La table en deux, en sept et à l’infini

Les couches du réseau de KleinThe shells of Klein’s lattice

Throughout, $E=\Q(\sqrt{-7})$, $\mathcal O_E=\Z[\alpha]$ with $\alpha=(-1+\sqrt{-7})/2$, $L=\Q(\zeta)$, $L_\infty$ is Klein’s lattice with Mumford’s form $h$, and $G_0$ is its group of isometries of determinant one. A native datum is a $G_0$-orbit of data built from $L_\infty$ and $h$: vectors and finite sets of vectors, sublattices, elements of $G_0$, or cells of $\Delta_2\times T_7$ at the vertex $(v_0,\Lambda_\infty)$. Its residue at 2 is the pair of its residues modulo $(\alpha)$ and $(\bar\alpha)$; at 7, its residue modulo $\sqrt{-7}$; at $\infty$, the point $[v]$ of $\Proj(L_\infty\otimes\C)$. A residue map is a description, its kernel at a datum is the stabilizer of the residue, and it forgets nothing exactly when it is a seam.

The shells come first. The vectors of norms 2 to 8 number 42, 56, 84, 168, 280, 336 and 462, as Elkies’ theta series $1+42q^2+56q^3+84q^4+168q^5+280q^6+336q^7+462q^8+\cdots$ says. Their $G_0$-orbits have stabilizers $C_4$ (the roots, and $\alpha$, $\bar\alpha$ and 2 times roots), $C_3$ (the vectors of norm 3, and $\alpha$ and $\bar\alpha$ times them) or trivial, and the pairs $\pm v$ have stabilizers $D_8$, $S_3$, $C_2$ or trivial. At norm 6 there are 280 vectors, the 112 imprimitive ones and one regular orbit of 168 primitive ones; a remark of Allcock and Kato counts $56+56$, so the group of linear and antilinear isometries has two orbits there, not one.

Les quinze objets dans le réseauThe fifteen objects in the lattice

Name the data as Allcock and Kato do, with the Fano plane read as $\Proj(L_\infty/\bar\alpha L_\infty)$. A frame is a triple of mutually orthogonal root pairs; an $a$-frame is one whose six roots have one residue modulo $\bar\alpha$, a $b$-frame one whose six roots have one residue modulo $\alpha$. A tetrahedron is a set of four vectors of norm 3 with pairwise inner product $-1$, of class $a$ or $b$ in the same way. A cyclotomic structure is an element $g\in G_0$ with $g+g^2+g^4=\alpha$, that is, an $\mathcal O_E$-algebra embedding $\Z[\zeta]\to\operatorname{End}(L_\infty)$ with $\zeta\mapsto g$; and for a Sylow 7-subgroup $P=\langle g\rangle$ the Mumford sublattice is $M_P=L_\infty(1-g)$.

Each root pair is orthogonal to exactly four others, so the frames are fourteen, seven of each class. The tetrahedra are twenty-eight, each summing to 0, and if $T$ is one then $-T$ is the other tetrahedron with the same residue, in the same orbit. The cyclotomic structures are exactly the 24 elements of trace $\alpha$. The Mumford sublattice does not depend on the generator, and the eight of them are the eight neighbours of $L_\infty$ in $T_7$: each has Gram determinant 7, no roots, 14 vectors of norm 3 and 42 of norm 7, a copy of Mumford’s lattice with $\mathcal O_L$ among them, and modulo $\sqrt{-7}$ it is the plane orthogonal to the isotropic point its Sylow subgroup fixes.

At the vertex $(v_0,\Lambda_\infty)$ the cells are data too: the fourteen neighbours at $(\alpha)$ have stabilizers $S_4^a$ and $S_4^b$, seven of each; the 21 chambers $D_8$; the eight neighbours at 7 $7{:}3$; the 112 squares, a neighbour at 2 with one at 7, two orbits with stabilizer $C_3$; and the 168 prisms, a chamber at 2 with an edge at 7, one regular orbit.

Ce que chaque place oublieWhat each place forgets

Each place forgets what its residue cannot see. At 2, where $-1\equiv1$, the residue of a vector forgets its sign, so the roots, the vectors of norms 3 and 5 and the tetrahedra have kernels $D_8$, $S_3$, $S_3$ and $S_4$. At 7 every native datum is read faithfully except those of norm 5. At $\infty$ the projective residue forgets scalars: the pairs, the frames, the ordered frame pairs and the cyclotomic structures are read faithfully, the vectors and the tetrahedra are not. Reading a vector of norm $n$ instead as a map of degree $n$ from the Klein quartic to the elliptic curve with complex multiplication by $\mathcal O_E$, as Elkies does, is faithful. A Mumford sublattice has trivial residue at 2 and at $\infty$, its index 7 being prime to 2.

