Automorphisms of Lattices. Application to Curves Jacques Martinet Universit´ e de Bordeaux, IMB February 16, 2020 Luminy, March 2019 Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 1 / 1
G -lattices Let E be an n -dimensional Euclidean space and let G be a finite subgroup of SO ( E ) . We consider for some groups G and some low dimensions the set L G of lattices invariant under G . The dual of a lattice Λ ∈ E is Λ ∗ = { x ∈ E | ∀ y ∈ Λ , x · y ∈ Z We shall then determine the automorphism groups of the various lattices in L G , characterize those which are isodual (that is, isometric to their dual), and pay special attention to symplectic lattices , those for which there exists an isoduality Λ → Λ ∗ such that u 2 = − Id. We shall often make use of an abuse of language, assuming only the weaker hypothesis “ u 2 = − k Id for some k > 0 ” and applying the definition above to a scaled copy of Λ . Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 2 / 1
(Complex) Abelian Varieties (1) These are the complex tori T := C g / Λ on which there exists g algebraically independent meromorphic functions, a property equivalent to the existence of a projective embedding, and also to the fact that they carry the structure of an algebraic variety, and above all, to the existence of Riemann form on T , that is a positive, definite Hermitian form on C g , the polarization , whose imaginary part is integral on the lattice. Such a form is well-defined by its real part, which gives C g the structure of a Euclidean space E (and also by its imaginary part, which is alternating). Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 3 / 1
(Complex) Abelian Varieties (1) These are the complex tori T := C g / Λ on which there exists g algebraically independent meromorphic functions, a property equivalent to the existence of a projective embedding, and also to the fact that they carry the structure of an algebraic variety, and above all, to the existence of Riemann form on T , that is a positive, definite Hermitian form on C g , the polarization , whose imaginary part is integral on the lattice. Such a form is well-defined by its real part, which gives C g the structure of a Euclidean space E (and also by its imaginary part, which is alternating). To x �→ i x corresponds ± u = u ± 1 ∈ End ( E ) with u 2 = − Id, and the integrality property above reads ⇒ u (Λ) ⊂ Λ ∗ . ∀ x , y ∈ Λ | x · u ( y ) ∈ Z ⇐ Now, given ( E , Λ) , a polarization is a linear map u ∈ End ( E ) such that u 2 = − Id and u (Λ) ⊂ Λ ∗ . and this is called principal when u (Λ) = Λ ∗ . We shall essentially consider Principally Polarized Abelian Varieties , PPAV for short. Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 3 / 1
(Complex) Abelian Varieties (2) Given Abelian varieties C g / Λ and C g ′ / Λ ′ , morphisms are linear maps which send Λ into Λ ′ , whence notions of isomorphisms and automorphisms . These notions must be considered with respect to maps u of square − Id: isomorphisms must commute with the u ’s, and one must check whether two distinct u ’s define isomorphic PPAV. Example 1: n=2. Λ = � e 1 , e 2 � , u = ( e 1 , e 2 ) �→ ( e ∗ 2 , − e ∗ 1 ) . � � Example 2: n=2+2. (Direct sums). (Λ 1 , u 1 ) , (Λ 2 , u 2 ) → (Λ 1 ⊥ Λ 2 ) , ( u 1 , u 2 ) . But if Λ 1 = Λ 2 there is also a twisted polarization, in general distinct, ..., however • one class on Z 2 m , in particular on Z 2 ⊥ Z 2 = Z 4 , • versus two on A 2 ⊥ A 2 , with commuting groups: normal: ( C 6 × C 6 ) ⋊ C 2 , of order 72 ; twisted: C 3 ⋊ D 4 , of order 24. Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 4 / 1
(Complex) Abelian Varieties (2) Given Abelian varieties C g / Λ and C g ′ / Λ ′ , morphisms are linear maps which send Λ into Λ ′ , whence notions of isomorphisms and automorphisms . These notions must be considered with respect to maps u of square − Id: isomorphisms must commute with the u ’s, and one must check whether two distinct u ’s define isomorphic PPAV. Example 1: n=2. Λ = � e 1 , e 2 � , u = ( e 1 , e 2 ) �→ ( e ∗ 2 , − e ∗ 1 ) . � � Example 2: n=2+2. (Direct sums). (Λ 1 , u 1 ) , (Λ 2 , u 2 ) → (Λ 1 ⊥ Λ 2 ) , ( u 1 , u 2 ) . But if Λ 1 = Λ 2 there is also a twisted polarization, in general distinct, ..., however • one class on Z 2 m , in particular on Z 2 ⊥ Z 2 = Z 4 , • versus two on A 2 ⊥ A 2 , with commuting groups: normal: ( C 6 × C 6 ) ⋊ C 2 , of order 72 ; twisted: C 3 ⋊ D 4 , of order 24. PPAV: 1. An elliptic curve E ; 2 normal. A product E ′ × E ′′ of elliptic curves ; 2 twisted. Jacobians , see below. Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 4 / 1
