Automorphisms of Lattices. Application to Curves Jacques Martinet - - PowerPoint PPT Presentation

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 LG 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 LG, 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 u2 = − Id. We shall often make use of an abuse of language, assuming only the weaker hypothesis “u2 = −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 := Cg/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 Cg, the polarization, whose imaginary part is integral on the lattice. Such a form is well-defined by its real part, which gives Cg 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 := Cg/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 Cg, the polarization, whose imaginary part is integral on the lattice. Such a form is well-defined by its real part, which gives Cg 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 u2 = − Id, and the integrality property above reads ∀ x, y ∈ Λ | x · u(y) ∈ Z ⇐ ⇒ u(Λ) ⊂ Λ∗ . Now, given (E, Λ), a polarization is a linear map u ∈ End(E) such that u2 = − 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 Cg/Λ and Cg′/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 = e1, e2, u = (e1, e2) → (e∗

2, −e∗ 1).

Example 2: n=2+2. (Direct sums).

• (Λ1, u1), (Λ2, u2)
• → (Λ1 ⊥ Λ2), (u1, u2).

But if Λ1 = Λ2 there is also a twisted polarization, in general distinct, ..., however

• one class on Z2m, in particular on Z2 ⊥ Z2 = Z4,
• versus two on A2 ⊥ A2, with commuting groups:

normal: (C6 × C6) ⋊ C2, of order 72 ; twisted: C3 ⋊ D4, of order 24.

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 4 / 1

(Complex) Abelian Varieties (2)

Given Abelian varieties Cg/Λ and Cg′/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 = e1, e2, u = (e1, e2) → (e∗

2, −e∗ 1).

Example 2: n=2+2. (Direct sums).

• (Λ1, u1), (Λ2, u2)
• → (Λ1 ⊥ Λ2), (u1, u2).

But if Λ1 = Λ2 there is also a twisted polarization, in general distinct, ..., however

• one class on Z2m, in particular on Z2 ⊥ Z2 = Z4,
• versus two on A2 ⊥ A2, with commuting groups:

normal: (C6 × C6) ⋊ C2, of order 72 ; twisted: C3 ⋊ D4, 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

• bject 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, E4, E6 for those (y2 = x3 + x,

y2 = x3 + 1) having larger automorphisms. Example 2, end. y2 = x6 + 1. Group: C3 ⋊ D4. C3: (x, y) → (ζ3x, y). D4: (x, y) → (−x, y) , (x, y) → (− 1

x , y x3 ) .

Thus the lattice A2 ⊥ A2 is associated with two distinct Abelian varieties, namely E6 × E6 and Jac(y2 = x6 + 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

• f y2 = x6 + 1 is not larger than 24.

Recall (for g ≥ 2) the dichotomy hyperelliptic curves ← →

• rdinary 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} × H0. 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, σ5 = 1, hence σ4 + σ3 + σ2 + σ + 1 = 0. Scale lattices to minimum 2. Let e ∈ S(Λ) (minimal) and let ei = σie. Then B := (e1, . . . , e4) is a basis for a sublattice Λ0 of Λ, of index bounded by γ5/2

5

< 3 and ≥ 5 if larger than 1 since Λ/Λ0 is a module aver Z[ζ5]. = ⇒ Λ = Z[G]e. Set t = e1 · e2 = ⇒ e1 · e3 = e1 · e4 = −1 − t. Matrices A = Gram(B) (at minimum 2) and Mσ = MatB(σ): A(t) =

• 2

t −t−1 −t−1 t 2 t −t−1 −t−1 t 2 t −t−1 −t−1 t 2

• ,

Mσ =

• 0 0 0 −1

1 0 0 −1 0 1 0 −1 0 0 1 −1

• .
• Domain. −1 ≤ t ≤ 0, symmetry w. r. − 1

2; 0, −1 represent A4, − 1 2 its scaled

dual.

• Automorphisms. On (− 1

2, 0), Aut(Λ) = D10.