No single place reads everything, but the places together read more. The vectors of norm 5 are read faithfully by 2 and 7 together. The ordered triangles of $a$-frames, and the pairs of a root pair with an orthogonal pair of norm 3, are read faithfully at each place: the stabilizer of $(\pm u,\pm w)$ is $D_8\cap S_3$, of order at most 2, and it contains $s_u$, which is $-1$ on $u^\perp\ni w$.

Le marquage arithmétiqueThe arithmetic marking

The sky is the conic of isotropic points of the ternary quadratic space $L_\infty/\sqrt{-7}L_\infty$, the link of $\Lambda_\infty$ in $T_7$. Of the bijections from it onto $\Proj^1(\F_7)$ that carry $G_0$ onto the Möbius group, half carry the elements of trace $\alpha$ into the class of $z\mapsto z+1$; under these the lattice has the character of Klein’s representation, the marking of the quartic column. So the book’s labelling is the arithmetic one read at $\bar\alpha$: the Fano plane of Part One is the reduction of Klein’s lattice at the prime of $E$ that does not contain the character value of $7A$. Two earlier statements agree with this independently of the lattice: the cyclic labellings for $\{0,4,6\}$ go onto $7A$, and at the prime $(2,\zeta^3+\zeta+1)$ of $\Q(\zeta)$, which lies over $(\alpha)$, the points of the Frobenius-fixed plane have class $S_4^b$.

Read at 7, the column of $\Proj^1(\F_7)$ is the lattice’s too. Write $b$ for $h\bmod\sqrt{-7}$ and $S$ for one of the two $G_0$-orbits of nonzero isotropic vectors, each of 24. The secant of a pair of norm 3, the two isotropic points orthogonal to its residue, is a bijection onto the 2-subsets. For a root $v$, the set $A(v)$ of isotropic points $c$ with $b(\tilde c,v)$ a nonzero square, $\tilde c\in S$ on $c$, is a bijection onto the orbit of $\{0,1,2,3\}$, and $-v$ gives the complement. A vector $w$ of norm 3 orders its secant $\{a,b\}$ so that $\det(\tilde a,\tilde b,w)$ is a nonzero square, a bijection onto the 56 ordered pairs, $-w$ giving the reversed pair. A tetrahedron $T$ goes to the set $A(T)$ of the first points of the ordered secants of its four vectors, the $a$-tetrahedra onto the orbit of $\{0,1,2,5\}$ and the $b$-tetrahedra onto that of $\{0,1,2,4\}$; and a Mumford sublattice goes to the isotropic point fixed by its Sylow subgroup. The square classes do not depend on the representatives, which change by squares.

Le Frobenius en deuxThe Frobenius at two

The prime 2 enters twice. The Klein quartic has good reduction there, its 24 points over $\F_8$ the reductions of its flexes, and $\F_8$ carries a Frobenius; the Fano plane is $\Proj(\F_8)$ over $\F_2$, and the octonion completion is built from the maps $v\mapsto w^xv^2$. A symmetry that normalizes the group gives two objects: if $\sigma\circ\mu(x)\circ\sigma^{-1}=\mu(cxc^{-1})$ on a marked set, then $\Phi=\mu(c)^{-1}\circ\sigma$ is an automorphism, the twisting element $c$ is unique when $G$ has trivial centre, and $\Phi(y)=c^{-1}y$ where $\sigma$ fixes $y$.

On the quartic, $\sigma_2\colon\zeta\mapsto\zeta^2$ satisfies $\sigma_2(\rho(x))=\rho(h^{-1}xh)$, so the twisting element $c=h^{-1}$ has power 2, and $\Phi=\rho(h)\circ\sigma_2$ is the flex-tangent map $\tau$ on the flexes, of power 4, and the identity on the bitangents, centres and flex triangles. The vectors fixed by the twisted Frobenius modulo a prime above 2 form a three-dimensional $\F_2$-space $V_0$, the two primes giving Fano planes whose points have classes $S_4^b$ and $S_4^a$, and each point of the quartic over $\F_8$ coordinatizes the dual plane, carrying $\tau$ to post-composition with the Frobenius of $\F_8$. In the octonion completion the Frobenius conjugates $M_x$ to $M_{2x}$ and multiplication by $w$ conjugates it to $M_{x-1}$; together they generate the normalizer of order 21 of the Singer group, and its Frobenius has power 2.

