Première partie · Le langage des suturesChapitre 5

Courte marche à travers la théorie de Galois

A short walk through Galois theory

Read from the draft of 3 October 2026

Plate 5.1The fifteen objects of the group of order 168, the connected objects of its Galois category. Only

G/1

and

G/G

, lit, are Galois objects, since the group is simple.

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10

How much of the language of seams is Galois theory already, and what does Galois theory not supply?

The first chapters set up objects, incarnations and seams for a finite group $G$ and proved the stabilizer principle. These notions are not new in kind. They are Galois theory in the form Grothendieck gave it, in which a Galois extension is replaced by a category of finite sets with an action and a functor that forgets the action. In that language an object is a connected object, a seam is an isomorphism, a marking is an identification of fibre functors, and a seam over an automorphism is a morphism twisted by an outer automorphism.

The chapter makes the correspondence exact. It then returns to the theorem of Galois’s last letter, where the group of order 168 first appears, and follows the outer automorphism of that group into arithmetic: there it becomes the Galois conjugation of $\sqrt{-7}$ , and it acts on the incarnations in Klein’s plane as seams over an automorphism. It ends with what Galois theory does not supply, which is most of the subject: which isomorphisms a theory singles out, and whether they cohere.

The same conjugation also measures what counting cannot hear and what finite sets cannot say. That is the Galois gap, and it has a chapter of its own, Chapter 17.

The central result · Objects are the connected objects of a Galois category

Let $\mathcal C_G$ be the category of finite $G$ -sets and $G$ -maps, and $F_G\colon\mathcal C_G\to\mathbf{FinSet}$ the functor that forgets the action.

(a) $(\mathcal C_G,F_G)$ satisfies Grothendieck’s axioms (G1)–(G6), and $g\mapsto(x\mapsto gx)$ is an isomorphism from $G$ onto $\Aut(F_G)$ .

(b) The connected objects of $\mathcal C_G$ are the objects of $G$ , its transitive sets. The object $G/H$ is Galois, that is, its automorphism group acts transitively on its fibre, if and only if $H$ is normal.

(c) A Galois category whose fundamental functor has automorphism group isomorphic to $G$ is equivalent to $\mathcal C_G$ , by an equivalence that carries the fundamental functor to $F_G$ .

Proof

(a) Limits, sums, and quotients by finite groups of $G$ -automorphisms are formed on the underlying sets. The image of a $G$ -map and its complement are $G$ -stable, so a $G$ -map is a surjection onto its image followed by the inclusion of a direct summand, and a bijective $G$ -map is a $G$ -isomorphism: $F_G$ is exact and reflects isomorphisms. Let $\eta$ be an automorphism of $F_G$ and put $g=\eta_G(1)$ , its value at the regular set. For $x\in X$ the map $f_x(k)=kx$ is a $G$ -map, and naturality gives $\eta_X(x)=\eta_X(f_x(1))=f_x(\eta_G(1))=gx$ . Conversely each $g$ defines an automorphism of $F_G$ , and composition corresponds to multiplication.

(b) A $G$ -set is the sum of its orbits, so it is connected exactly when it is transitive. The automorphism group of $G/H$ is $N_G(H)/H$ , acting freely on $G/H$ , and transitively exactly when $N_G(H)=G$ . (c) is Grothendieck’s theorem (SGA 1, Exposé V).

Status

The Galois-categorical reading of floors one and two is classical, and the chapter says so: Galois categories after SGA 1, the Galois theory of fields in this form after Szamuely, categorical Galois theory after Borceux and Janelidze, non-abelian $H^1$ and twisted forms after Serre, and gerbes after Giraud. Its proofs are short and complete. The counts in the gerbe of twisted seams, the involutive seams of the double cover and the Galois action on Klein’s quartic were checked by machine, the last in $\Q(\zeta_7)$ , $\Q(\zeta_{21})$ and $\Q(\zeta_{28})$ .

Galois’s statement is classical, and Gierster gave its first complete proof in 1881. The chapter adds three things: Galois’s own construction made precise, a second proof of necessity by characters, which was not found in the sources checked, and the fields of definition of the three resolvents. All three are proved, with the normalizers and genera checked by machine. Rigidity and the fields of definition of the modular coverings are Serre’s account of classical results, with the character tables certified exactly, and the realization of the outer automorphism by complex conjugation is classical, after Elkies.

Les étages un et deux comme catégorie galoisienneFloors one and two as a Galois category

Plate 5.1The fifteen objects of the group of order 168, the connected objects of its Galois category. Only

G/1

and

G/G

, lit, are Galois objects, since the group is simple.

Grothendieck lists six axioms for a category $\mathcal C$ with a functor $F$ to finite sets: finite limits, finite sums and quotients by finite groups of automorphisms exist; every morphism is a strict epimorphism followed by a monomorphism onto a direct summand; and $F$ is exact and reflects isomorphisms. A pair satisfying them is equivalent to the finite continuous $\pi$ -sets with their forgetful functor, for $\pi=\Aut(F)$ . Only finite $\pi$ occur here.

Read in this language, the first floor is the category $\mathcal C_G$ itself. An object of $G$ is a connected object, an incarnation is a connected object as some theory presents it, and a seam is an isomorphism between two of them. The Galois objects are the quotients by normal subgroups. The group of order 168 is simple, so among its fifteen objects exactly two are Galois: the regular object $G/1$ , of size 168, and the point $G/G$ .

