A moduli interpretation for the non-split Cartan modular curve

Modular curves like X_0(N) and X_1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL_2(Z), they allow for a more arithmetic description as a solution to a moduli problem. This description turns out to be very useful in many applications. We wish to give such a moduli description for two modular curves: those associated to non-split Cartan subgroups and their normaliser in GL_2(F_p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures of p-torsion points on elliptic curves over number fields. Some classical results about the geometry of those curves can be proven using this moduli description. For instance, we can count the number of elliptic points, describe the cusps and the degeneracy maps. We also give a moduli-theoretic interpretation and a new proof of a result of Chen.


Introduction
Let p be an odd prime. Let Y (p) be the affine modular curve classifying elliptic curves with full level p structure. The completed modular curve X(p) classifies generalised elliptic curves with full level p structure. Those two curves admit integral models over the ring of integers of the cyclotomic field Q(ζ p ). See [DR73] and [KM85]. The modular curve X(p) comes equipped with a natural action by GL 2 (F p ). For any subgroup H of GL 2 (F p ) the quotient X(p)/H defines an algebraic curve X H over Q(ζ p ) det(H) . Hence the points on X H over an algebraically closed fieldk of characteristic different from p are H-orbits ofk-points of X(p). However in some interesting cases, there is a nice description of a moduli problem for X H too.
As an example, we explain the case when H is the Borel subgroup B = ( * * 0 * ) in GL 2 (F p ). First, the points in Y (p)(k) arek-isomorphism classes of pairs E, (P, Q) where E/k is an elliptic curve and (P, Q) form a basis of E [p]. For a fixed E, all the pairs (P , Q ) in the B-orbit of (P, Q) are such that P is in the subgroup C generated by P . Hence thek-points on the quotient curve Y B can be identified withk-isomorphism classes of pairs (E, C) with E again an elliptic curve defined overk and C a cyclic subgroup of order p in E [p]. The latter description is now independent of the initial choice of the Borel subgroup B in GL 2 (F p ) and only uses the geometry of E. The curve X B is usually denoted by X 0 (p).
Another example is the quotient by the split Cartan subgroup which consists of diagonal matrices in GL 2 (F p ). The corresponding curve is denoted here by X sp (p) and it parametrisesk-isomorphism classes E, (A, B) of generalised elliptic curves E endowed with two distinct cyclic subgroups A and B of order p in E. For its normaliser S, we find the curve X S = X + sp (p) which classifies generalised elliptic curves with an unordered pair {A, B} of cyclic subgroups A and B of order p. All these cases are easy to describe because the subgroups H can be defined as the stabiliser of some object under a natural action of GL E[p] .
In view of Serre's problem to classify the possible Galois module structure of the p-torsion of an elliptic curve over a number field, there are two further modular curves of importance. The aim of this paper is to give a good moduli description for those, namely when H is a non-split Cartan subgroup or a normaliser of a non-split Cartan subgroup in GL 2 (F p ). We will denote the corresponding modular curves by X nsp (p) and X + nsp (p) respectively. See the start of Section 2 for detailed definitions. These curves have been studied for instance by Ligozat [Lig77], Halberstadt [Hal98], Chen [Che98,Che00], Merel and Darmon [Mer99,DM97] and Baran [Bar10].
In our description, the modular curve X + nsp (p) will classify elliptic curves endowed with a level structure that we call a necklace. Roughly speaking, a necklace is a regular (p + 1)-gon whose corners, called pearls, are all cyclic subgroups of order p in E and such that there is an element in PGL E[p] that turns this necklace by one pearl. This will not depend on the choice of a non-split Cartan subgroup.
In Section 2, we will define these necklaces in detail and we will give an alternative and more geometric description using the cross-ratio in P E [p] . The following Section 3 shows how classical results about the geometry of X nsp (p) can be proven using this moduli interpretation. For instance, we can count the number of elliptic points, describe the cusps and the degeneracy maps. In Section 4, we reprove a result by Chen. He shows [Che98,Che00] that there is an isogeny between the Jacobian of X + sp (p) and the product of the Jacobians of X 0 (p) and X + nsp (p). The proof of this theorem is not entirely new, however we believe that it gives a better geometric vision of the maps involved and the representation-theoretic proof. We conclude the paper with some numerical data related to Chen's Theorem.
Since the prime p is fixed throughout the paper, we will now omit it from the notations and only write X nsp and X + nsp . It is to note that there should be no real difficulty in generalising our moduli description to composite levels N . With view on the problem of Serre to classify the Galois structure of p-torsion subgroups of elliptic curves over Q, prime levels are maybe the most interesting.