In the lattice’s own coordinates the Frobenius needs no twist. The eigenlines $e_\lambda$, $\lambda\in\{\zeta,\zeta^2,\zeta^4\}$, of the 24 cyclotomic structures are pairwise orthogonal and are the 24 flexes; the tangent at $e_\lambda$ is $e_\lambda\vee e_{\lambda^2}$, and $\sigma_2$ applied to coordinates carries $e_\lambda$ to $e_{\lambda^2}$. So $\tau=\sigma_2$ on the flexes, and the polarity $e\mapsto e^\perp$ of $h$ is $T\circ\tau$, $T$ sending a flex to its tangent: the three seams from the flexes to the flex tangents, $T$, $T\circ\tau$ and $T\circ\tau^2$, form a torsor for $\langle\tau\rangle\cong C_3$, and the lattice supplies the polarity. The automorphism group $N(C_7)/C_7\cong C_3$ of the object of size 24 is $\mathrm{Gal}(L/E)$, the decomposition group of the prime $\mathfrak P=(2,\zeta^3+\zeta+1)$ over $(\alpha)$, which is inert in $L$, and at the totally ramified $\sqrt{-7}$ the inertia group. Modulo $\mathfrak P$ the eigenlines become the 24 points of $\Proj^2(\F_8)$ off the lines of $\Proj^2(\F_2)$, and $\sigma_2$ the Frobenius $x\mapsto x^2$. In Klein’s coordinates, whose matrices are not defined over $E$, the twisting element is the cost of that choice.

Les sutures naturelles du réseauThe natural seams of the lattice

Which seams does the lattice prefer? The 336 antilinear isometries of $L_\infty$, the bijections with $c(\lambda x)=\bar\lambda c(x)$ and $h(cx,cy)=\overline{h(x,y)}$, form a coset of $\{\pm1\}\times G_0$, and each induces the outer automorphism of $G_0$. Complex conjugation $b_0$ of $L$ preserves $L_\infty=(1-\zeta)^{-1}\Z[\zeta]$ and is one of them. The arithmetic symmetry group of the lattice, its linear and antilinear isometries normalizing $G_0$, is $A=\{\pm1\}\times A_1$ with $A_1=G_0\rtimes\langle b_0\rangle\cong\PGL(2,7)$, of order 672. It acts on the sky through $\PGL(2,7)$ with kernel $\{\pm1\}$, $b_0$ going to an involution outside $\PSL(2,7)$; on the incidence graph of the Fano plane, the link at 2, by collineations and a polarity; and on Klein’s plane by $\rho(G)$ and complex conjugation of Klein’s coordinates. A seam between two incarnations built from the lattice is natural when it commutes with every element of $A$ that preserves both. By the stabilizer principle applied to $A$, an incarnation stable under a group $A'$ between $G_0$ and $A$ has a form, the class of its $A'$-stabilizer among the complements of $N(H)/H$; natural seams exist exactly between incarnations of the same form, and then they form a torsor under the natural automorphisms, the part of $N(H)/H$ fixed by the stabilizer. A form is projective when $-1$ acts trivially and a vector form otherwise. For Klein’s lattice: $C_3$ and $C_4$ have four forms each, two of them projective; $C_7$ has one, on which all of $C_3$ is natural; the rigid rows have one; $V_4^a$, $V_4^b$, $A_4^a$, $A_4^b$, relative to $\{\pm1\}\times G_0$, have one projective form, on which all of $N(H)/H$ is natural; the object of size 84 has two forms, on which only the involution of quotient class $C_4$ is natural; and the regular object has a projective form with natural automorphisms $S_3$ and a vector form with $\{\pm1\}$.