• 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, σ5 = 1, hence σ4 + σ3 + σ2 + σ + 1 = 0. Scale lattices to minimum 2. Let e ∈ S(Λ) (minimal) and let ei = σie. Then B := (e1, . . . , e4) is a basis for a sublattice Λ0 of Λ, of index bounded by γ5/2

5

< 3 and ≥ 5 if larger than 1 since Λ/Λ0 is a module aver Z[ζ5]. = ⇒ Λ = Z[G]e. Set t = e1 · e2 = ⇒ e1 · e3 = e1 · e4 = −1 − t. Matrices A = Gram(B) (at minimum 2) and Mσ = MatB(σ): A(t) =

• 2

t −t−1 −t−1 t 2 t −t−1 −t−1 t 2 t −t−1 −t−1 t 2

• ,

Mσ =

• 0 0 0 −1

1 0 0 −1 0 1 0 −1 0 0 1 −1

• .
• Domain. −1 ≤ t ≤ 0, symmetry w. r. − 1

2; 0, −1 represent A4, − 1 2 its scaled

dual.

• Automorphisms. On (− 1

2, 0), Aut(Λ) = D10.

• Duality. Guess a possible Moebius transform exchanging 0, − 1

2, then check

that it it works. Guess. α(t) = 2t+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.

tP A(α(t)) P = 5 1−t−t2 2+t

A(t)−1 for P = 0 −1 0 −1

1 0 −1 1 1

• .

= ⇒ 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

Dimension 4, group of order 5 (or) 10 (Second slide)

Check.

tP A(α(t)) P = 5 1−t−t2 2+t

A(t)−1 for P = 0 −1 0 −1

1 0 −1 1 1

• .

= ⇒ At most one isodual lattice in the range [− 1

2, 0]: the fixed point of α.

Existence. y2 = x5 + 1 ! = ⇒ Exactly one curve over C having an automorphism of order 5. A matrix calculation shows that only a C10 commutes with u = ⇒ Aut(y2 = x5 + 1) = C10. Checks.

tP =−P shows the symplectic property:

indeed, returning to computations in the natural basis, we obtain (P A)2 = −k · Id .

• Remark. The conjugate of A(t0) is not positive definite.

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 8 / 1

Dimension 4, group of order 5 (or) 10 (Second slide)

Check.

tP A(α(t)) P = 5 1−t−t2 2+t

A(t)−1 for P = 0 −1 0 −1

1 0 −1 1 1

• .

= ⇒ At most one isodual lattice in the range [− 1

2, 0]: the fixed point of α.

Existence. y2 = x5 + 1 ! = ⇒ Exactly one curve over C having an automorphism of order 5. A matrix calculation shows that only a C10 commutes with u = ⇒ Aut(y2 = x5 + 1) = C10. Checks.

tP =−P shows the symplectic property:

indeed, returning to computations in the natural basis, we obtain (P A)2 = −k · Id .

• Remark. The conjugate of A(t0) is not positive definite.
• Question. Is this general ?

Compare weakly eutatic lattices ...

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 8 / 1

Automorphisms of curves

Let g ≥ 2. Then automorphism groups of curves are finite, of order bounded from above by the Hurwitz bound 84(g − 1), attained on sparse values of g, e.g., g = 3. (Otherwise, | Aut | ≤ 48(g − 1), e.g., g = 2,...) Finite groups which occur as subgroups of some curve of genus g ≥ 2 are considered in detail in Breuer’s lecture Notes of the L.M.S. For each g ≥ 2 finitely presented groups are listed by generators and relations, such that exactly their finite quotients occur as an automorphism group of some curve of genus g ≥ 2.

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 9 / 1

Automorphisms of curves

Let g ≥ 2. Then automorphism groups of curves are finite, of order bounded from above by the Hurwitz bound 84(g − 1), attained on sparse values of g, e.g., g = 3. (Otherwise, | Aut | ≤ 48(g − 1), e.g., g = 2,...) Finite groups which occur as subgroups of some curve of genus g ≥ 2 are considered in detail in Breuer’s lecture Notes of the L.M.S. For each g ≥ 2 finitely presented groups are listed by generators and relations, such that exactly their finite quotients occur as an automorphism group of some curve of genus g ≥ 2. Warning : AN is not THE !