Notations
The following is a list of modular curves that appear in this paper and the notations we frequently use. The definitions will be given later. See also Sections 3.1 and 4.1 for degeneracy maps and correspondences between them.
Level structure Normaliser of a non-split Cartan N Necklace v Matrices in GL(F p ) will be written as · · · · . Instead the coset of matrices in PGL(F p ) will be represented by matrices of the form · · · · .
2 The moduli problem of necklaces 2.1 Non-split Cartan subgroups and their modular curves We refer to [Ser97] for definitions and results about non-split Cartan subgroups and Dixon's classification of maximal subgroups of GL 2 (F p ) and just briefly recall some facts. The group GL 2 (F p ) acts on the right on P 1 (F p 2 ) by (x : y) a b c d = (ax + cy : bx + dy). Any non-split Cartan subgroup of GL 2 (F p ) can be defined as the stabiliser H α of (1 : α) in P 1 (F p 2 ) \ P 1 (F p ) for a choice of α ∈ F p 2 \ F p . We see that H α has order p 2 − 1 as the action of GL 2 (F p ) is transitive on P 1 (F p 2 ) \ P 1 (F p ).
Alternatively, we can consider the basis (1, α) of F p 2 as a F p -vector space. Then we claim that H α is equal to the image of the map i α : F × p 2 → GL 2 (F p ) sending β to the matrix which represents the multiplication by β on F p 2 written in basis (1, α). 1 Indeed, let β = x + yα ∈ F × p 2 with x, y ∈ F p . If X 2 − tX + n is the minimal polynomial of α over F p , then x −ny y x + ty and so (1 : α)i α (β) = (x + yα : −ny + (x + ty)α) = (β : βα) = (1 : α). So the image of i α is contained in H α and they are equal because they are of the same size.
Given a choice of a non-split Cartan subgroup H, we define the modular curve X nsp as the quotient X H . Note that the quotient does not depend on the choice of H as these subgroups are all conjugate. However the description of points on X nsp as H-orbits do.
The normaliser N of a non-split Cartan subgroup H in GL 2 (F p ) contains H with index 2. It can be viewed as adding the image under i α of the conjugation map in Gal(F p 2 /F p ) on F × p 2 . The corresponding quotient X N will be denoted by X + nsp .

Necklaces
Let γ be a multiplicative generator of F × p 2 . For any 2-dimensional F p -vector space V , we define C γ to be the conjugacy class in PGL(V ) of all elements h which have a representative in GL(V ) whose characteristic polynomial is equal to the minimal polynomial of γ. In other words, all representatives of h ∈ C γ have an eigenvalue in F × p · γ. Ifγ is the conjugate of γ over F p , then Cγ = C γ . If a basis of V is chosen then C γ consist of all classes of matrices i α (γ) as α runs through F p 2 \ F p . Note that the class of i α (γ) = iᾱ(γ) = N(γ) −1 i α (γ) −1 is equal to the inverse of class of i α (γ). In particular, in any non-split Cartan subgroup in PGL(V ), there are exactly two generators h and h −1 that belong to C γ . As γ varies, we obtain the 1 2 ϕ(p + 1) conjugacy classes of elements of order p + 1. Letk be an algebraically closed field of characteristic 0 or different from p and let E/k be an elliptic curve. We know that there exists p + 1 cyclic subgroups of order p in E[p]. We will consider lists (C 0 , C 1 , . . . , C p ) of those cyclic subgroups and will say that two such lists are equivalent if we can obtain one from the other by a cyclic permutation; so (C 0 , C 1 , . . . , C p ) and (C 1 , C 2 , . . . , C p , C 0 ) are equivalent.
Definition. An equivalence class If h is such an element, then we must also have h(C p ) = C 0 as h is of order p + 1. Note also that if (C 0 , C 1 , . . . , C p ) is an oriented necklace with a certain h ∈ C γ , then so is ( Let us consider the dependence on γ; so to be more precise, we will call it now an oriented γ-necklace.
Lemma 1. Let γ and γ be two generators of F × p 2 . There is a canonical bijection between oriented γnecklaces and oriented γ -necklaces.
Proof. Since F × p 2 is cyclic, there exists an integer k ∈ [0, p 2 − 1] such that γ = γ k and such that k is coprime to p + 1. In particular C γ is the set of all h k with h ∈ C γ . So the requested bijection is given by with the index taken modulo p + 1.
As a consequence, we may now fix a choice of γ for the rest of the paper. In a picture, we arrange the subgroups C 0 , . . . , C p like pearls on a necklace that can be turned around the neck using the automorphism h of P(E[p]). If we allow the necklace to be worn in both directions, we get the notion of a necklace without orientation: Definition. Let w denote the involution defined by w(C 0 , C 1 , . . . , C p ) = (C p , C p−1 , . . . , C 0 ) which changes the orientation of an oriented necklace. A necklace is a w-orbit of oriented necklaces {v, w(v)}.
Lemma 2. Fix a generator γ of F × p 2 . Let C 0 , C 1 , and C 2 be three distinct cyclic subgroups of order p in E[p]. Then there exists a unique element h ∈ C γ in PGL(E[p]) such that h(C 0 ) = C 1 and h(C 1 ) = C 2 .
Proof. Choose generators P 0 and P 1 in C 0 and C 1 respectively and consider them as a basis of E[p]. Write t and n for the trace and the norm of γ. Our class of matrices must contain a matrix of the form ( 0 y x t ) for some x and y in F × p if we want h(C 0 ) = C 1 and tr(h) = t. As y varies the points yP 0 + tP 1 form an affine line and hence there is a unique y ∈ F × p such that yP 0 + tP 1 belongs to C 2 . Finally, we have no choice but to set x = ny −1 if we also want det(h) = n.
This lemma implies that for any triple (C 0 , C 1 , C 2 ) of distinct cyclic subgroups of E[p], there is a unique oriented necklace of the form (C 0 , C 1 , C 2 , . . . ). We will denote it by C 0 → C 1 → C 2 . Similarly there is a unique necklace with consecutive pearls C 0 , C 1 , C 2 , which we denote by There is a natural action of PGL(E[p]) on the set of oriented necklaces by setting g · (C 0 , . . . C p ) = g(C 0 ), . . . , g(C p ) for g in PGL(E[p]). If h ∈ C γ is such that h(C i ) = C i+1 then ghg −1 ∈ C γ can be used to show that g(C 0 ), . . . , g(C p ) is indeed an oriented necklace. Since the action of PGL(E[p]) on P(E[p]) is simply 3-transitive, Lemma 2 implies that the action of PGL(E[p]) is transitive on oriented necklaces. By definition, for every oriented necklace v, there exists h ∈ C γ fixing it. Therefore, the group generated by h, which is a non-split Cartan subgroup in PGL(E[p]) will belong to the stabiliser of v. It is clear that this is equal to the stabiliser of v. We have shown: ). The set of oriented γ-necklaces is isomorphic as a G-set to G/H where H is any non-split Cartan group in G. Similarly, the set of γ-necklaces is G-isomorphic to G/N for the normaliser of a non-split Cartan group N in G. In particular, there are exactly p(p − 1) oriented necklaces and p(p − 1)/2 necklaces.