The book’s named seams sort accordingly. Natural for all of $A$: the seam by the multiplier 2 from ordered pairs to $3A$, the seam by the multiplier $\iota$ from imaginary points to $4A$, the tangent and residual-point seams between flexes and flex tangents, hence $\tau$ and the polarity, the seam from centres with a bitangent to the arcs of the Coxeter graph, the residue maps at 7 of the pairs of norm 3, the root pairs and the Mumford sublattices, and every seam of a rigid object stable under $A$, among them the ten among the five incarnations of the twenty-eight. Natural only for $\{\pm1\}\times G_0$: the square-class seams on the line, the seam by $\omega$ on the tangent, the point rule and the bracket rule on the Coxeter edges, the reduction of contact points at the prime over the Bianchi prime, and the involutions of quotient classes $V_4^a$ and $V_4^b$ of the object of size 84. Natural for a subgroup of index two not containing $-1$: the ordered secant of a vector of norm 3, natural for $A_1$, and the four-set $A(v)$ of a root, natural for the stabilizer $\langle G_0,-b_0\rangle$ of the orbit $S$ it uses. Every seam of the last two kinds joins incarnations of different forms and is natural for the stabilizer of the convention it uses.

Loops through the places close up inside these groups. Read at 2 through the incidence graph, at $\infty$ through Klein’s plane and at 7 through the sky, the holonomies of the loops $2\to\infty\to7\to2$ of natural seams form exactly the natural automorphisms of the incarnation at 2 for $C_2$, $C_3$, $C_7$ and 1, of orders 2, 2, 3 and 6; for $C_4$ no natural seam joins the directed 8-cycles at 2 to the eigenvectors for $i$ at $\infty$, and the cyclic forms close up through 7 and $G_0$ instead. The twisted natural self-seams of an incarnation form a group of order $336\,|\Aut_A|$ in which the untwisted ones have index 2: the Galois class, realized at 2 by the polarity, at $\infty$ by complex conjugation and at 7 by an odd Möbius map, as the type law says for a split, a complex and a ramified place. Not every natural seam has a construction. There are six natural seams from the pairs of norm 7 to the orbit of $(1{:}2{:}5)$, and no native datum with that orbit as residue was found; every point of Klein’s plane fixed by complex conjugation and by no element of $\rho(G)$ has an orbit of the one projective form of the regular object, so these orbits form a continuum, any two joined by six natural seams. A statement that every natural seam is induced can therefore hold only for a notion of induced seam that contains the stabilizer principle for $A$ itself, and then it says nothing. The general theory of natural seams of an arithmetic source is the subject of the chantier on reciprocity.

Le revêtement double dans le réseau de WeilThe double cover in the Weil lattice

The double cover has a lattice of its own. Let $\tilde G=\SL(2,7)$ and let $E_8^a$ be the hermitian lattice of the Weil quartet of the next chapter, on which $\tilde G$ acts by the signed permutations of a cross of sixteen vectors $\pm v_c$, one pair over each point $c$ of $\Proj^1(\F_7)$; its roots outside the cross lie over pairs of points, the two points fixed by the image of their stabilizer. Every $\tilde G$-orbit of nonzero vectors is a new object, since $-I$ acts as $-1$. The odd lift of $C_7$ is never the stabilizer of a vector, so the object of size 48 is a forced gap of the vectors. Sets of vectors fill it: for a root $r$ over a pair not containing $\infty$, the seven roots $Ur$, $U$ the unipotent odd lift of $C_7$ fixing $v_\infty$, have stabilizer exactly $U$, and their orbit is the object of size 48.

So the four new objects are data of $E_8^a$: the cross, the other two orbits of roots, the vectors of norm 2 or 3 with trivial stabilizer, and the seven-sets $Ur$. With the fifteen objects of Klein’s lattice, $L_\infty\oplus E_8^a$ over $\Z[\alpha]$, an integral form of the Weil representation $3\oplus4$, carries all nineteen objects of $\tilde G$.

The sign of $-I$ is seen differently at each place. At 2, $x-(-x)=2x=\lambda\bar\lambda x$ lies in $\lambda E_8^a$ and in $\bar\lambda E_8^a$, so $-I$ lies in the kernel of every residue and no new object is read faithfully there. At 7, $E_8^a/\sqrt{-7}E_8^a\cong\F_7^4$ has no invariant line, plane or hyperplane, and the residues of the cross lie on eight lines any four of which are independent: the module is $\mathrm{Sym}^3V$, $V$ the natural module, and the cross is the twisted cubic over the eight points of $\Proj^1(\F_7)$. There $-I$ lies in no residue kernel, every residue of a new object is again a new object, and the roots, the vectors of norm 2 and the seven-sets $Ur$ are read faithfully. At $\infty$ the linear residue in the Weil quartet is faithful, and the projective one has kernel $\{\pm I\}$ times the stabilizer.