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 9 / 1

Automorphisms of curves

Let g ≥ 2. Then automorphism groups of curves are finite, of order bounded from above by the Hurwitz bound 84(g − 1), attained on sparse values of g, e.g., g = 3. (Otherwise, | Aut | ≤ 48(g − 1), e.g., g = 2,...) Finite groups which occur as subgroups of some curve of genus g ≥ 2 are considered in detail in Breuer’s lecture Notes of the L.M.S. For each g ≥ 2 finitely presented groups are listed by generators and relations, such that exactly their finite quotients occur as an automorphism group of some curve of genus g ≥ 2. Warning : AN is not THE ! In the next slide we shall consider Weil’s dichotomy on 2-dimensional PPAV in connection with characteristic polynomials of elements of order 3 or 4 in 2-dimensional lattices. As a consequence we obtain:

• Thm. If Aut(C) contains a C3 (resp. a C4), it contains a D6 (resp. a D4).

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 9 / 1

Automorphisms of curves

Let g ≥ 2. Then automorphism groups of curves are finite, of order bounded from above by the Hurwitz bound 84(g − 1), attained on sparse values of g, e.g., g = 3. (Otherwise, | Aut | ≤ 48(g − 1), e.g., g = 2,...) Finite groups which occur as subgroups of some curve of genus g ≥ 2 are considered in detail in Breuer’s lecture Notes of the L.M.S. For each g ≥ 2 finitely presented groups are listed by generators and relations, such that exactly their finite quotients occur as an automorphism group of some curve of genus g ≥ 2. Warning : AN is not THE ! In the next slide we shall consider Weil’s dichotomy on 2-dimensional PPAV in connection with characteristic polynomials of elements of order 3 or 4 in 2-dimensional lattices. As a consequence we obtain:

• Thm. If Aut(C) contains a C3 (resp. a C4), it contains a D6 (resp. a D4).
• Cor. SL2(3) exists only inside GL2(3).

[2-Sylow are H8 and F∗

9 · Gal(F9/F3), respectively; the latter group contains

groups of type D4, H8, C8.]

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 9 / 1

Dimension 4, Automorphisms of Order 3 and 4

If σ ∈ GL(E) is of order 3 then (a): χσ(X) = (X 2 + X + 1)(X − 1)2, or (b): χσ(X) = (X 2 + X + 1)2. If σ ∈ GL(E) is of order 4 then (a): χ±σ(X) = (X 2 + 1)(X − 1)2, or (b): χσ(X) = (X 2 + 1)2. In cases (a), E splits canonically. These cases correspond to product of elliptic curves. In cases (b), the representation is isotypic. These cases correspond to Jacobians. I do not know completely the correspondence lattices ← → curves. For larger groups, lattices Λ and curves C are as follows: Either Λ = A2 ⊥ A2 and C is y2 = x6 + 1, or Λ = D4 and C is y2 = x(x4 + 1). This latter curve is known a the Bolza curve. Its automorphism group was found by Bolza in 1887 (American J. Math. 10, 47–70).

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 10 / 1

C8 and the Bolza curve

n = 4,G = σ, σ4 = − Id. Basis for Λ: B = (ei) with ei = σi−1e, for a convenient e ∈ S(Λ).

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 11 / 1

C8 and the Bolza curve

n = 4,G = σ, σ4 = − Id. Basis for Λ: B = (ei) with ei = σi−1e, for a convenient e ∈ S(Λ). Matrix description: A(t) = 2

t 0 −t t 2 t t 2 t −t 0 t 2

• and P =

1 0 −1 0 1 −1 0

• ,

tP A(t)P = A(−t) , A(t) A(−t) = 2(2 − t2) I4 , and tP = −P .

= ⇒ Symplectic lattices. Domain: 0 ≤ t ≤ 1, but no isoduality commutes with σ for t ∈ (0, 1). t = 0: scaled copy of Z4 ← → E4 × E4. t = 1: D4 = (Z4)even. Identifying D4 with the set of even usual quaternions identifies D∗

4 with the order M of Hurwitz’s quaternions and shows the

existence of a unique symplectic structure on D4; and using left multiplication by M× identifies its automorphism group with the double cover

• S4 → S4
• ≃
• GL2(3) → PGL2(3)
• .
• S4 , not

S4.