Moduli description
Let H be a non-split Cartan subgroup in G = GL 2 (F p ) and write N for its normaliser. Letk be an algebraically closed field of characteristic different from p. The moduli space Y (p)(k) consists ofkisomorphism classes of pairs E, (P, Q) where E is an elliptic curve overk and (P, Q) is an F p -basis of E[p]. The group GL 2 (F p ) acts on a pair on the right as usual E, (P, Q) · a b c d = E, (aP +cQ, bP +dQ) . A point in Y H (k) is an orbit under this action by the non-split Cartan subgroup H. For a given elliptic curve E/k, the H-orbits of triples is a G-set isomorphic to G/H. Corollary 3 has shown us that the set of oriented necklaces on E is also isomorphic to G/H. Hence we have: Proposition 4. Let H be a non-split Cartan subgroup in GL 2 (F p ). There is a bijection between the points in Y H (k) and the set ofk-isomorphism classes of pairs (E, v) composed of an elliptic curve E/k together with an oriented necklace v in E. Similarly Y N (k) consists of pairs (E, v) where v is a necklace in E.
We will from now on informally say that Y + nsp and Y nsp are coarse moduli spaces for the moduli problem of elliptic curves endowed with a necklace and an oriented necklaces respectively. In order to make this absolutely precise, we would have to define necklaces for elliptic curves over arbitrary schemes and that is very cumbersome to do. There is a similar problem for the split Cartan subgroup, too. However, later in Section 3.3, we give the description of k-rational points for fields k which are not algebraically closed.
With the correct definition one could now show that the scheme X nsp over Z 1 p admits a moduli problem in the form of necklaces. As usual one would define X nsp over Z as the normalisation over the j-line. We do not address here what the fibre at p of X + nsp and X nsp could look like, but we hope that this new moduli interpretation might be helpful.

The cross-ratio
The choice of basis does not matter as the cross-ratio is PGL ] for all 0 i p with the index taken modulo p + 1. Hence we can attach a cross-ratio to each necklace. As described above, the action of PGL E[p] on oriented necklaces is transitive and hence this cross-ratio [C 0 , C 1 ; C 2 , C 3 ] is the same for all oriented γ-necklaces.
Proposition 5. Let γ be a generator of F × p 2 of trace t and norm n. Set ξ γ = t 2 /(n − t 2 ). Then a list (C 0 , C 1 , . . . , C p ) of all distinct cyclic subgroups of order p in E represents a γ-necklace if and only if This provides a new possibility of defining necklaces by-passing completely the use of the automorphism group of E[p], but only relying on the projective geometry of P(E[p]).
Proof. We only need to compute the cross-ratio for one necklace. We take the basis such that h = [ 0 −n 1 t ] is in C γ . The necklace now contains the consecutive pearls (1 : 0), (0 : 1), (−n : t) and (−nt : −n + t 2 ) from which we obtain the above cross-ratio ξ γ .
Since to each triple (C 0 , C 1 , C 2 ) there is a unique C 3 such that [C 0 , C 1 ; C 2 , C 3 ] = ξ γ , we have a second proof of Lemma 2.

Relation to other descriptions
We recall a different description of the H α -orbits of points in X(p) where α is a choice in F p 2 \ F p . See [Mer99]. Let E/k be an elliptic curve. Choose a basis P 0 , P 1 of E[p] and identify E[p] with F p 2 via P 0 → 1 and P 1 → α. Any basis (P, Q) of E[p] is equal to (P 0 , P 1 )g for some g ∈ GL 2 (F p ). Consider the GL 2 (F p )-equivariant map which sends (P 0 , P 1 ) to (1 : α) ∈ P 1 (F p 2 ) \ P 1 (F p ). Since the action of H α is now just the multiplication on F p 2 , it induces a well defined GL 2 (F p )-equivariant map from the set of H α -orbits of basis (P, Q) to P 1 (F p 2 ) \ P 1 (F p ). This is a GL 2 (F p )-equivariant bijection.
This leads now to a moduli problem description of X nsp . Each point in Y nsp (k) withk an algebraically closed field of characteristic different from p is ak-isomorphism class of (E, C) where E/k is an elliptic curve and C is an element in P E[p]⊗F p 2 \P E[p] . The group PGL E[p] acts on the left on P E[p]⊗F p 2 by its action on E[p].
We will now give an explicit PGL E[p] -equivariant bijection between the set of oriented γ-necklaces of E and P E[p] ⊗ F p 2 \ P E[p] . Write n and t for the norm and trace of the fixed element γ in F p 2 . Consider the map Note that such a basis exists because neither n nor t could be zero when γ is a multiplicative generator of F p 2 . We have to show that this map is well-defined. Let h be a generator in the stabiliser of v which belongs to C γ . In the basis (P, Q) this element h is represented by the matrix 0 −n 1 t . Now This shows that the line in P E[p] ⊗ F p 2 does not depend on the choices made in the construction. It also is evident from this that the stabiliser of v is equal to the stabiliser of the image. From the construction we see that the map is PGL E[p] -equivariant. Since the actions are transitive, it follows that it is surjective and hence bijective.
Since we have no geometric object linked to E which can be thought of directly as an element in E[p] ⊗ F p 2 , we believe that the moduli problem of necklaces has its advantages.
3 Describing the geometry and arithmetic with necklaces