Definition

A category equivalent to the category of finite continuous $\pi$ -sets, for a profinite group $\pi$ , is a Galois category, and a functor $F$ to finite sets satisfying Grothendieck’s axioms is a fundamental functor, or fibre functor. For a finite group $G$ , the category $\mathcal C_G$ of finite $G$ -sets and $G$ -maps, with the functor $F_G$ that forgets the action, is one, and $\Aut(F_G)\cong G$ .

Les corps comme incarnationsFields as incarnations

Plate 5.2Left, the three cube roots of 2: complex conjugation fixes

\sqrt[3]2

and exchanges the other two. Right, the fields between

\Q

and

\Q(\sqrt[3]2,\omega)

, each over the subgroup of

S_3

that fixes it; the three cubic fields, gold, are one rigid object, joined by exactly one

\Q

-isomorphism each.

Let $K/k$ be a finite Galois extension with group $G$ . The functor $A\mapsto\operatorname{Hom}_k(A,K)$ sends a finite étale algebra split by $K$ to a finite $G$ -set, and the field $K^H$ to $G/H$ : the $k$ -embeddings of $K^H$ into $K$ are the restrictions of the elements of $G$ , and two restrictions agree exactly when the elements differ on the right by an element of $H$ . So the fields between $k$ and $K$ are incarnations of the objects of $G$ , and the fundamental theorem of Galois theory is the stabilizer principle.

The smallest non-abelian case shows every feature. In $K=\Q(\sqrt[3]2,\omega)$ , with group $S_3$ , the three cubic fields $\Q(\sqrt[3]2)$ , $\Q(\omega\sqrt[3]2)$ and $\Q(\omega^2\sqrt[3]2)$ are fixed by the three subgroups of order two. They are three incarnations of $S_3/C_2$ , which is rigid because $C_2$ is its own normalizer, so between two of them there is exactly one $\Q$ -isomorphism, such as $\sqrt[3]2\mapsto\omega\sqrt[3]2$ , and these compose coherently. The field $\Q(\omega)=K^{C_3}$ has automorphism group $N(C_3)/C_3\cong C_2$ , complex conjugation, and the description $\Q(\omega)\subset K$ forgets the three cube roots of 2, the fibre $C_3$ of $S_3\to S_3/C_3$ .

Proposition(Fields as incarnations) proved

Let $K/k$ be a finite Galois extension with group $G$ .

(a) The functor $A\mapsto\operatorname{Hom}_k(A,K)$ is an anti-equivalence from the finite étale $k$ -algebras split by $K$ onto $\mathcal C_G$ . It sends the field $K^H$ to $G/H$ .

(b) The objects of $G$ correspond to the fields between $k$ and $K$ , up to $k$ -isomorphism. The stabilizer principle becomes the fundamental theorem of Galois theory, and $\Aut_k(K^H)\cong N_G(H)/H$ . The seams between two incarnations of one object become the $k$ -isomorphisms between two fields, a torsor under the automorphism group of either.

(c) A description $G/H\to G/L$ , $H\le L$ , corresponds to the inclusion $K^L\subseteq K^H$ . Its kernel at the base point is $L=\operatorname{Gal}(K/K^L)$ . What it forgets there is $\operatorname{Hom}_{K^L}(K^H,K)$ , the set of conjugates over $K^L$ of a primitive element of $K^H$ , a copy of $L/H$ .

Proof

(a) is the form of Galois’s theorem given by Grothendieck. (b) An anti-equivalence preserves isomorphisms and reverses composition, so $\Aut_k(K^H)$ is the opposite group of $\Aut_G(G/H)$ , which is isomorphic to it. (c) Restriction $\operatorname{Hom}_k(K^H,K)\to\operatorname{Hom}_k(K^L,K)$ is the projection $gH\mapsto gL$ . Its fibre over the inclusion of $K^L$ consists of the embeddings of $K^H$ that are the identity on $K^L$ , and by the Galois theory of $K/K^L$ , whose group is $L$ , they form a copy of $L/H$ .

Marquages et sutures torduesMarkings and twisted seams

Plate 5.3The polarity carries each line of the Fano plane to its pole. It is no seam, since the stabilizer of a point fixes no line; it is a seam over the outer automorphism

g\mapsto(g^{\mathsf T})^{-1}

A theory that supplies a finite group $\Gamma$ acting on a set $Y$ meets $G$ through a marking $\mu\colon G\to\Gamma$ . Restriction along $\mu$ is an exact functor $\mu^*\colon\mathcal C_\Gamma\to\mathcal C_G$ with $F_G\circ\mu^*=F_\Gamma$ , and the marked set is $\mu^*(Y)$ . Conversely a functor $H\colon\mathcal C_\Gamma\to\mathcal C_G$ with an isomorphism $\eta\colon F_G\circ H\to F_\Gamma$ determines exactly one marking for which every $\eta_Y$ is a $G$ -isomorphism, and replacing $\eta$ by $\gamma\circ\eta$ conjugates the marking by $\gamma$ . So a change of marking by an inner automorphism is a change of base point: the maps $y\mapsto\mu(h)y$ form an isomorphism of functors $\mu^*\to(\mu\circ\iota_h)^*$ . This is the finite case of SGA 1, Exposé V, Corollary 6.3.

An automorphism $\alpha$ of $G$ twists every $G$ -set, $Y\mapsto Y^\alpha$ , and a seam over $\alpha$ from $X$ to $Y$ is a morphism $X\to\alpha^*Y$ . The polarity of the Fano plane, which sends the line $\{v:u\cdot v=0\}$ to the point $u$ , is the first example: it is no seam, since the stabilizer of a point fixes no line, but it is a seam over the outer automorphism $g\mapsto(g^{\mathsf T})^{-1}$ . Only the class of $\alpha$ in $\operatorname{Out}(G)$ matters, and the proposition says exactly in what sense.