• Jacques Martinet (Universit´

e de Bordeaux, IMB) February 16, 2020 11 / 1

Dimension 6 : |G| = 7 or 9

Write G = σ, σ of order 7 or 9. In both cases we have a hexagonal domain D, with 6 extremal matrices, A1, B1, A2, B2, A3, B3 each of the A’s and B’s representing one lattice. The automorphism g → g2 / g−2 of G divides D into three sub-domains. n = 7: A ← → A(1)

6 , B ←

→ A(2)

6 .

n = 9: A ← → E6 , B ← → E∗

6 .

Craig’s lattices A(i)

n , n = p − 1, p an odd prime: 1 p TrQ(ζp)/Q(zz)

restricted to Pi, P = (1 − ζp) .

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 12 / 1

Dimension 6 : |G| = 7 or 9

Write G = σ, σ of order 7 or 9. In both cases we have a hexagonal domain D, with 6 extremal matrices, A1, B1, A2, B2, A3, B3 each of the A’s and B’s representing one lattice. The automorphism g → g2 / g−2 of G divides D into three sub-domains. n = 7: A ← → A(1)

6 , B ←

→ A(2)

6 .

n = 9: A ← → E6 , B ← → E∗

6 .

Craig’s lattices A(i)

n , n = p − 1, p an odd prime: 1 p TrQ(ζp)/Q(zz)

restricted to Pi, P = (1 − ζp) . Up to scale, i → i + p−1

2

is a period, and i → p+1

2

− i is a duality. = ⇒ If p ≡ 3 mod 4, A((p+1)/4)

n

is isodual, indeed symplectic: use multiplication by the Gauss sum

• i mod p

i

p

• ζi

p ,

= (ζ + ζ2 + ζ4) − (ζ3 + ζ5 + ζ6) if p = 7 .

• Remarks. 1. A(1)

n

= An ; 2 (Craig). min A(i)

n ≥ 2i ;

3 (Elkies). Equality holds if i = p+1

4

(based on his theory of Mordell-Weil lattices).

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 12 / 1

More on |G| = 7

Gram matrix. A =   

2 t1 t2 t3 t3 t2 t1 2 t1 t2 t3 t3 t2 t1 2 t1 t2 t3 t3 t2 t1 2 t1 t2 t3 t3 t2 t1 2 t1 t2 t3 t3 t2 t1 2

   where t1+t2+t3 = −1 . Orbits under ±σ. Three at vertices, two on edges, one in Int(D).

• Automorphisms. Aut = D14 and Aut+ = C14, except for A6 and A∗

6 (2 × S7)

and A(2)

6

(2 × (GL3(2) · 2 ≃ 2 × PSL2(7)).

• Duality. 3 to 1 from Ai to A0: (− 1

3, − 1 3, − 1 3), representing A∗ 6 indeed the

barycenter of the Ai’s (and of the Bi’s), 1 to 1 on Bi, 2 to 1 on edges, 1 to 1 in Int(D) except 1 to 3 at A0 and 1 to 2 on the images of sets A — B — A of edges.

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 13 / 1

More on |G| = 7

Gram matrix. A =   

2 t1 t2 t3 t3 t2 t1 2 t1 t2 t3 t3 t2 t1 2 t1 t2 t3 t3 t2 t1 2 t1 t2 t3 t3 t2 t1 2 t1 t2 t3 t3 t2 t1 2

   where t1+t2+t3 = −1 . Orbits under ±σ. Three at vertices, two on edges, one in Int(D).

• Automorphisms. Aut = D14 and Aut+ = C14, except for A6 and A∗

6 (2 × S7)

and A(2)

6

(2 × (GL3(2) · 2 ≃ 2 × PSL2(7)).

• Duality. 3 to 1 from Ai to A0: (− 1

3, − 1 3, − 1 3), representing A∗ 6 indeed the

barycenter of the Ai’s (and of the Bi’s), 1 to 1 on Bi, 2 to 1 on edges, 1 to 1 in Int(D) except 1 to 3 at A0 and 1 to 2 on the images of sets A — B — A of edges. These are arcs of hyperbola connecting the Bi’s to A0. Example: t2

1 − 4t1t2 − 3t2 2 − 3t1 − 8t2 − 3 = 0 ,

• r

t1 = −3t2−50t+25

−6t2+40t+50, t2 = 17t2−20t−25 −6t2+40t+50 .