Degeneracy maps
Let A be the group of scalars in GL 2 (F p ) and consider the associated modular curve X A . Because the group PGL 2 (F p ) acts sharply 3-transitive on P 1 (F p ), the curve X A represents the moduli problem associating to each elliptic curve E a triple of distinct cyclic subgroups (C 0 , C 1 , C 2 ) of order p in E, which is also called a projective frame in P(E[p]).
The map π A : X(p) → X A can be chosen to be the following. Let n and t be the norm and trace of our fixed generator γ in F p 2 . To each basis (P, Q) of the p-torsion of an elliptic curve E, we associate the triple P , Q , −n P + t Q . From the fact that t = 0, it is clear that this gives a map X(p) → X A . Next we describe the map π nsp : X A → X nsp . We have a natural choice to send the triple (C 0 , C 1 , C 2 ) to the unique oriented necklace C 0 → C 1 → C 2 given by Lemma 2. Similarly, we will send it to the necklace C 0 − C 1 − C 2 to define the map π + nsp : X A → X + nsp . The advantage of our choices is that π nsp • π A provides an explicit bijection between the set of orbits of isomorphism classes (E, (P, Q)) under a particular non-split Cartan subgroup H 0 and the set of isomorphism classes (E, v) of elliptic curves endowed with an oriented necklace. Let H 0 be the nonsplit Cartan subgroup in GL 2 (F p ) generated by the matrix This insures that the map which sends an orbit (E, (P, is well defined and it gives the expected bijection. Under the map (1) in Section 2.5, identifying necklaces with elements in , the degeneracy map above can also be described as sending the basis (P, This provides an explicit bijection between the set of orbits of isomorphism classes (E, (P, Q)) under H 0 and the set of isomorphism classes (E, C) of elliptic curves endowed with an element We will see later in Section 4.7 another naturally defined degeneracy mapπ + nsp : X A → X + nsp .

Cusps
Proposition 6. The modular curve X nsp has p − 1 cusps, each ramified of degree p over the cusp ∞ in X(1).
Proof. In order to determine the structure of the cusps, we use the Tate curve E q over Q((q)). Formally, one can deduce the proposition using Theorem 10.9.1 in [KM85] from the fact that a non-split Cartan subgroup of PGL 2 (F p ) acts transitively on P 1 (F p ) and that it contains no non-trivial element from any Borel subgroup. In particular, the formal completion of X nsp along the cusps is the formal spectrum of , where α p = q and ζ is a p-th root of unity. However, we can also view it on the necklaces of E q . The Tate curve has a distinguished cyclic subgroup µ p of order p. Any oriented necklace v can be turned in such a way that C 0 = µ p . The two following pearls C 1 and C 2 have each a generator which is a p-th root of q, say αζ i and αζ j , respectively, where 0 i = j < p. From the action of the inertia group of the extension Q((q))[α, ζ] over Q((q)), we see that all the p necklaces with a given i − j ∈ F × p meet at the same cusp in the special fibre at (q).
The cusps are not defined over Q but over the cyclotomic field Q(µ p ) only, forming one orbit under the action of the Galois group, despite the fact that X nsp is defined over Q. See Appendix A.5 in [Ser97]. As a consequence there are p − 1 choices of embeddings X nsp → Jac(X nsp ), all defined over Q(µ p ) only and none of them is a canonical choice.
With the same proof we show that X + nsp has (p − 1)/2 cusps defined over the maximal real subfield of Q(µ p ).

Galois action
Let k be a field of characteristic different from p and write G k for its absolute Galois group. For any σ in G k and point x ∈ Y nsp (k), represented by the pair (E, v), we define σ(x) in the obvious way as thek-isomorphism class of the pair shows that j(E) ∈ k and hence we may assume that E is defined over k.
For every σ ∈ G k there is an automorphism

A lemma on antipodal pearls and cross-ratios
Let E an elliptic curve over an algebraically closed fieldk of characteristic different from p. The following definition and lemma will be used in many places later on.
Definition. Let v = (C 0 , C 1 , . . . , C p ) be a necklace in E. Two pearls C i and C j are called antipodal in v if i ≡ j + p+1 2 (mod p + 1). In other words if they are diametrically opposed when we represent the necklace as a regular (p + 1)-gon. 1 0 ] is an element of order two without a fixed point in P 1 (F p ). Hence it belongs to a unique non-split Cartan subgroup H. Then the necklace v whose stabiliser is the normaliser of H is a necklace such that A B ∈ v and C D ∈ v as g(A) = B and g(C) = D. Conversely, if we have such a necklace v for A, B, C, D, then the unique element of order 2 which preserves the orientation on v, must send A to B and C to D. Hence it is of the form g = [ 0 d 1 0 ]. However if it has no fixed points in P 1 (F p ), then d has to be a non-square in F × p . Finally, we have to count how many necklaces have A B ∈ v. By the above proof, this is the same as to count how many matrices g = [ 0 d 1 0 ] belong to a non-split Cartan subgroup. That is p−1 2 as there are that many non-squares d in F × p .