Proposition(Seams over automorphisms and

\operatorname{Out}(G)

) proved

For $\alpha\in\Aut(G)$ the twist $\alpha^*\colon Y\mapsto Y^\alpha$ is an exact autoequivalence of $\mathcal C_G$ with $F_G\circ\alpha^*=F_G$ , and a seam over $\alpha$ from $X$ to $Y$ is a morphism $X\to\alpha^*Y$ .

(a) The morphisms of functors $\alpha^*\to\beta^*$ are the families $y\mapsto cy$ for the $c\in G$ with $\beta=\iota_c\circ\alpha$ . In particular $\alpha^*\cong\id$ if and only if $\alpha$ is inner.

(b) The automorphisms of the identity functor of $\mathcal C_G$ are the actions of the elements of the centre $Z(G)$ .

(c) Every exact autoequivalence $E$ of $\mathcal C_G$ with $F_G\circ E\cong F_G$ is isomorphic to some $\alpha^*$ , and $\alpha\mapsto\alpha^*$ induces an anti-isomorphism from $\operatorname{Out}(G)$ onto the group of isomorphism classes of such autoequivalences. When $Z(G)=1$ , two of them are isomorphic in at most one way.

Proof

(a) Let $\theta\colon\alpha^*\to\beta^*$ be a morphism of functors and $c=\theta_G(1)$ , its value at the regular $G$ -set. The right multiplications $r_k(x)=xk$ are $G$ -maps, and the twists do not change them, so naturality gives $\theta_G(k)=ck$ . That $\theta_G$ is a $G$ -map $\alpha^*G\to\beta^*G$ says $c\,\alpha(g)k=\beta(g)\,ck$ , that is, $\beta=\iota_c\circ\alpha$ . Naturality along $f_y(k)=ky$ gives $\theta_Y(y)=cy$ , and conversely $y\mapsto cy$ is a $G$ -map when $\beta=\iota_c\alpha$ . (b) is (a) with $\alpha=\beta=\id$ .

(c) By the proposition on markings, with $\Gamma=G$ , $E\cong\mu^*$ for an endomorphism $\mu$ of $G$ . As $\mu^*$ is essentially surjective, the regular $G$ -set is some $\mu^*(Y)$ , whose stabilizers contain $\ker\mu$ ; so $\mu$ is injective, hence an automorphism, and $\mu^*$ is the twist by it. Since $(Y^\alpha)^\beta=Y^{\alpha\beta}$ the assignment reverses composition, and by (a) $\alpha^*\cong\beta^*$ exactly when $\beta\in\iota(G)\alpha$ . Two isomorphisms $\alpha^*\to\beta^*$ differ by an automorphism of $\alpha^*$ , which by (a) is the action of a central element.

Monodromie, torseurs et gerbesMonodromy, torsors and gerbes

Plate 5.4The gerbe of twisted seams at 168 is neutral. Each of the nine objects whose class the outer automorphism fixes has overgroups of index 2 in

\PGL(2,7)

meeting

G

in its stabilizer: 28 for the regular object, 2 or 1 for the others. The other six objects come in pairs the outer automorphism exchanges.

Over a connected graph, a family of incarnations of an object $X$ joined by seams is a principal $A$ -covering, $A=\Aut_G(X)$ : the fibre over a vertex is the set of alignments, and the seams are the transition maps. Its class is its holonomy, a homomorphism from the fundamental group to $A$ up to conjugation, and the family is coherent exactly when the class is trivial. Allowing seams over automorphisms, that is, changes of marking, replaces $A$ by the group $\Aut^{\mathrm{tw}}(X)$ of pairs $(\beta,f)$ with $f$ a seam over $\beta$ from $X$ to $X$ .

When $Z(G)=1$ , $G$ sits in $\Aut^{\mathrm{tw}}(X)$ as the normal subgroup of pairs $(\iota_g,\,x\mapsto gx)$ , meeting $A$ trivially, and $E_X=\Aut^{\mathrm{tw}}(X)/G$ is an extension of $\operatorname{Out}(G)_{[H]}$ by $A$ : an $A$ -gerbe, in the sense of Springer and Giraud. It is neutral, the extension splits, exactly when the action of $G$ on $X$ extends to $\Aut(G)_{[H]}$ with each $\beta$ acting by a seam over $\beta$ . For $\PSL(2,7)$ , with $\beta_0$ the conjugation by $z\mapsto-z$ , the extension splits for all fifteen objects, so every object whose stabilizer class is fixed by $\Aut(G)$ is the restriction of a $\PGL(2,7)$ -set. For the nine classes 1, $C_2$ , $C_3$ , $C_4$ , $S_3$ , $C_7$ , $D_8$ , $7{:}3$ , $G$ , the subgroups $H'\le\PGL(2,7)$ with $H'\cap G=H$ and $|H':H|=2$ number 28,2,2,2,1,1,1,1,1. The action of $\SL(2,7)$ on each of its four new objects likewise extends to the matrices of determinant $\pm1$ .

The book’s non-neutral gerbes appear at the double cover. For $\SL(2,7)\to G$ , the automorphisms of a new object that lie over those of its quotient form a central extension by $\{\pm I\}$ , which splits for the objects of sizes 16 and 48 and not for those of sizes 112 and 336. A cycle of seams whose monodromy on the ordered pairs of points is the exchange has, on the new object, monodromy of order 4 squaring to $-I$ . And since the coverings of the next sections have the centreless group $\PSL(2,p)$ , their field of moduli is a field of definition (Coombes and Harbater, Dèbes and Douai).

