An overwiev of a conjecture of Kitaoka Marcin Mazur An overwiev of a conjecture of Kitaoka – p. 1/2
V is a finite dimensional Q -vector space. An overwiev of a conjecture of Kitaoka – p. 2/2
V is a finite dimensional Q -vector space. β : V × V − → Q is a positive definite symmetric bilinear form (i.e. an inner product). An overwiev of a conjecture of Kitaoka – p. 2/2
V is a finite dimensional Q -vector space. β : V × V − → Q is a positive definite symmetric bilinear form (i.e. an inner product). q is the associated quadratic form. An overwiev of a conjecture of Kitaoka – p. 2/2
V is a finite dimensional Q -vector space. β : V × V − → Q is a positive definite symmetric bilinear form (i.e. an inner product). q is the associated quadratic form. A lattice is a finitely generated subgroup of V which contains a basis of V . An overwiev of a conjecture of Kitaoka – p. 2/2
V is a finite dimensional Q -vector space. β : V × V − → Q is a positive definite symmetric bilinear form (i.e. an inner product). q is the associated quadratic form. A lattice is a finitely generated subgroup of V which contains a basis of V . A positive lattice is a pair ( L, β ) , where L is a lattice in a finite dimensional vector space with an inner product β . An overwiev of a conjecture of Kitaoka – p. 2/2
V is a finite dimensional Q -vector space. β : V × V − → Q is a positive definite symmetric bilinear form (i.e. an inner product). q is the associated quadratic form. A lattice is a finitely generated subgroup of V which contains a basis of V . A positive lattice is a pair ( L, β ) , where L is a lattice in a finite dimensional vector space with an inner product β . It is often more convenient to use q instead of β . An overwiev of a conjecture of Kitaoka – p. 2/2
Let ( L, β ) be a positive lattice. An overwiev of a conjecture of Kitaoka – p. 3/2
Let ( L, β ) be a positive lattice. Consider a number field k with the ring of integers R . An overwiev of a conjecture of Kitaoka – p. 3/2
Let ( L, β ) be a positive lattice. Consider a number field k with the ring of integers R . We can extend β to a symmetric R -bilinear form β R on RL = R ⊗ Z L by setting β R ( a ⊗ v, b ⊗ w ) = abβ ( v, w ) . An overwiev of a conjecture of Kitaoka – p. 3/2
Let ( L, β ) be a positive lattice. Consider a number field k with the ring of integers R . We can extend β to a symmetric R -bilinear form β R on RL = R ⊗ Z L by setting β R ( a ⊗ v, b ⊗ w ) = abβ ( v, w ) . Question. Let ( L 1 , β 1 ) , ( L 2 , β 2 ) be positive lattices such that RL 1 and RL 2 are isometric. Does it follow that L 1 and L 2 are isometric? An overwiev of a conjecture of Kitaoka – p. 3/2
Let ( L, β ) be a positive lattice. Consider a number field k with the ring of integers R . We can extend β to a symmetric R -bilinear form β R on RL = R ⊗ Z L by setting β R ( a ⊗ v, b ⊗ w ) = abβ ( v, w ) . Question. Let ( L 1 , β 1 ) , ( L 2 , β 2 ) be positive lattices such that RL 1 and RL 2 are isometric. Does it follow that L 1 and L 2 are isometric? Exercise : Consider the positive lattices ( Z 2 , 2 x 2 + 3 y 2 ) and ( Z 2 , x 2 + 6 y 2 ) . These are not isometric but after tensoring with the Gaussian integers Z [ i ] they become isometric via � � 2 i, 3 isometry given by the matrix . 1 , − i An overwiev of a conjecture of Kitaoka – p. 3/2
However, if we restrict our attention to totally real number fields k then the situation is much more optimistic. An overwiev of a conjecture of Kitaoka – p. 4/2
However, if we restrict our attention to totally real number fields k then the situation is much more optimistic. Conjecture A (Kitaoka) If k is totally real and σ : RL 1 − → RL 2 is an isometry then σ ( L 1 ) = L 2 . An overwiev of a conjecture of Kitaoka – p. 4/2
First attempt An overwiev of a conjecture of Kitaoka – p. 5/2
First attempt Tensor product of positive lattices. Definition. Let ( L 1 , β 1 ) , ( L 2 , β 2 ) be positive lattices. Define ( L 1 , β 1 ) ⊗ ( L 2 , β 2 ) := ( L 1 ⊗ L 2 , β ) , where β ( m 1 ⊗ m 2 , n 1 ⊗ n 2 ) = β 1 ( m 1 , n 1 ) β 2 ( m 2 , n 2 ) . An overwiev of a conjecture of Kitaoka – p. 5/2
First attempt Tensor product of positive lattices. Definition. Let ( L 1 , β 1 ) , ( L 2 , β 2 ) be positive lattices. Define ( L 1 , β 1 ) ⊗ ( L 2 , β 2 ) := ( L 1 ⊗ L 2 , β ) , where β ( m 1 ⊗ m 2 , n 1 ⊗ n 2 ) = β 1 ( m 1 , n 1 ) β 2 ( m 2 , n 2 ) . Conjecture B (Kitaoka) Let ( M, β ) , ( M 1 , β 1 ) , ( M 2 , β 2 ) be positive lattices. If ( M, β ) ⊗ ( M 1 , β 1 ) and ( M, β ) ⊗ ( M 2 , β 2 ) are isometric then so are ( M 1 , β 1 ) and ( M 2 , β 2 ) . An overwiev of a conjecture of Kitaoka – p. 5/2