Elliptic points
We proceed to count elliptic points using our moduli description. Our results in Propositions 9 and 12 below agree with the more general calculations by Baran in Proposition 7.10 in [Bar10]. Assume for this that p > 3. Consider the canonical coverings X nsp −→ X(1) and X + nsp −→ X(1). An elliptic point on X nsp or X + nsp is a point in the fibre of a point in X(1) represented by an elliptic curve E with Aut(E) = {±1}. Hence, an elliptic point on X nsp can be represented by a pair (E, v) such that there is an automorphism on E that induces a non-trivial element g ∈ PGL E[p] which fixes v. Consider the involution w on X nsp which reverses the orientation of the oriented necklaces. An elliptic point on X + nsp can be viewed as a pair (E, {v, wv}) with an automorphism g ∈ PGL E[p] and an oriented necklace v such that either g(v) = v or g(v) = w(v). In the latter case, we say that v and its necklace {v, w(v)} is flipped by g.
First note that if (E, ·) is a elliptic point then j(E) = 1728 and g is of order two or j(E) = 0 and g is of order three. These are elliptic curves with complex multiplication and E[p] becomes a free End(E)/pmodule of rank 1. So if g is of order 2 and p ≡ 3 (mod 4) or if g is of order 3 and p ≡ 2 (mod 3), then End(E)/p ∼ = F p 2 and hence g belongs to a unique non-split Cartan subgroup of PGL E[p] . Instead, if g is of order 2 and p ≡ 1 (mod 4) or if g is of order 3 and p ≡ 1 (mod 3) then End(E)/p ∼ = F p ⊕ F p and therefore g belongs to a unique split Cartan subgroup as it will have exactly two fixed points.

Fixed oriented necklaces
Let (E, v) be an elliptic point on X nsp with the oriented necklace v fixed by g. Then g is in the non-split Cartan subgroup stabilising v. Hence by the above, p ≡ 3 (mod 4) if g has order 2 and p ≡ 2 (mod 3) if g has order 3. Conversely, if these congruence conditions are satisfied then g is in a unique non-split Cartan subgroup which is the stabiliser of exactly two oriented necklaces, namely v and wv. This gives the following result: Proposition 9. For r = 2 and 3, let e r be the number of elliptic points in X nsp with g of order r. Then In the cases where e r = 2 the two corresponding oriented necklaces are in the same w-orbit.

Flipped necklaces
Let (E, v) be an elliptic point on X + nsp where the necklace v = { v, w v} is flipped by g. If g were of order 3, we would have v = g 3 ( v) = w( v). Hence g is of order 2. The involution g is in a split Cartan subgroup if p ≡ 1 (mod 4) and in a non-split Cartan subgroup if p ≡ 3 (mod 4).
Lemma 10. Suppose that p ≡ 1 (mod 4) and let A, A denote the two fixed points of g in P(E[p]). A necklace v is flipped by g if and only if A and A are antipodal in v. Consequently, there are p−1 2 necklaces flipped by g.

Proof.
Let v = (C 0 , C 1 , C 2 , . . . , C p ) be a flipped oriented necklace where the indices are taken modulo p + 1. It follows that if A = C k then A = g(A ) = C p+1−k , so k = (p + 1)/2 and A and A are antipodals in v = { v, w v}. Moreover from g(C k ) = C p+1−k , we see that g will act on v, represented as a regular (p + 1)-gon, as the reflection through the axis passing through A and A .
Conversely, let v be a necklace in which A A . Let B B be two other antipodal pearls in v. Let h be the element of order 2 in the normaliser of the non-split Cartan subgroup stabilising v. As it exchanges antipodal pairs in v, we have h(A) = A and h(B) = B .
Since hgh −1 is also an involution that fixes A and A , it follows that hgh −1 = g as there is a unique involution fixing two given points. Therefore hg(B) = gh(B) = g(B ) which implies that g(B) g(B ) ∈ v.
As g sends antipodal pairs in v to antipodal pairs in g(v), we also have A A and g(B) g(B ) in g(v). Hence, by Lemma 8, either The first case is excluded because g does not belong to a non-split Cartan subgroup if p ≡ 1 (mod 4).
The end of the proof follows from the fact that there are (p − 1)/2 necklaces such that A and A are antipodal, again by Lemma 8.
Lemma 11. Suppose that p ≡ 3 (mod 4). Let A ∈ P E[p] . Consider the map sending a necklace v to the pearl antipodal to A in v. This is a bijection between necklaces v flipped by g and the set of pearls B ∈ {A, g(A)} such that A, B; g(A), g(B) is a non-square in F p .
Proof. Let v be a necklace flipped by g. As above, from g( v) = w( v) we get g(C k ) = C p+1−k for all k and one can see that, since p ≡ 3 (mod 4), g will act on v as a reflection through an axis that does not pass through a corner of the regular (p + 1)-gon. Let B be antipodal to A in v. So B = A. If B were equal to g(A), then A and B would be on the line orthogonal to the axis of reflection of g. But this would imply that p + 1 ≡ 2 (mod 4), and hence B = g(A). Finally, since g flips v, we see that g(A) g(B) ∈ v. By Lemma 8, it follows that A, B; g(A), g(B) is a non-square modulo p. The same lemma also shows that our map v → B is injective.
Conversely, suppose that B ∈ {A, g(A)} is such that the crossratio A, B; g(A), g(B) is a non-square modulo p. Since A, B, g(A) and g(B) are all distinct, Lemma 8 applies to show that there is a necklace v with A B and g(A) g(B). Now g(v) has also g(A) g(B) and A B. The same lemma now shows that g(v) = v. If the orientation of v were fixed rather than flipped, then g(A) would be B. Hence our map is surjective, too.
Proposition 12. For r = 2 or 3, let e + r be the number of elliptic points with g of order r in X + nsp . Then Proof. The number of elliptic points for X + nsp is the sum of the number of fixed and the number of flipped necklaces. In Proposition 9 we counted the fixed ones. We have already counted the flipped necklaces for p ≡ 1 (mod 4) in Lemma 10. Now suppose p ≡ 3 (mod 4) and let A ∈ P E[p] . By Lemma 11, we must count how many pearls B there are such that B ∈ {A, g(A)} and A, B; g(A), g(B) is a non-square in  : 1). Hence A, B; g(A), g(B) = 1 + b 2 . So we have to count the number of b ∈ F × p such that 1 + b 2 is a non-square. One finds that there are p+1 2 such b by counting the cases when 1 + b 2 is a square using that there are p + 1 points on a projective conic a 2 + b 2 = c 2 .