Theorem(Seam monodromy is non-abelian

H^{1}

) proved

Let $\mathcal G$ be a connected graph with base vertex $v$ and $\pi=\pi_1(\mathcal G,v)$ , and let $X$ be an object with stabilizer $H$ and automorphism group $A=\Aut_G(X)\cong N_G(H)/H$ .

(a) The gauge classes of seam systems for $X$ over $\mathcal G$ correspond to $H^1(\pi,A)=\operatorname{Hom}(\pi,A)/A$ , the first non-abelian cohomology set for the trivial action. They also correspond to the isomorphism classes of principal $A$ -coverings of $\mathcal G$ : to a system corresponds the covering whose fibre over a vertex $w$ is the set of alignments $\Iso_G(X,Y_w)$ , with $A$ acting by precomposition and the seams as transition maps. The system is coherent exactly when its class is trivial.

(b) Let $\Aut^{\mathrm{tw}}(X)$ be the group of pairs $(\beta,f)$ , with $\beta\in\Aut(G)$ and $f$ a seam over $\beta$ from $X$ to $X$ , an extension $1\to A\to\Aut^{\mathrm{tw}}(X)\to\Aut(G)_{[H]}\to1$ . With an incarnation of $X$ at each vertex, a seam over an automorphism on each edge, and changes of marking as gauge transformations, such systems correspond to $H^1(\pi,\Aut^{\mathrm{tw}}(X))$ . Their image in $H^1(\pi,\Aut(G)_{[H]})$ is the monodromy of the markings, and since $\pi$ is free, every class there arises.

Proof

(a) The correspondence with $\operatorname{Hom}(\pi,A)/A$ is the gauge theorem of Chapter 4. A principal $A$ -covering of a connected graph is determined up to isomorphism by its holonomy, a homomorphism $\pi\to A$ up to conjugation, and in a gauge the transition maps of the covering of alignments are the link variables. (b) The gauge theorem uses only that link variables compose in a group, so it applies with $\Aut^{\mathrm{tw}}(X)$ in place of $A$ . A homomorphism from a free group lifts along a surjection, generator by generator.

Les lettres conjointesGalois’s conjugate letters

Plate 5.5Galois’s conjugate letters on the projective lines over

\F_5

\F_7

and

\F_{11}

: each matching pairs every point with its lettre conjointe, and its stabilizer is

A_4

S_4

A_5

, of index

p

. The conjugate of 1 is a square only for

p=5

Galois’s letter to Auguste Chevalier of 29 May 1832 ends its account of the theory of equations with the modular equations of elliptic functions. For a prime $p$ the modular equation of degree $p+1$ has its roots $x_k$ indexed by $k\in\Proj^1(\F_p)$ , and Galois gives its group as the $(p+1)p(p-1)$ substitutions $x_k\mapsto x_{(ak+b)/(ck+d)}$ ; those in which $ad-bc$ is a square form $\PSL(2,p)$ . He asks whether the degree can be lowered to $p$ , that is, whether this group has a subgroup of index $p$ , and answers: “Ainsi, pour les cas de $p=5$ , 7, 11, l’équation modulaire s’abaisse au degré $p$ .” In the higher cases, he adds, the reduction is impossible. An article of April 1830 had claimed the reduction only for $p=5$ , and the letter corrects it, as Liouville notes in his edition of 1846. Gierster gave the first complete proof in 1881, and Dickson’s list of subgroups gives the proof of Galois’s window in Chapter 2.

Part (b) is Galois’s own step. Write $M$ for the conjugate of 1; then the conjugate of $m^2$ is $m^2M$ . If $M$ were a square, taking $m^2=M$ would give $M^2=1$ , so $M=-1$ would be a square, which, he says, can happen only for $p=5$ . For $p=7$ and $p=11$ he writes down the matchings of (c), with $M=3$ and $M=2$ , which are not squares.

Proposition(Galois’s conjugate letters) proved

Let $p\ge5$ be prime, $G=\PSL(2,p)$ acting on $P=\Proj^1(\F_p)$ , and $H\le G$ a subgroup of index $p$ .

(a) $H$ is transitive on $P$ . The stabilizer $H_y$ of a point is cyclic of order $(p-1)/2$ and fixes exactly one other point $y'$ , Galois’s lettre conjointe of $y$ . The pairs $\{y,y'\}$ form an $H$ -invariant perfect matching $M_H$ of $P$ , and $H$ is the stabilizer of $M_H$ in $G$ . So the subgroups of index $p$ are the stabilizers of the perfect matchings of $P$ whose $G$ -orbit has $p$ members.

(b) If $\infty'=0$ , then $(m^2k)'=m^2k'$ for all $m,k\in\F_p^\times$ .

(c) Galois’s matchings are $\infty0$ , 13, 26, 45 for $p=7$ and $\infty0$ , 12, 36, 48, $5\,10$ , 97 for $p=11$ . For $p=5$ the matching $\infty0$ , 14, 23 has the same property. Their stabilizers are groups $S_4$ , $A_5$ and $A_4$ of index $p$ . In the marking of the seam table, Galois’s matching for $p=7$ has stabilizer of class $S_4^b$ : it is a line of the Fano plane.

