a non-special straight line over a n.f. L , then | ∆ 1 | , | ∆ 2 | ≤ c eff ([ L : Q ]) . ( ∆ i discriminant of the CM-order End � τ i , 1 � ) ◮ In particular: all CM-points belonging to non-special straight lines defined over Q can (in principle) be listed explicitly.

Kühne’s “uniformity observation” Kühne (2013): ◮ If ( j ( τ 1 ) , j ( τ 2 )) belongs to a non-special straight line over a n.f. L , then | ∆ 1 | , | ∆ 2 | ≤ c eff ([ L : Q ]) . ( ∆ i discriminant of the CM-order End � τ i , 1 � ) ◮ In particular: all CM-points belonging to non-special straight lines defined over Q can (in principle) be listed explicitly. ◮ Bajolet (2014): software to determine all CM-points on a given line.

CM-Points on Straight Lines Special straight lines :

CM-Points on Straight Lines Special straight lines : ◮ vertical x 1 = j ( τ 1 ) and horizontal x 2 = j ( τ 2 ) lines;

CM-Points on Straight Lines Special straight lines : ◮ vertical x 1 = j ( τ 1 ) and horizontal x 2 = j ( τ 2 ) lines; ◮ x 1 = x 2 (which is Y 0 ( 1 ) ).

CM-Points on Straight Lines Special straight lines : ◮ vertical x 1 = j ( τ 1 ) and horizontal x 2 = j ( τ 2 ) lines; ◮ x 1 = x 2 (which is Y 0 ( 1 ) ).

CM-Points on Straight Lines Special straight lines : ◮ vertical x 1 = j ( τ 1 ) and horizontal x 2 = j ( τ 2 ) lines; ◮ x 1 = x 2 (which is Y 0 ( 1 ) ). Obvious cases ( j ( τ 1 ) , j ( τ 2 )) belongs to a non-special straight line over Q in one of the following cases:

CM-Points on Straight Lines Special straight lines : ◮ vertical x 1 = j ( τ 1 ) and horizontal x 2 = j ( τ 2 ) lines; ◮ x 1 = x 2 (which is Y 0 ( 1 ) ). Obvious cases ( j ( τ 1 ) , j ( τ 2 )) belongs to a non-special straight line over Q in one of the following cases: ◮ j ( τ 1 ) , j ( τ 2 ) ∈ Q ;

CM-Points on Straight Lines Special straight lines : ◮ vertical x 1 = j ( τ 1 ) and horizontal x 2 = j ( τ 2 ) lines; ◮ x 1 = x 2 (which is Y 0 ( 1 ) ). Obvious cases ( j ( τ 1 ) , j ( τ 2 )) belongs to a non-special straight line over Q in one of the following cases: ◮ j ( τ 1 ) , j ( τ 2 ) ∈ Q ; ◮ j ( τ 1 ) � = j ( τ 2 ) , Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = K , [ K : Q ] = 2.

CM-Points on Straight Lines Special straight lines : ◮ vertical x 1 = j ( τ 1 ) and horizontal x 2 = j ( τ 2 ) lines; ◮ x 1 = x 2 (which is Y 0 ( 1 ) ). Obvious cases ( j ( τ 1 ) , j ( τ 2 )) belongs to a non-special straight line over Q in one of the following cases: ◮ j ( τ 1 ) , j ( τ 2 ) ∈ Q ; ◮ j ( τ 1 ) � = j ( τ 2 ) , Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = K , [ K : Q ] = 2.

CM-Points on Straight Lines Special straight lines : ◮ vertical x 1 = j ( τ 1 ) and horizontal x 2 = j ( τ 2 ) lines; ◮ x 1 = x 2 (which is Y 0 ( 1 ) ). Obvious cases ( j ( τ 1 ) , j ( τ 2 )) belongs to a non-special straight line over Q in one of the following cases: ◮ j ( τ 1 ) , j ( τ 2 ) ∈ Q ; ◮ j ( τ 1 ) � = j ( τ 2 ) , Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = K , [ K : Q ] = 2. (Can be easily listed.)

CM-Points on Straight Lines Special straight lines : ◮ vertical x 1 = j ( τ 1 ) and horizontal x 2 = j ( τ 2 ) lines; ◮ x 1 = x 2 (which is Y 0 ( 1 ) ). Obvious cases ( j ( τ 1 ) , j ( τ 2 )) belongs to a non-special straight line over Q in one of the following cases: ◮ j ( τ 1 ) , j ( τ 2 ) ∈ Q ; ◮ j ( τ 1 ) � = j ( τ 2 ) , Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = K , [ K : Q ] = 2. (Can be easily listed.) Theorem (A., B., Pizarro; May 2014) If a CM-points belongs to a non-special straight line over Q then we have one of the two cases above.