Genus
From the above, we can now proceed to compute the genus of our modular curves. Of course, we find the well-known formulae, as for instance in Appendix A.5 to [Ser97], [Bar10] or [Che98]. The reader can also find tables for the genus of X nsp and X + nsp for small primes p in [Bar10]. The Riemann-Hurwitz formula applied to the modular curve X H associated to a subgroup of finite index H of GL 2 (F p ) and with the canonical morphism X H → X(1) of degree d, gives the following formula for the genus g(X H ) of X H : where e r is the number of elliptic points in X H of order r and e ∞ is the number of cusps.
With the results of Sections 3.5 and 3.2, a straightforward computation gives the following.
Proposition 13. The genera of X nsp and X + nsp are g(X nsp ) = 1 12 With the same method, one can compute the genus of other modular curves, for instance X 0 and X + sp . The classical results (see for instance [Shi94] and [Che98]) for their genus are Then one can verify easily the relation noticed by Birch following Chen's calculation of genus and confirmed by Chen's isogeny g X + nsp + g X 0 = g X + sp . (2)

Hecke operators
Let be any prime distinct from p. Denote by X + 0,nsp ( , p) = X 0 ( ) × X(1) X + nsp (p). We recall how the Hecke correspondence T is defined through the following two natural degeneracy maps ρ and ρ : X + 0,nsp ( , p) −→ X + nsp (p). The modular curve X + 0,nsp ( , p) parametrises isomorphism classes E, (f, v) of elliptic curves E endowed with an -isogeny f : E → E and a necklace v. Let ρ be the map obtained by forgetting the -structure and ρ the map which sends (E, (f, v)) to E , f (v) . The image f (v), defined as f (C 0 ), f (C 1 ), . . . , f (C p ) when v = (C 0 , . . . , C p ), is indeed a necklace on E = f (E) since = p.
The correspondence T on X + nsp is now defined as ρ * • ρ * . It induces an endomorphism on Pic(X + nsp ) by Picard functoriality. On the divisor (z) with the point z in X + nsp represented by (E, v), it is defined as where the sum runs over all isogenies f from E of degree . One can verify in a classical manner that these correspondences and this moduli-theoretic description coincide with the Hecke operators defined by double coset (see for instance [DI95]). In a similar way, one can check that the definition using double coset gives T p = 0. This is conform with Chen's theorem (see our Theorem 15) since the cuspforms on X + nsp correspond to cuspforms on the new part of X + 0 (p 2 ).

A pairing
Given two necklaces v and w in E, we set It is the number of antipodal pearls that v and w have in common. We can extend it linearly to all v Zv regarded as an abelian group with an action by PGL E[p] .
First, we note that we are left to prove that the pairing is positive, takes value 0 or 1 on distinct necklaces, and is non-degenerate. In this section, we only give the proof of the two first facts. The proof of non-degeneracy will be given in Section 4.5 and numerical examples are in Section 5.
Proof. The statement that v, w ∈ {0, 1} for v = w is a direct consequence of Lemma 8: If v, w > 2, then there are four distinct A, B, C, D with both A B and C D in v and w, contradicting the lemma.
Let u = a v v be an element in Z v. We have where {A,B} is the sum running over all unordered pairs of distinct cyclic subgroups of E[p]. Hence the pairing is positive. The non-degeneracy of the pairing will be shown in Section 4.5.