Proof

(a) The stabilizer $B=G_\infty$ has order $p(p-1)/2$ , its unipotent radical $U$ has order $p$ , and $B/U$ is cyclic of order $(p-1)/2$ . Since $|H||B|/|G|=(p-1)/2$ , $|H\cap B|\ge(p-1)/2$ . As $|H|=(p^2-1)/2$ is prime to $p$ , $H\cap U=1$ , so $H\cap B$ embeds in $B/U$ ; hence $|H\cap B|=(p-1)/2$ and $HB=G$ , that is, $H$ is transitive on $P$ . A nontrivial element of $H\cap B$ is $z\mapsto\lambda z+\mu$ with $\lambda\ne1$ , so it fixes $\infty$ and exactly one finite point, and every nontrivial power of a generator commutes with it and fixes the same point $x=\infty'$ . Then $H_x\supseteq H_\infty$ , and the two have the same order, so they are equal. Hence $y\mapsto y'$ is a fixed-point-free involution commuting with $H$ , and $M_H$ is $H$ -invariant. Its stabilizer contains $H$ and is proper, since $G$ is 2-transitive on $P$ , and $|G:H|=p$ is prime.

(b) If $\infty'=0$ , then $H_\infty=H_0$ is the pointwise stabilizer of 0 and $\infty$ , which is $\{z\mapsto m^2z\}$ ; it lies in $H$ and so preserves $M_H$ . (c) was checked by machine. For $p=7$ the matching lies in the orbit of $\{01,25,3\infty,46\}$ , which the seam table assigns to $S_4^b$ , and its stabilizer has that class.

Une seconde preuve, par les caractèresA second proof, by characters

Plate 5.6At

h^j

, for

h

a generator of a non-split torus of order

n=(p+1)/2

, the number of fixed points is

1-2\cos(2\pi j/n)

, set beside each

n

-th root of unity:

(5,2,2)

(7,1,3,1)

(11,0,2,3,2,0)

. At

p=13

n=7

, and

2\cos(2\pi/7)

is not an integer.

Galois’s theorem allows an action of $\PSL(2,p)$ on $p$ points only for $p=5$ , 7, 11. Necessity has a short proof by characters. A transitive group of prime degree is 2-transitive or solvable, so the permutation character is $1+\chi$ with $\chi$ irreducible of degree $p-1$ ; for $p\ge5$ such a $\chi$ is cuspidal, and on a non-split torus its values are $-\omega-\omega^{-1}$ for a character $\omega$ of the torus. A number of fixed points is an integer, and that is enough.

For $p=5$ , 7, 11 the permutation characters were computed directly. On a generator $h$ of the torus $C$ , $(\pi(h^j))_{0\le j<n}$ is $(5,2,2)$ , $(7,1,3,1)$ and $(11,0,2,3,2,0)$ , and $\pi-1$ is the irreducible character of degree $p-1$ . The orders 3, 4, 6 are those of the rotations that preserve a lattice in the plane, and the same window reads through the binary polyhedral groups as $(p^2-1)/2\in\{12,24,60\}$ . This argument was not found in the sources checked.

Proposition(A second proof of necessity, by characters) proved

Let $p\ge5$ , suppose $G=\PSL(2,p)$ has a subgroup $H$ of index $p$ , and let $\pi$ be the permutation character of $G$ on $G/H$ . Let $C\le G$ be the image of a non-split torus of $\SL(2,p)$ , cyclic of order $n=(p+1)/2$ . Then $\pi=1+\chi$ , where $\chi$ is a cuspidal character of degree $p-1$ , attached to a character $\omega$ of $C$ that is faithful, and

\pi(h)=1-\omega(h)-\omega(h)^{-1}\qquad(h\in C,\ h\ne1).

Consequently $2\cos(2\pi/n)$ is an integer, so $n\in\{3,4,6\}$ and $p\in\{5,7,11\}$ .

Proof

$G$ acts transitively on the $p$ points of $G/H$ and is not solvable. By Burnside’s theorem a transitive group of prime degree is 2-transitive or solvable, so the action is 2-transitive and $\pi=1+\chi$ with $\chi$ irreducible of degree $p-1$ . The irreducible representations of $\SL(2,p)$ of degree $p-1$ , for $p\ge5$ , are the cuspidal $\pi_\omega$ , where $\omega$ is a character of the norm-one subgroup of $\F_{p^2}^\times$ with $\omega^2\ne1$ , and on an element with eigenvalues $z^{\pm1}\notin\F_p$ the trace of $\pi_\omega$ is $-\omega(z)-\omega(z)^{-1}$ . Since $\chi$ is trivial on $-I$ , $\omega$ factors through $C$ .

If $\omega(h)=1$ for some $h\ne1$ in $C$ , then $\pi(h)=-1$ , which is impossible; so $\omega$ is faithful on $C$ . For a generator $h$ of $C$ , $\omega(h)$ is then a primitive $n$ -th root of unity, and $\pi(h)$ is an integer, so $2\cos(2\pi/n)\in\Z$ . This forces $n\in\{1,2,3,4,6\}$ , and $n\ge3$ for $p\ge5$ .

Les trois résolvantesThe three resolvents

Plate 5.7The three resolvents: under each of Galois’s matchings, the genus of the curve

X(p)/H

, which is 0 every time, and the field over which the resolvent of degree

p

is defined,

\Q

\Q(\sqrt{-7})

\Q(\sqrt{-11})

Each prime of the window carries a geometry on its $p$ points and a resolvent of degree $p$ . For $p=5$ the stabilizer is $A_4$ , the points are the five letters, or $\Proj^1(\F_4)$ , Galois’s matching is $\infty0$ , 14, 23, and the resolvent is defined over $\Q$ . For $p=7$ there are two classes of $S_4$ , the points or the lines of the Fano plane, Galois’s matching $\infty0$ , 13, 26, 45 has class $S_4^b$ , and the field is $\Q(\sqrt{-7})$ . For $p=11$ there are two classes of $A_5$ , the points or the blocks of the biplane, and the field is $\Q(\sqrt{-11})$ .