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 13 / 1

PPAV for |G = 7| (first slide)

We must now classify symplectic lattices in D. One of them, with representative any of the matrices Bi, is A(2)

n , for which

Aut+ is of index 2 in 2 × GL2(3). Any other isodual lattice must lie in Int(D), off the arcs of hyperbolas. We choose a sub-domain D′ ⊂ D, and in D′, we guess on one example the set of minimal vectors of one lattice; this will then hold in the whole connected components of the complementary set of the hyperbolas. It turns out that we may choose ±{e∗

3, −e∗ 1 + e∗ 4, −e∗ 2 + e∗ 5, −e∗ 3 + e∗ 6, −e∗ 4, e∗ 1 − e∗ 5, e∗ 2 − e∗ 6}

(seven vectors adding to zero). We then calculate the corresponding parameters u1 = e∗

3 · (−e∗ 1 + e∗ 4), u2 = e∗ 3 · (−e∗ 2 + e∗ 5) and

u3 = e∗

3 · (−e∗ 3 + e∗ 6) (of course, u3 = −u1 − u2 − 1), obtaining

u1 = −5t2

1 −8t1t2+t2 2 +t1−2t2+1

t2

1 +3t1t2+4t2 2 +4t1+6t2−3

and u2 =

2t2

1 +6t2t1+t2 2 +t1+5t2+1

t2

1 +3t1t2+4t2 2 +4t1+6t2−3 .

Solving the system {u1 = t1, u2 = t2}, we find t1 = −(1 + 6θ + θ2)/2 = −0.176... t2 = θ = −0.109... , where θ = −1 − 2 cos(4π/7), the only choice for which (t1, t2) ∈ D. Let Λ0 be this lattice, defined over Q(ζ7 + ζ−1

7 ).

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 14 / 1

PPAV for |G| = 7 (second slide)

We now now that there are at most two PPAV invariant underG, exactly two if the isodual lattice above is actually of symplectic type. No need to prove this:

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 15 / 1

PPAV for |G| = 7 (second slide)

We now now that there are at most two PPAV invariant underG, exactly two if the isodual lattice above is actually of symplectic type. No need to prove this: Look at the projective curves H (hyperelliptic) and K (Klein’s quartic): H : y2z5 = x7 + z7 , K : x3y + y3z + z3x , with their respective automorphisms of order 7 (ζ = ζ7) σ1 : (x, y, z) → (ζx, y, z) and σ2 : (x, y, z) → (ζx, ζ4y, ζ2z).

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 15 / 1

PPAV for |G| = 7 (second slide)

We now now that there are at most two PPAV invariant underG, exactly two if the isodual lattice above is actually of symplectic type. No need to prove this: Look at the projective curves H (hyperelliptic) and K (Klein’s quartic): H : y2z5 = x7 + z7 , K : x3y + y3z + z3x , with their respective automorphisms of order 7 (ζ = ζ7) σ1 : (x, y, z) → (ζx, y, z) and σ2 : (x, y, z) → (ζx, ζ4y, ζ2z). Observe that K has an automorphism of order 3, whereas |Aut(Λ0)| = 28 and |Aut+(Λ0)| = 14. This shows that there are two curves having an automorphism of order 7: H, with lattice Λ0 and Aut(H) = C14, and K, with lattice A(2)

n . In this latter case, the Hurwitz bound shows that

Aut(K) has index 4 in Aut(A(2)

n ),

hence is equal to GL3(2), since this group is simple. Of course all that concerns K was known to Klein !

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 15 / 1

|G| = 9

This time lattices having a dual containing two orbits of minimal vectors lie on six arcs of conics connecting consecutive vertices. Their complementary set in Int(D) is the union of six, pairwise equivalent connected components ... End of slide REMOVED At the date of the talk, because of a scaling error. I had missed a lattice defined (up to scale) over Q(ζ9) corresponding to the ordinary curve of genus 3 X 3Y + Y 3Z + Z 4 .

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 16 / 1

THE END

Jacques Martinet (Universit´ e de Bordeaux, IMB) February 16, 2020 17 / 1