Jacobian varieties We consider connected, compact, 1-dimensional analytic varieties (alias (compact) Riemann surfaces ), alias complex curves. With such an object X one associates first an integer g ≥ 0, the genus (a topological invariant), next a PPAV of complex dimension g , its Jacobian Jac ( X ) , and (once chosen some x ∈ X ) a map ϕ : X → Jac ( X ) with ϕ ( x ) = 0. Characterization: for any Abelian variety A and any map f : X → A with f ( x ) = 0, there exists a unique homomorphism of A.V. F : Jac ( X ) → A such that f = F ◦ ϕ . [More generally, whatever dim X , this formalism generalizes to Albanese varieties ]. Notation. We write E for “generic” elliptic curves, E 4 , E 6 for those ( y 2 = x 3 + x , y 2 = x 3 + 1) having larger automorphisms. Example 2, end . y 2 = x 6 + 1. Group: C 3 ⋊ D 4 . C 3 : ( x , y ) �→ ( ζ 3 x , y ) . x , y D 4 : ( x , y ) �→ ( − x , y ) , ( x , y ) �→ ( − 1 x 3 ) . Thus the lattice A 2 ⊥ A 2 is associated with two distinct Abelian varieties, namely E 6 × E 6 and Jac ( y 2 = x 6 + 1 ) . Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 5 / 1
Torelli’s theorem Torelli (1913); Weil (1957); for a a really convenient formulation, see Serre, in an appendix to a paper by Kristin Lauter. This establishes that Jacobians reflect isomorphisms between curves, and justifies that the order of the group of y 2 = x 6 + 1 is not larger than 24. Recall (for g ≥ 2) the dichotomy hyperelliptic curves ← → ordinary curves. On the size of automorphisms, in a crude form, Torelli tells us that automorphism groups of hyperelliptic (resp.ordinary) curves are in one-to-one (resp. one-to-two) correspondence with the automorphism groups of their Jacobians. Indeed if X is ordinary, H = Aut ( Jac ( X )) splits as a product {± Id } × H 0 . Weil considers apart low genera. In particular his proof shows that PPAV of complex dimension 2 share out among two, mutually exclusive types, namely products of elliptic curves and Jacobians — thus products of (one or two) Jacobians. [This has been extended to dimension 3 (Ort–Ueno), but is not true in higher dimensions.] Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 6 / 1
Dimension 4, group of order 5 (or) 10 (First slide) Write G = � σ � , σ 5 = 1, hence σ 4 + σ 3 + σ 2 + σ + 1 = 0. Scale lattices to minimum 2. Let e ∈ S (Λ) (minimal) and let e i = σ i e . Then B := ( e 1 , . . . , e 4 ) is a basis for a sublattice Λ 0 of Λ , of index bounded by γ 5 / 2 < 3 and ≥ 5 if larger 5 than 1 since Λ / Λ 0 is a module aver Z [ ζ 5 ] . = ⇒ Λ = Z [ G ] e . Set t = e 1 · e 2 = ⇒ e 1 · e 3 = e 1 · e 4 = − 1 − t . Matrices A = Gram ( B ) (at minimum 2) and M σ = Mat B ( σ ) : � � � � 2 t − t − 1 − t − 1 0 0 0 − 1 t 2 t − t − 1 1 0 0 − 1 A ( t ) = , M σ = . − t − 1 t 2 t 0 1 0 − 1 − t − 1 − t − 1 t 2 0 0 1 − 1 Domain. − 1 ≤ t ≤ 0, symmetry w. r. − 1 2 ; 0 , − 1 represent A 4 , − 1 2 its scaled dual. Automorphisms. On ( − 1 2 , 0 ) , Aut (Λ) = D 10 . Duality. Guess a possible Moebius transform exchanging 0 , − 1 2 , then check that it it works. Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 7 / 1
Dimension 4, group of order 5 (or) 10 (First slide) Write G = � σ � , σ 5 = 1, hence σ 4 + σ 3 + σ 2 + σ + 1 = 0. Scale lattices to minimum 2. Let e ∈ S (Λ) (minimal) and let e i = σ i e . Then B := ( e 1 , . . . , e 4 ) is a basis for a sublattice Λ 0 of Λ , of index bounded by γ 5 / 2 < 3 and ≥ 5 if larger 5 than 1 since Λ / Λ 0 is a module aver Z [ ζ 5 ] . = ⇒ Λ = Z [ G ] e . Set t = e 1 · e 2 = ⇒ e 1 · e 3 = e 1 · e 4 = − 1 − t . Matrices A = Gram ( B ) (at minimum 2) and M σ = Mat B ( σ ) : � � � � 2 t − t − 1 − t − 1 0 0 0 − 1 t 2 t − t − 1 1 0 0 − 1 A ( t ) = , M σ = . − t − 1 t 2 t 0 1 0 − 1 − t − 1 − t − 1 t 2 0 0 1 − 1 Domain. − 1 ≤ t ≤ 0, symmetry w. r. − 1 2 ; 0 , − 1 represent A 4 , − 1 2 its scaled dual. Automorphisms. On ( − 1 2 , 0 ) , Aut (Λ) = D 10 . Duality. Guess a possible Moebius transform exchanging 0 , − 1 2 , then check that it it works. Guess . α ( t ) = 2 t + 1 t − 2 . Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 7 / 1
Dimension 4, group of order 5 (or) 10 (Second slide) Check . t P A ( α ( t )) P = 5 1 − t − t 2 A ( t ) − 1 2 + t for � 0 − 1 0 − 1 � 1 0 0 0 P = . 0 0 0 − 1 1 0 1 0 At most one isodual lattice in the range [ − 1 = ⇒ 2 , 0 ] : the fixed point of α . Existence . Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 8 / 1
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