In every case the resolvent curve $X(p)/H$ has genus 0. On the seven points, for $p=7$ , the branch cycles have indices 2, 4 and 6, so $2g-2=-14+12$ .

Proposition(The three resolvents) proved

Let $p\in\{5,7,11\}$ and let $H\le\PSL(2,p)$ have index $p$ .

(a) The covering $X(p)/H\to X(1)$ , of degree $p$ , has genus 0. The genera of $X(p)$ and of $X_0(p)$ are 0,3,26 and 0,0,1.

(b) $\PGL(2,p)$ has a subgroup of index $p$ if and only if $p=5$ . For $p=7$ and $p=11$ the subgroups of index $p$ of $\PSL(2,p)$ are their own normalizers in $\PGL(2,p)$ . For $p=5$ the normalizer of $A_4$ in $\PGL(2,5)\cong S_5$ is a group $S_4$ not contained in $\PSL(2,5)$ .

(c) Galois’s resolvent of degree $p$ is defined over $\Q$ for $p=5$ . For $p=7$ and $p=11$ it is defined only over $\Q(\sqrt{-7})$ and $\Q(\sqrt{-11})$ , and the nontrivial automorphism of that field exchanges the two resolvents attached to the two classes of $H$ .

Proof

(a) Riemann–Hurwitz, with the branch cycles $x_0,y_0,z_0$ of the modular covering acting on $G/H$ , on $G$ and on $\Proj^1(\F_p)$ ; checked by machine. (b) A subgroup $K$ of index $p$ in $\PGL(2,p)$ has order $p^2-1$ , which does not divide $|\PSL(2,p)|$ , so $K\cap\PSL(2,p)$ has index 2 in $K$ . It is then a subgroup of index $p$ in $\PSL(2,p)$ and is normal in $K$ . For $p=7$ and $p=11$ its normalizer in $\PGL(2,p)$ is itself, checked by machine, which is a contradiction. For $p=5$ , $S_4$ works.

(c) The splitting field of the modular equation over $\Q(j)$ is a $\PGL(2,p)$ -extension with constant field $\Q(\sqrt{p^*})$ . For $p=5$ the fixed field of $S_4$ is a degree-5 extension of $\Q(j)$ whose constant field is $\Q$ , because $S_4$ maps onto $\PGL(2,5)/\PSL(2,5)$ ; geometrically it is $X(5)/A_4$ . For $p=7$ and $p=11$ , suppose the resolvent were defined over $\Q$ as a covering of the $j$ -line. Its arithmetic monodromy group would normalize the geometric one, $\PSL(2,p)$ acting on $p$ points, in $S_p$ , and that normalizer is $\PSL(2,p)$ itself: $H$ is self-normalizing, and its class is not fixed by the outer automorphism. So the Galois closure would be a regular $\PSL(2,p)$ -covering over $\Q$ of rigid type $C$ or $\delta(C)$ , which the modular coverings theorem excludes. The exchange is the rigidity theorem’s part (e).

Rigidité et revêtements modulairesRigidity and the modular coverings

Plate 5.8Serre’s triple on

\Proj^1(\F_7)

x_0

y_0

and

z_0

, of orders 2, 3 and 7, drawn as the permutations they are, fixed points ringed, with

x_0y_0z_0=1

. There are exactly 168 such triples in

2A\times3A\times7A

, all conjugate.

Over $\Q(j)$ the group of Galois’s modular equation is $\PGL(2,p)$ , and it becomes $\PSL(2,p)$ once $\sqrt{p^*}$ is adjoined, $p^*=(-1)^{(p-1)/2}p$ . The modern form of this is the rigidity method of Belyi, Thompson, Matzat, and Malle and Matzat, in Serre’s account: a rigid triple of conjugacy classes determines a Galois covering of the line branched at three points, and the character values on the triple determine its field of definition. Throughout, $p\in\{5,7,11\}$ , $pA$ is the class of $z\mapsto z+1$ , $pB$ that of $z\mapsto z+r$ with $r$ a non-square, $\delta$ is conjugation by $\mathrm{diag}(r,1)$ , and $C=(2A,3A,pA)$ . Serre’s triple is $x_0=\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$ , $y_0=\left(\begin{smallmatrix}0&-1\\1&-1\end{smallmatrix}\right)$ , $z_0=\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$ , the images of $S$ , $(TS)^{-1}$ and $T$ , which generate $\PSL(2,\Z)$ .

On $(2A,3A,pA,pB)$ only the characters of degree $(p\pm1)/2$ are irrational: their values on $pA$ and $pB$ are $(1\mp\sqrt5)/2$ for $p=5$ , $(-1\mp\sqrt{-7})/2$ for $p=7$ and $(-1\pm\sqrt{-11})/2$ for $p=11$ , exchanged between the two classes. So the covering $X(p)\to X(1)$ , branched over $j=1728$ , 0, $\infty$ , with inertia generators conjugate to $x_0$ , $y_0$ , $z_0$ , is the only $G$ -covering of type $C$ ; its genus is 0, 3, 26. As a $G$ -covering it is defined over $\Q(\sqrt{p^*})$ , after Hecke, and over no smaller field: an automorphism of $\overline\Q$ that negates $\sqrt{p^*}$ carries it to the covering of type $(2A,3A,pB)=\delta(C)$ , its twist by $\delta$ . The fields of definition of the $p+1$ cyclic subgroups of order $p$ of an elliptic curve with invariant $j$ generate a Galois extension of $\Q(j)$ with group $\PGL(2,p)$ , acting on them as on $\Proj^1(\F_p)$ , with constant field $\Q(\sqrt{p^*})$ and group $\PSL(2,p)$ over $\Q(\sqrt{p^*})(j)$ . This is the group of Galois’s modular equation.