Chen's isogeny 4.1 Definitions and statement
In [Che98], Chen proved that Jac(X + nsp ) = Jac(X + 0 (p 2 )) new . Edixhoven and de Smit [dSE00, Edi96] found a different and rather elegant proof. Finally Chen gave in [Che00] an explicit description of his morphism Jac(X + sp ) → Jac(X + nsp ) × Jac(X 0 ). With our new moduli description this morphism can be described yet in another manner. Letk be an algebraically closed field of characteristic different from p. In Section 3.6, we have given the definitions of the modular curves X 0 and X + sp . The points in X sp (k) can be represented ask-isomorphism classes of the form E, {A, B} where {A, B} is a unordered pair of distinct cyclic subgroups of order p in E. Let y = E, {A, B} be a point in X + sp . We define ϕ(y) on the divisor (y) to be the sum of (E, v) where v runs over all necklaces in which the pearls A and B are antipodal. This extends linearly to a map on the Jacobians.
Further we define the maps ψ, µ, and λ in the diagram on the right as follows. If y is the point E, {A, B} as above, then µ(y) is the sum of the points (E, A) and (E, B) in X 0 . If x = (E, A) with A a cyclic subgroup of order p in E is a point on X 0 , then we set ψ(x) equal to the sum of E, {A, B} where B runs through all cyclic subgroups of order p in E distinct from A. Finally if z = (E, v) is a point on X nsp for some necklace v, then λ(z) is the sum of all E, {A, B} where A and B run through all pairs of antipodal pearls in the necklace v. All these three correspondences extend linearly to the corresponding Jacobians. As explained in Section 4.7, those correspondences comes from degeneracy maps (where we replace π + nsp byπ + nsp which will be defined in Section 4.7). We proceed to reprove Chen's result. Even if we believe that our proof is simpler and conceptually better visualised than the original proof in [Che00], we have to emphasise that it is mostly a reformulation or translation of Chen's proof into our new language. The crucial argument at the end is the same.
where the last sum runs over all necklaces v in which A and B are antipodal. Now A will appear in each necklace once and for each necklace there is a unique B which is antipodal to A in v. Hence ϕ • ψ(x) is equal to the sum over all possible necklaces of E. But is the pullback of a divisor (E) on X(1) by the natural projection j + nsp : X + nsp → X(1). Since X(1) ∼ = P 1 has trivial Jacobian, we find ϕ • ψ = 0 on the Jacobians. This proof was already noted by Merel on page 189 in [Mer99].
Next, for a point z = (E, v) in X + nsp , we have where B in the sums denotes the unique pearl which is antipodal to A in v and where j 0 : X 0 → X(1). Hence µ • λ = 0 on the Jacobians. Finally, we obtain and hence µ • ψ = [p − 1] on the Jacobian of X 0 .

Making use of antipodal pearls
We deduce from the earlier Lemma 8 the following result: Corollary 18. For every E, {A, B} ∈ X + sp , we have  Proposition 20. Let j sp : X + sp → X(1) be the natural projection. The relation holds for all E, {A, B} ∈ X + sp .
Proof. This is just the combination of Corollary 18, Lemma 19, the equality

Representation theoretic argument
The main argument in [Edi96,dSE00] that an isogeny must exist between the Jacobians, and even some information about its degree, is directly deduced from the Brauer relation between certain permutation representation. Denote by B, S, and N a Borel subgroup, a normaliser of a split Cartan subgroup and a normaliser of a non-split Cartan subgroup of a group G isomorphic to PGL 2 (F p ), respectively. Then (see [Edi96,dSE00]) We fix an elliptic curve E over an algebraically closed field of characteristic different from p. Write From the fact that the middle line (for T ) in table 2.2 in [Edi96] only contains 0 and 1, we see that V ⊗ C decomposes into a sum of distinct irreducible C[G]-modules. We denote by χ W the character and e W = 1/|G| · g∈G χ W (g) g −1 the idempotent associated to such an irreducible C[G]-submodule W of V . For f a Q[G]-endomorphism of V , Schur's Lemma implies that f | W is the multiplication by a scalar c W (f ) ∈ C. Let K be the cyclotomic field Q(ζ p−1 , ζ p+1 ). A look at the character table (for instance  table 2.1 in [Edi96]) of G ∼ = PGL 2 (F p ) shows that all values of characters are contained in K. Since c W (f ) = 1/ dim(W ) · tr(e W • f ), we see that c W (f ) belongs to Q(χ W ) ⊂ K.
We will now consider these scaling factors for the Q[G]-endomorphisms in equation (3). Let W be an irreducible complex representation which appears in the decomposition of V ⊗ C but not in the image of ψ⊗C : A C (E, A) → V ⊗C. Then c W (ψ•µ) = 0. By the Brauer relation, since A Q (E, A) ∼ = Q[G/B], the representation W also appears in the decomposition of U ⊗ C. Then similarly c W (j) = 0. Hence the equation (3) gives However, since c W (α) ∈ K, it can not be equal to ± √ p. This shows that c W (λ • ϕ) = 0 for all irreducible W which do not appear in the image of ψ. Therefore the map ϕ is a G-isomorphism from V / im ψ into the non-trivial part of U . Moreover, the map ϕ • λ : U → U has the same scalar factors c W (ϕ • λ) = c W (λ • ϕ) = 0 and on the trivial part it is the scalar multiplication by (p 2 − 1)/4 = 0. It follows that ϕ • λ is a G-automorphism of U .

End of proof of Proposition 14
We compute We deduce from the above representation theoretic input that the pairing in Section 3.8 is non-degenerate.
This concludes the proof of Proposition 14. It is to note that the non-degeneracy of the pairing is equivalent to the difficult part of the proof of Chen's isogeny in Theorem 15. It would be nice to find a purely combinatorial proof of the non-degeneracy of this pairing.