Consider the trace form tr = tr k/ Q on R . The pair ( R, tr ) is a positive lattice. An overwiev of a conjecture of Kitaoka – p. 6/2
Consider the trace form tr = tr k/ Q on R . The pair ( R, tr ) is a positive lattice. If ( L, β ) is a positive lattice and β R is the extension of β to RL then ( RL, tr ◦ β R ) is a positive lattice naturally isometric to ( R, tr ) ⊗ ( L, β ) . An overwiev of a conjecture of Kitaoka – p. 6/2
Consider the trace form tr = tr k/ Q on R . The pair ( R, tr ) is a positive lattice. If ( L, β ) is a positive lattice and β R is the extension of β to RL then ( RL, tr ◦ β R ) is a positive lattice naturally isometric to ( R, tr ) ⊗ ( L, β ) . Any isometry of RL 1 and RL 2 induces isometry of ( R, tr ) ⊗ ( L 1 , β 1 ) and ( R, tr ) ⊗ ( L 2 , β 2 ) . Conjecture B implies that ( L 1 , β 1 ) and ( L 2 , β 2 ) are isometric. An overwiev of a conjecture of Kitaoka – p. 6/2
Minimal vectors An overwiev of a conjecture of Kitaoka – p. 7/2
Minimal vectors Definition. Let ( L, β ) be a positive lattice. An overwiev of a conjecture of Kitaoka – p. 7/2
Minimal vectors Definition. Let ( L, β ) be a positive lattice. min( L ) = min { q ( v ) : v ∈ L, v � = 0 } An overwiev of a conjecture of Kitaoka – p. 7/2
Minimal vectors Definition. Let ( L, β ) be a positive lattice. min( L ) = min { q ( v ) : v ∈ L, v � = 0 } M ( L ) = { v ∈ L : q ( v ) = min( L ) } . An overwiev of a conjecture of Kitaoka – p. 7/2
Minimal vectors Definition. Let ( L, β ) be a positive lattice. min( L ) = min { q ( v ) : v ∈ L, v � = 0 } M ( L ) = { v ∈ L : q ( v ) = min( L ) } . Exercise. M ( R, tr ) = {− 1 , 1 } , min( R, tr ) = [ k : Q ] . An overwiev of a conjecture of Kitaoka – p. 7/2
Minimal vectors Definition. Let ( L, β ) be a positive lattice. min( L ) = min { q ( v ) : v ∈ L, v � = 0 } M ( L ) = { v ∈ L : q ( v ) = min( L ) } . Exercise. M ( R, tr ) = {− 1 , 1 } , min( R, tr ) = [ k : Q ] . Definition. A positive lattice ( M, γ ) is of E -type if M ( M ⊗ L ) = M ( M ) ⊗ M ( L ) for any positive lattice L . An overwiev of a conjecture of Kitaoka – p. 7/2
Theorem. Fix a positive integer n . An overwiev of a conjecture of Kitaoka – p. 8/2
Theorem. Fix a positive integer n . Suppose that either ( R, tr ) is of E -type or every positive lattice of rank at most n is of E -type. An overwiev of a conjecture of Kitaoka – p. 8/2
Theorem. Fix a positive integer n . Suppose that either ( R, tr ) is of E -type or every positive lattice of rank at most n is of E -type. If L 1 , L 2 are positive lattices of rank at most n and σ : RL 1 − → RL 2 is an isometry, then σ ( L 1 ) = L 2 . An overwiev of a conjecture of Kitaoka – p. 8/2
Theorem. Fix a positive integer n . Suppose that either ( R, tr ) is of E -type or every positive lattice of rank at most n is of E -type. If L 1 , L 2 are positive lattices of rank at most n and σ : RL 1 − → RL 2 is an isometry, then σ ( L 1 ) = L 2 . Idea of the proof. An overwiev of a conjecture of Kitaoka – p. 8/2
Theorem. Fix a positive integer n . Suppose that either ( R, tr ) is of E -type or every positive lattice of rank at most n is of E -type. If L 1 , L 2 are positive lattices of rank at most n and σ : RL 1 − → RL 2 is an isometry, then σ ( L 1 ) = L 2 . Idea of the proof. Let v ∈ M ( R ⊗ L 1 ) . An overwiev of a conjecture of Kitaoka – p. 8/2
Theorem. Fix a positive integer n . Suppose that either ( R, tr ) is of E -type or every positive lattice of rank at most n is of E -type. If L 1 , L 2 are positive lattices of rank at most n and σ : RL 1 − → RL 2 is an isometry, then σ ( L 1 ) = L 2 . Idea of the proof. Let v ∈ M ( R ⊗ L 1 ) . By assumption, v ∈ L 1 . An overwiev of a conjecture of Kitaoka – p. 8/2
Theorem. Fix a positive integer n . Suppose that either ( R, tr ) is of E -type or every positive lattice of rank at most n is of E -type. If L 1 , L 2 are positive lattices of rank at most n and σ : RL 1 − → RL 2 is an isometry, then σ ( L 1 ) = L 2 . Idea of the proof. Let v ∈ M ( R ⊗ L 1 ) . By assumption, v ∈ L 1 . Clearly σ ( v ) ∈ M ( R ⊗ L 2 ) . An overwiev of a conjecture of Kitaoka – p. 8/2
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