Theorem(Rigidity for

\PSL(2,p)

) proved

Let $p\in\{5,7,11\}$ and $G=\PSL(2,p)$ .

(a) On $2A$ , $3A$ , $pA$ , $pB$ the irreducible characters take the values of Seams’ table, irrational only for the degrees $(p\pm1)/2$ .

(b) The equation $xyz=1$ has exactly $|G|$ solutions in $2A\times3A\times pA$ . All of them generate $G$ , and they are conjugate: $C$ is strictly rigid. Serre’s triple is one of them.

(c) The field of rationality of $C$ , the fixed field in $\Q(\mu_N)$ of $\{k:C_i^k=C_i\}$ with $N$ the exponent of $G$ , is $\Q(\sqrt{p^*})$ . It is generated by the values of the irreducible characters on the classes of $C$ .

(d) $|\Aut(G)|=|\Sigma(2A,3A,pA)|+|\Sigma(2A,3A,pB)|=2|G|$ , and $\operatorname{Out}(G)$ is generated by the class of $\delta$ .

(e) Let $k_0$ be a non-square modulo $p$ that is 1 modulo the other primes dividing $N$ . The power map $g\mapsto g^{k_0}$ and $\delta$ permute the classes in the same way, fixing $2A$ and $3A$ and exchanging $pA$ and $pB$ . Moreover $\chi(g^{k_0})=\chi(\delta g)=\sigma(\chi(g))$ for every irreducible $\chi$ , where $\sigma$ negates $\sqrt{p^*}$ .

Proof

(b) is Serre’s. Every part was checked by machine in exact arithmetic: the character tables were found by Burnside’s algorithm and certified exactly, the triples were counted by enumeration, and the Gauss sum $\sum_a(\tfrac ap)x^a$ squares to $p^*$ modulo the cyclotomic polynomial $\Phi_p$ . For (d): an automorphism is determined by the image of $(x_0,y_0,z_0)$ , which lies in $\Sigma(2A,3A,pA)$ or $\Sigma(2A,3A,pB)$ , because $2A$ and $3A$ are the only classes of their orders. Conversely $\Sigma(2A,3A,pA)$ is the conjugation orbit of the triple, and $\Sigma(2A,3A,pB)=\delta(\Sigma(2A,3A,pA))$ .

L’action de Galois sur la quartique de KleinThe Galois action on Klein’s quartic

Plate 5.9Galois’s matching for

p=7

, the conic with

\bar\alpha

, has class

S_4^b

. Complex conjugation acts on the projective line as

z\mapsto-z

, a reflection in this drawing, and carries it to a matching of class

S_4^a

, the conic with

\alpha

Klein’s representation $\rho$ is defined over $\Q(\zeta)$ , $\zeta=e^{2\pi i/7}$ , and $\sigma_a\colon\zeta\mapsto\zeta^a$ acts on the points, lines and conics of its plane coordinatewise. It maps the matrices $\rho(G)$ onto themselves, with $\sigma_a\circ\rho=\rho\circ\alpha_a$ for $\alpha_a$ the conjugation by $z\mapsto az$ : inner exactly when $a$ is a square, so that complex conjugation $\sigma_6$ acts as the outer automorphism, $z\mapsto-z$ . On each incarnation in the column of the Klein quartic whose stabilizer class $\Aut(G)$ fixes, $\sigma_a$ acts as a seam over $\alpha_a$ . On the rigid ones it is the unique such seam, and carried through the unique seams to the projective line it is $z\mapsto az$ : on pairs of points for the bitangents, on points for the flex triangles, on involutions for the centres, and on perfect matchings for the conics.

The conics are where the outer automorphism shows. The conic $x^2+y^2+z^2+\bar\alpha(xy+yz+zx)=0$ , with $\bar\alpha=\zeta^3+\zeta^5+\zeta^6$ , has stabilizer of class $S_4^b$ , and it is the one the unique seam attaches to Galois’s matching $\infty0$ , 13, 26, 45; its conjugate, with $\alpha=\zeta+\zeta^2+\zeta^4$ , has class $S_4^a$ , and a non-square $a$ exchanges the two orbits of seven conics. No rule invariant under $\Aut(G)$ says which seven-point set is the set of points of the Fano plane. Over $\Q(\sqrt{-7})$ there is one, the choice of a square root of $-7$ , and complex conjugation exchanges the two answers: the bridge between points and lines, refuted for a fixed marking, is built over the outer automorphism by Galois conjugation. The realization by complex conjugation is classical, after Elkies; the same sign of $\sqrt{-7}$ is one of the two orientations of Chapter 13, and its realization in the residues at the primes is the type law of Chapter 6.