Complex Multiplication Lattices j -invariant Complex Multiplication Class Field Theory The Class Number Theorem of André Special Points and Special Curves Theorem of André CM-Points on Straight Lines Kühne’s “uniformity observation” CM-Points on Straight Lines The Proof Equality of CM-fields The Proof Proof of Theorem ECMF Discriminants with Class Group Annihilated by 2 Proof of Theorem ECMF

Equality of CM-fields Theorem ECMF (based on ideas of André, Edixhoven and Kühne) Assume that L = Q ( j ( τ 1 )) = Q ( j ( τ 2 )) .

Equality of CM-fields Theorem ECMF (based on ideas of André, Edixhoven and Kühne) Assume that L = Q ( j ( τ 1 )) = Q ( j ( τ 2 )) . ◮ If Q ( τ 1 ) � = Q ( τ 2 ) then L is the table: Field L ∆ Cl (∆) Q − 3 , − 4 , − 7 , − 8 , − 11 , − 12 , − 16 , − 19 , − 27 , − 28 , − 43 , − 67 , − 163 trivial √ Q ( 2 ) − 24 , − 32 , − 64 , − 88 Z / 2 Z √ Q ( 3 ) − 36 , − 48 Z / 2 Z √ Q ( 5 ) − 15 , − 20 , − 35 , − 40 , − 60 , − 75 , − 100 , − 115 , − 235 Z / 2 Z √ Q ( 13 ) − 52 , − 91 , − 403 Z / 2 Z √ Q ( 17 ) − 51 , − 187 Z / 2 Z √ √ ( Z / 2 Z ) 2 Q ( 2 , 3 ) − 96 , − 192 , − 288 √ √ ( Z / 2 Z ) 2 Q ( 3 , 5 ) − 180 , − 240 √ √ ( Z / 2 Z ) 2 Q ( 5 , 13 ) − 195 , − 520 , − 715 √ √ ( Z / 2 Z ) 2 Q ( 2 , 5 ) − 120 , − 160 , − 280 , − 760 √ √ ( Z / 2 Z ) 2 Q ( 5 , 17 ) − 340 , − 595 √ √ √ ( Z / 2 Z ) 3 Q ( 2 , 3 , 5 ) − 480 , − 960

Equality of CM-fields Theorem ECMF (based on ideas of André, Edixhoven and Kühne) Assume that L = Q ( j ( τ 1 )) = Q ( j ( τ 2 )) . ◮ If Q ( τ 1 ) � = Q ( τ 2 ) then L is the table: Field L ∆ Cl (∆) Q − 3 , − 4 , − 7 , − 8 , − 11 , − 12 , − 16 , − 19 , − 27 , − 28 , − 43 , − 67 , − 163 trivial √ Q ( 2 ) − 24 , − 32 , − 64 , − 88 Z / 2 Z √ Q ( 3 ) − 36 , − 48 Z / 2 Z √ Q ( 5 ) − 15 , − 20 , − 35 , − 40 , − 60 , − 75 , − 100 , − 115 , − 235 Z / 2 Z √ Q ( 13 ) − 52 , − 91 , − 403 Z / 2 Z √ Q ( 17 ) − 51 , − 187 Z / 2 Z √ √ ( Z / 2 Z ) 2 Q ( 2 , 3 ) − 96 , − 192 , − 288 √ √ ( Z / 2 Z ) 2 Q ( 3 , 5 ) − 180 , − 240 √ √ ( Z / 2 Z ) 2 Q ( 5 , 13 ) − 195 , − 520 , − 715 √ √ ( Z / 2 Z ) 2 Q ( 2 , 5 ) − 120 , − 160 , − 280 , − 760 √ √ ( Z / 2 Z ) 2 Q ( 5 , 17 ) − 340 , − 595 √ √ √ ( Z / 2 Z ) 3 Q ( 2 , 3 , 5 ) − 480 , − 960 ◮ If Q ( τ 1 ) = Q ( τ 2 ) then ∆ 1 / ∆ 2 ∈ { 1 , 4 , 1 / 4 } or ∆ 1 , ∆ 2 ∈ {− 3 , − 12 , − 27 } .

The Proof May assume: ◮ τ i = − b i + √ ∆ i ; 2 a i

The Proof May assume: ◮ τ i = − b i + √ ∆ i ; 2 a i ◮ Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = L

The Proof May assume: ◮ τ i = − b i + √ ∆ i ; 2 a i ◮ Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = L ◮ [ L : Q ] = h (∆ 1 ) = h (∆ 2 ) ≥ 3

The Proof May assume: ◮ τ i = − b i + √ ∆ i ; 2 a i ◮ Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = L ◮ [ L : Q ] = h (∆ 1 ) = h (∆ 2 ) ≥ 3 ◮ j ( τ 1 ) � = j ( τ 2 ) .

