Une loi de réciprocité

The composite $\tau$ of the tangent seam and the residual-point seam is an automorphism of the flexes of power 4. So the cycle flexes $\to$ flex tangents $\to$ flexes along these two natural seams has monodromy of order 3, and $\tau$ turns each flex triangle cyclically. Under every seam between the flexes and the cyclic labellings, $\tau$ corresponds to Hall’s multiplier 2, and both correspond to the fourth-power map on $7A$ and $7B$.

Let $\mathcal G$ be a connected graph with fundamental group $\pi$, and $X$ an object with stabilizer $H$ and automorphism group $A\cong N_G(H)/H$. The gauge classes of seam systems for $X$ over $\mathcal G$ correspond to $H^1(\pi,A)=\operatorname{Hom}(\pi,A)/A$, and to the principal $A$-coverings of $\mathcal G$ whose fibre over a vertex is the set of alignments there. The system is coherent exactly when its class is trivial. With seams over automorphisms, the classes are $H^1(\pi,\Aut^{\mathrm{tw}}(X))$.

The vectors of norm 2 of $L_\infty$ form 21 pairs $\pm v$, and those of norm 3 form 28. (1) In Klein’s plane, $s_v(x)=-x+h(x,v)v$ for $v$ of norm 2 runs through the 21 involutions of $G_0$, and $\mathbb P(v^\perp)$ for $v$ of norm 3 through the 28 bitangents of the quartic. (2) At 2, $v\mapsto(v\bmod\alpha,\ (v\bmod\bar\alpha)^\perp)$ carries the pairs of norm 2 onto the 21 flags of the Fano plane and those of norm 3 onto its 28 antiflags. (3) At 7, reduction carries them onto the 21 points inside the conic and the 28 outside it, hence onto the 28 pairs of points of the sky. These maps commute with $G_0$, so the bitangents, the antiflags and the pairs of points of the sky are three reductions of one set of pairs of vectors, joined by the seams of the object of size 28.

Let $L$ be a $G$-lattice over $\mathcal O_K$ with good reduction at $\mathfrak P$, residue $V_{\mathfrak P}=L/\mathfrak PL$, and $(\sigma,\alpha)$ a twisted Galois symmetry. Then $L$ has good reduction at $\sigma\mathfrak P$ and $(V_{\sigma\mathfrak P})^{\alpha}\cong\bar\sigma V_{\mathfrak P}$. In particular: if $\sigma\notin D(\mathfrak P)$ there is a $\bar\sigma$-semilinear seam over $\alpha$ from the residue at $\mathfrak P$ to the residue at $\sigma\mathfrak P$ (split type); if $\sigma\in D(\mathfrak P)\setminus I(\mathfrak P)$, $\alpha$ is realized on $V_{\mathfrak P}$ by a $\bar\sigma$-semilinear map (inert type); if $\sigma\in I(\mathfrak P)$, by a linear map normalizing the image of $G$, not in $k^\times\rho_{\mathfrak P}(G)$ when $G$ is perfect and $\alpha$ is outer (ramified type). If moreover $\sigma\circ\chi=\bar\chi$, then $\alpha$ carries every residue of good reduction to its dual.

Each of the fifteen data above is a single $G_0$-orbit whose stabilizers form the class of its object. Moreover each root pair is orthogonal to exactly four others, the frames are exactly fourteen, seven $a$-frames and seven $b$-frames; the tetrahedra are exactly twenty-eight and each sums to 0; the cyclotomic structures are exactly the 24 elements of trace $\alpha$; and the eight Mumford sublattices $L_\infty(1-g)$ are the eight neighbours of $L_\infty$ in the tree at 7.

Every incarnation in the book’s seam tables, and each of the five incarnations of the object of size 28, is a $G_0$-set of data of the lattice at one place. For the six rigid objects every seam is induced. For $C_3$, $C_4$, $C_7$, $A_4^a$, $A_4^b$, $V_4^a$ and $V_4^b$ the natural automorphisms realize all of $N(H)/H$, so all seams are induced once one is. For $G/C_2$ at most two of the four seams between two incarnations stable under the antilinear isometries are induced, and they differ by the involution of quotient class $C_4$. For the regular object, the natural automorphisms of such an incarnation form a group of order at most 6; on the vectors of norm 5 they are $\pm1$.

The objects $G/C_2$ and $G/1$ are the rows on which the arithmetic symmetry group acts nontrivially on $N(H)/H$, so that even between incarnations of one form only some seams are natural: two of four for $G/C_2$, six (projective) or two (vector) of 168 for $G/1$. Their exception is a residual symmetry, a torsor that is never a point, not a cohomological obstruction: $G/C_2$ has one projective form and $G/1$ one projective form. The obstruction does occur, at the rows $C_3$ and $C_4$, which the lattice counts as induced because induction allows auxiliary choices: points of contact and eigenvectors for $i$ have the dihedral form, ordered pairs, imaginary points and the classes $3A$, $4A$ the cyclic one.

Let $Y$ be an incarnation of an object among the points, lines or conics of the plane of the quartic. If some $\sigma\in\operatorname{Gal}(\overline\Q/\Q)$ with $\sigma(\sqrt{-7})=-\sqrt{-7}$ maps $Y$ onto itself, then the stabilizer class of $Y$ is fixed by $\Aut(G)$. Hence the objects with stabilizers $V_4^a$, $V_4^b$, $A_4^a$, $A_4^b$, $S_4^a$, $S_4^b$ have no incarnation there defined over $\Q$ as a set.

Let $A'$ be a group of arithmetic symmetries with $G\le A'$, preserving the class $[H]$ and an incarnation $X$, and let $B$ be the stabilizer in $A'$ of a point with $G$-stabilizer $H$. The $A'$-sets that are transitive $G$-sets of class $[H]$ correspond to the classes of complements of $N_G(H)/H$ in $N_{A'}(H)/H$, the pointed set $H^1(A'/G,N_G(H)/H)$: the forms. A natural seam $X\to Y$ exists if and only if $X$ and $Y$ have the same form; the natural seams then form a torsor under $(N_G(H)/H)^B$, the natural automorphisms; and a canonical seam exists exactly when the forms agree and that group is trivial. Forms exist if and only if the extension of $A'/G$ by $N_G(H)/H$ splits.