The Galois orbits have sizes 1,1,1,3,6,6,6 on the flexes and flex tangents, 1,3,6,6,6,6 on the bitangents and their poles, 3,6,6,6 on the centres and axes, 1,1,6 on the flex triangles, and 2,6,6 on the fourteen conics and the fourteen self-polar triangles, each orbit meeting both classes equally. On the flexes, $\rho(G)$ and the $\sigma_a$ generate $\PGL(2,7)\times C_3$ , of order 1008, the $C_3$ generated by the twisted Frobenius of Chapter 10; the quotient by it is $\PGL(2,7)$ with constant field $\Q(\sqrt{-7})$ . The automorphism groups of objects are realized by Galois groups too: $\operatorname{Gal}(\Q(\zeta_{21})/\Q(\zeta))$ exchanges the two points of contact of each bitangent, and $\operatorname{Gal}(\Q(\zeta_{28})/\Q(\zeta))$ inverts the eigenvectors for $i$ of the elements of order 4.

Corollary(Galois-stable incarnations) proved

Let $Y$ be an incarnation of an object among the points, lines or conics of the plane of the quartic. If $Y$ is mapped onto itself by some $\sigma\in\operatorname{Gal}(\overline\Q/\Q)$ with $\sigma(\sqrt{-7})=-\sqrt{-7}$ , then its stabilizer class is fixed by $\Aut(G)$ . Hence the six 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 that is defined over $\Q$ as a set. Their incarnations come in pairs exchanged by the conjugation of $\sqrt{-7}$ .

Proof

$\sigma$ restricts to some $\sigma_a$ with $a$ a non-square, so it is a seam over the outer automorphism $\alpha_a$ from $Y$ to $Y$ . A seam over $\alpha$ from $Y$ to $Y$ exists only if $\st(Y)=\alpha^{-1}(\st(Y))$ .

Ce que la théorie de Galois ne fournit pasWhat Galois theory does not supply

seam theorythe study of seams

Existencestabilizer classNumberN(H)/HConsistencyrigidity, monodromyMarkingsOut(G)Forgettingdescription, kernelImpossibilitynegative space

6Les continus

continuumspinor systemcommit algebra

5Les complétions

completionorientation

4Les doubles vies

lifedouble lifedictionarytype law

3Ce qui est su

bridgestatusrefuteddescriptionkernelGalois gap

Negative spaceabsenceforced gapwindowimprintreduction

2Les sutures

seamseam groupoidrigid objectcoherenceseam systemseam monodromygaugepowerquotient classtwisting elementseam over an automorphismGalois category

1L’incarnation

objectstabilizer classmarkingincarnationalignmentnew object

0Le fonds classique

G-sets and G-maps · stabilizers and normalizers · orbitals · permutation isomorphism · Gassmann equivalence · the table of marks

Plate 5.10The tower of concepts with what Galois theory supplies lit: the grammar of floors one and two, descriptions and their kernels, and Galois’s own absence with its window. Natural seams and their coherence, the statuses of bridges, and the floors from the double lives up are left unlit.

Galois theory supplies the grammar of floors one and two. Objects are connected objects of a Galois category, seams are its isomorphisms, markings are identifications of fibre functors, seams over automorphisms are morphisms twisted by the outer automorphisms, and monodromy is a class in non-abelian $H^1$ . It supplies one absence of floor three, Galois’s own, and an arithmetic layer in which the outer automorphism of the group of order 168 is the Galois conjugation of $\sqrt{-7}$ .

It does not supply the rest. It does not say which isomorphisms a theory singles out, or whether those natural seams cohere: a Galois category contains every isomorphism and prefers none, so the monodromy theorem of Chapter 4 is not a theorem of Galois theory. It does not supply the double lives, buildings and continua of floors four to six. An exceptional isomorphism is a coincidence among finite groups, not a Galois correspondence, and a building comes from a completion, not from a field extension, although both meet Galois theory at decomposition groups and residue fields. Nor does it identify the group of order 168 with the Galois group of a field whose places are the objects of the seam table. The group arises here as a reduction, $\PSL(2,\mathcal O)/\Gamma(\mathfrak p)$ , of a discrete subgroup of $\PSL(2,\C)$ . The Galois groups that act in this chapter are small: $\operatorname{Gal}(\Q(\sqrt{-7})/\Q)$ acts through $\operatorname{Out}(G)$ , the cyclotomic groups above it act through the automorphism groups $N_G(H)/H$ of the objects, and the Frobenius elements that occur, at 2 in Chapter 10 and here, are seams over automorphisms, symmetries of incarnations.

Remark(What the language of seams adds)

Everything in the Galois-categorical reading is classical: Galois categories, the Galois theory of fields in this form, categorical Galois theory, non-abelian $H^1$ and twisted forms, and gerbes. What the language of seams adds is an emphasis, not a theorem. Several theories meet one connected object through different pointed Galois categories $(\mathcal C_\Gamma,F_\Gamma)$ , joined by markings, and the subject is the isomorphisms that those theories single out and whether they cohere. A Galois category contains every isomorphism between two incarnations and prefers none. Which ones a construction picks out, and whether the picks commute around a cycle, are data that the Galois category does not contain. Likewise the statuses of a bridge record knowledge, not structure.

Galois theory gives seam theory its grammar and one arithmetic fact: at 168 the outer automorphism is the conjugation of $\sqrt{-7}$ , which exchanges the two orbits of conics in Klein’s plane and with them the points and the lines of the Fano plane. What it leaves to seam theory is the choice: which isomorphisms a theory singles out, and whether they cohere around a cycle.

The conjugation returns three times. In the residues at the primes it is the type law of Chapter 6; at the double cover the gerbes stop being neutral, in Chapter 11; and in Chapter 17 it measures exactly what counting cannot hear and what finite sets cannot say.