The Proof May assume: ◮ τ i = − b i + √ ∆ i ; 2 a i ◮ Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = L ◮ [ L : Q ] = h (∆ 1 ) = h (∆ 2 ) ≥ 3 ◮ j ( τ 1 ) � = j ( τ 2 ) .

The Proof May assume: ◮ τ i = − b i + √ ∆ i ; 2 a i ◮ Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = L ◮ [ L : Q ] = h (∆ 1 ) = h (∆ 2 ) ≥ 3 ◮ j ( τ 1 ) � = j ( τ 2 ) . Crucial steps: ◮ Q ( τ 1 ) = Q ( τ 2 ) ;

The Proof May assume: ◮ τ i = − b i + √ ∆ i ; 2 a i ◮ Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = L ◮ [ L : Q ] = h (∆ 1 ) = h (∆ 2 ) ≥ 3 ◮ j ( τ 1 ) � = j ( τ 2 ) . Crucial steps: ◮ Q ( τ 1 ) = Q ( τ 2 ) ; ◮ a 1 = a 2 = 1;

The Proof May assume: ◮ τ i = − b i + √ ∆ i ; 2 a i ◮ Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = L ◮ [ L : Q ] = h (∆ 1 ) = h (∆ 2 ) ≥ 3 ◮ j ( τ 1 ) � = j ( τ 2 ) . Crucial steps: ◮ Q ( τ 1 ) = Q ( τ 2 ) ; ◮ a 1 = a 2 = 1;

The Proof May assume: ◮ τ i = − b i + √ ∆ i ; 2 a i ◮ Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = L ◮ [ L : Q ] = h (∆ 1 ) = h (∆ 2 ) ≥ 3 ◮ j ( τ 1 ) � = j ( τ 2 ) . Crucial steps: ◮ Q ( τ 1 ) = Q ( τ 2 ) ; ◮ a 1 = a 2 = 1; Consequences: ◮ ∆ 1 = ∆ , ∆ 2 = ∆ or 4 ∆ .

The Proof May assume: ◮ τ i = − b i + √ ∆ i ; 2 a i ◮ Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = L ◮ [ L : Q ] = h (∆ 1 ) = h (∆ 2 ) ≥ 3 ◮ j ( τ 1 ) � = j ( τ 2 ) . Crucial steps: ◮ Q ( τ 1 ) = Q ( τ 2 ) ; ◮ a 1 = a 2 = 1; Consequences: ◮ ∆ 1 = ∆ , ∆ 2 = ∆ or 4 ∆ . √ √ √ ◮ τ 1 = − b 1 + ∆ , τ 2 = − b 2 + ∆ or τ 2 = − b 2 + 2 ∆ 2 2 2

The Proof May assume: ◮ τ i = − b i + √ ∆ i ; 2 a i ◮ Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = L ◮ [ L : Q ] = h (∆ 1 ) = h (∆ 2 ) ≥ 3 ◮ j ( τ 1 ) � = j ( τ 2 ) . Crucial steps: ◮ Q ( τ 1 ) = Q ( τ 2 ) ; ◮ a 1 = a 2 = 1; Consequences: ◮ ∆ 1 = ∆ , ∆ 2 = ∆ or 4 ∆ . √ √ √ ◮ τ 1 = − b 1 + ∆ , τ 2 = − b 2 + ∆ or τ 2 = − b 2 + 2 ∆ 2 2 2 ◮ In the first case j ( τ 1 ) = j ( τ 2 )

The Proof May assume: ◮ τ i = − b i + √ ∆ i ; 2 a i ◮ Q ( j ( τ 1 )) = Q ( j ( τ 2 )) = L ◮ [ L : Q ] = h (∆ 1 ) = h (∆ 2 ) ≥ 3 ◮ j ( τ 1 ) � = j ( τ 2 ) . Crucial steps: ◮ Q ( τ 1 ) = Q ( τ 2 ) ; ◮ a 1 = a 2 = 1; Consequences: ◮ ∆ 1 = ∆ , ∆ 2 = ∆ or 4 ∆ . √ √ √ ◮ τ 1 = − b 1 + ∆ , τ 2 = − b 2 + ∆ or τ 2 = − b 2 + 2 ∆ 2 2 2 ◮ In the first case j ( τ 1 ) = j ( τ 2 ) ◮ In the second case ( j ( τ 1 ) , j ( τ 2 )) ∈ Y 0 ( 2 ) .