End of proof of Theorem 15
In Section 4.4, we have shown that the map ϕ has finite kernel and cokernel. In other words the map ϕ • λ : Div(X + nsp ) → Div(X + nsp ) has finite kernel and cokernel in each fibre. Since the size of them is independent of the fibre, the above map has kernel and cokernel of finite exponent. Now consider the induced map ϕ • λ : Jac(X + nsp ) → Jac(X + nsp ) on the Jacobian. If [D] is a divisor class in Jac(X + nsp ), then there is a multiple [mD] which is in the image of ϕ • λ. Therefore the map ϕ • λ has finite cokernel on the Jacobians. Comparing the dimensions it follows that it has finite kernel, too. This implies that ϕ has finite cokernel and λ has finite kernel in the sequences in Theorem 15.
To conclude we have to verify that the sequences have finite cohomology in the middle term. This can be deduced from counting the dimension together with all the known parts of the theorem: We know from (2) in Section 3.6 that the dimension of Jac(X + sp ) is equal to the sum of the dimensions of Jac(X 0 ) and Jac(X + nsp ). Since ϕ has finite cokernel, its kernel has now the same dimension as Jac(X 0 ), which is also the dimension of the image of ψ. Because the sequence is a complex, we have im ψ ⊂ ker ϕ and the quotient is finite because they have the same dimension. The argument for the second sequence is similar. This concludes the proof of Theorem 15.

Relation to Chen's computations
To relate our proof to the previous proof in [Che00], we need first to establish a translation. The first difference is that we work with PGL 2 (F p ) rather than with GL 2 (F p ), but that does not make any real difference.
Fix an elliptic curve E over an algebraically closed field of characteristic different from p. Let us fix two distinct subgroups A 0 and B 0 in E. Further we choose a necklace v 0 in which A 0 is antipodal to B 0 . Let B be the stabiliser of A 0 in G = PGL E[p] , which is a Borel subgroup, let S be the stabiliser of {A 0 , B 0 }, which is the normaliser of a split Cartan subgroup, and let N be the stabiliser of v 0 , which is the normaliser of a non-split Cartan. Then we define three G-isomorphisms The important about of the exact choices here is that N ∩ S contains 4 elements. Had we taken "adjacent" rather than "antipodal" pearls in the necklace, we would only have 2 elements. Compare with remarque 3 in [dSE00].
Recall from [Che00] that for each double coset HgH for some subgroups H and H of G and g ∈ G, there is a G-morphism Θ(HgH ) : Q[G/H] → Q[G/H ] sending H to the sum s∈Ω sH such that s∈Ω sH = HgH is a disjoint union.
Lemma 21. We have Proof. We only illustrate the first equality as the proof is very similar for all of the first four equalities. The map Θ(B1S) sends B = ι −1 0 (A 0 ) to the sum of sS where s runs over a system Ω of representatives of B/(B ∩ S). The quotient group is the group of elements in G fixing A 0 modulo the subgroup of elements also fixing B 0 . So Now it is clear that equation (3) is exactly what Chen proves in Proposition 8.6 and Proposition 8.7. His proof is a computation in double coset operators. He then goes on to give formulae for the values of c W (λ • ϕ) in terms of character sums. However, his final argument that they are non-zero can be shortened as we did in Section 4.4 without making the values more explicit.
Finally, we wish to point out that Chen also describes the maps using the degeneracy morphisms. See his Theorem 2 in [Che00]. For instance, let consider the usual degeneracy morphisms π 0 : X A −→ X 0 defined by (E, (A, B, C)) → (E, A) and π + sp : X A −→ X + sp defined by (E, (A, B, C) → (E, {A, B}), it is easy to see from our definitions that (p − 1) · ψ = (π + sp ) * • (π 0 ) * and (p − 1) · µ = (π 0 ) * • (π + sp ) * hold as maps on divisors. To explain ϕ and λ, we have to replace π + nsp by another degeneracy map. Let ε be a non-square in F p . We defineπ + nsp : X A → X + nsp by sending E, (A, B, C) to the following necklace v in E. First there exist a unique D distinct from A, B, and C such that [A, B; C, D] = ε. Then, by Lemma 8 there is a unique v such that A B ∈ v and C D ∈ v. It is also this lemma which shows that this map is PGL E[p] -equivariant.
Lemma 22. We have 4 · ϕ = (π + nsp ) * • (π + sp ) * and 4 · λ = (π + sp ) * • (π + nsp ) * . Let v be a necklace with A B ∈ v. We wish to determine how often v appears in the above sum, that is to say how many X ∈ {A, B} are there such that v =π + nsp E, (A, B, X) . In other words, we wish to count the X such that [A, B; X, X ] = ε where X is the antipodal pearl to X in v. We can choose a basis of E[p] such that A = (1 : 0), B = (0 : 1) and the subgroups X ∈ {A, B} are X = (1 : a) for some a ∈ F × p . The involution in the stabiliser of v is then represented by a matrix g = [ 0 d 1 0 ] with d non-square and the antipodal pearl to X in v is X = (da : 1). It follows that [A, B; X, X ] = da 2 . Since ε and d are non-squares, the equation da 2 = ε has two solutions in F × p . Hence there are two pearls X ∈ {A, B} such that v =π + nsp E, (A, B, X) and consequently

Proof. Let
The second equality follows from an analogous argument.

Examples
We add some numerical examples for small primes, mainly on the eigenvalues of the pairing in Section 3.8.
It is now easy to read off the pairing ·, · defined in Section 3.8. Let v be the first necklace in the list. Of course, we have v, v = 3. On the one hand, we have v, w = 0 for w being any of the necklaces from the second to the seventh and, on the other hand, v, w = 1 when w is any of the last three necklaces. The resulting matrix ( v, w ) v,w is non-singular. Its eigenvalues are 6, four times 1, and five times 4.