Counting elliptic curves with prescribed level structures over number fields

Harron and Snowden counted the number of elliptic curves over $\mathbb{Q}$ up to height $X$ with torsion group $G$ for each possible torsion group $G$ over $\mathbb{Q}$. In this paper we generalize their result to all number fields and all level structures $G$ such that the corresponding modular curve $X_G$ is a weighted projective line $\mathbb{P}(w_0,w_1)$ and the morphism $X_G\to X(1)$ satisfies a certain condition. In particular, this includes all modular curves $X_1(m,n)$ with coarse moduli space of genus $0$. We prove our results by defining a size function on $\mathbb{P}(w_0,w_1)$ following unpublished work of Deng, and working out how to count the number of points on $\mathbb{P}(w_0,w_1)$ up to size $X$.


Introduction
Let E be an elliptic curve over a number field K. The Mordell-Weil theorem says that E(K) is isomorphic to Z r × E(K) tor for some r ≥ 0, where E(K) tor is the (finite) torsion subgroup of E(K). It is a natural question which groups appear as E(K) tor , and moreover how often each one of these groups appears. Harron and Snowden [11] studied this question and answered it in the case K = Q. The aim of this paper is to study the same problem, but to both allow K to be any number field and to answer the more general question how often a prescribed G-level structure appears.
To make this question more precise, let n be a positive integer, let G be a subgroup of GL 2 (Z/nZ), and let K be a number field. We say that an elliptic curve E over K admits a G-level structure if there exists a (Z/nZ)basis of E[n](K) such that the Galois representation ρ E,n : Gal(K/K) → GL 2 (Z/nZ) defined by this basis has image contained in G. We write We will define a size function S K from the set of isomorphism classes of elliptic curves over K to R >0 ; see Definition 7.1. We define a function N G,K : R >0 → Z ≥0 by N G,K (X) = #{E ∈ E G,K | S K (E) 12 ≤ X}.
Let X G be the moduli stack of generalized elliptic curves with G-level structure. This is a one-dimensional proper smooth geometrically connected algebraic stack over the fixed field K G of the action of G on Q(ζ n ) given by (g, ζ n ) → ζ det g n . We consider cases where X G is a weighted projective line P(w 0 , w 1 ) over K G . We can now state our main result (which is also stated in a slightly different form in Theorem 7.6).
Theorem 1.1. Let n be a positive integer, and let G be a subgroup of GL 2 (Z/nZ). Assume that the stack X G over K G is isomorphic to P(w) K G , where w = (w 0 , w 1 ) is a pair of positive integers, and let e(G) be the reduced degree of the canonical morphism X G → X(1) (see Definition 4.2). Furthermore, assume e(G) = 1 or w = (1, 1) holds. Then for every finite extension K of K G , we have As all modular curves X G = X 1 (m, n) with coarse moduli space of genus 0 satisfy the assumptions of Theorem 1.1, our result generalizes [11,Theorem 1.2], where this statement was proved in the case where K = Q and where G is one of the 15 groups corresponding to the torsion groups from Mazur's theorem.
A recent result of Pizzo, Pomerance and Voight [15] is N G,Q (X) ∼ X 1/2 for G such that X G = X 0 (3). Moreover, they determined the constant in front of the leading term of the function N G,Q (X) as well as the first two lower-order terms. This result falls outside of the reach of our results, as X 0 (3) is not a weighted projective line (cf. Remark 7.4).
Similarly, Pomerance and Schaefer [16] proved that N G,Q (X) ∼ X 1/3 for G such that X G = X 0 (4), and determined the constants in front of the leading term and the first lower-order term. Our result implies N G,K ≍ X 1/3 for all number fields K; for K = Q, this follows from the sharper results of [16].
Cullinan, Kenney and Voight [4, Theorem 1.3.3] proved a sharper version of Theorem 1.1 in the special case where X G is a projective line (i.e. isomorphic to P 1 = P(1, 1)) and K = Q. More precisely, they give an asymptotic expression for N G,Q (X) containing an effectively computable leading coefficient and an error term.
Boggess and Sankar [2] determined the growth rate of the number of elliptic curves over Q with a cyclic n-isogeny for n ∈ {2, 3, 4, 5, 6, 8, 9, 12, 16, 18}. Out of these, only the cases n = 2 and n = 4 (for which a more precise result was already proved in [11,16]) correspond to weighted projective lines and are therefore generalized to number fields by Theorem 1.1.
Remark 1.2. The 12-th power in the definition of N G,K (X) is included for easier comparison with the height function in [11]; see Remark 7.2. Remark 1.3. Our result gives a more conceptual interpretation of d(G); cf. [11, §1.2]. Namely, we show that d(G) can be expressed in terms of the pair of positive integers (w 0 , w 1 ) for which X G is isomorphic to the weighted projective line with weights (w 0 , w 1 ), and e(G), an invariant (similar to the degree) of the morphism X G → X(1).
We also remark that our result shows how in certain cases one can count points in the image of a morphism of stacks, partially answering a question in [11,Remark 1.5].
It is known that P(w) is a proper smooth algebraic stack over Z, and in fact a complete smooth toric Deligne-Mumford stack in the sense of Fantechi, Mann and Nironi [10,Example 7.27]. For every ring R, there is a groupoid of R-points of P(w). We will mostly be interested in the set of isomorphism classes of this groupoid, which we call the set of R-points of P(w) and denote by P(w)(R). Given a field L, the set P(w)(L) can be described as The image in P(w)(L) of an element x ∈ L n+1 \ {0} will be denoted by [x].
Example 2.2. If w = (m) with m a positive integer, then P(m) is canonically isomorphic to the classifying stack of the group scheme µ m . If L is a field, then we have

Size functions
Let w be an (n+1)-tuple as above, let K be a number field, and let O K be its ring of integers. On the set P(w)(K), we define a size function similarly to Deng [7]; see Definition 3.4 below. We do not restrict to weighted projective spaces that are "well-formed" in the sense of [7]. Definition 3.1. For x ∈ K n+1 , the scaling ideal of x, denoted by I w (x), is the intersection of all fractional ideals a of O K satisfying x ∈ a w 0 × · · · × a wn . Similarly, for an (n + 1)-tuple (b 0 , . . . , b n ) of fractional ideals of O K , the scaling ideal of (b 0 , . . . , b n ), denoted by I w (b 0 , . . . , b n ), is the intersection of all fractional ideals a of O K satisfying b i ⊆ a w i for all i.

Remark 3.2.
For all x ∈ K n+1 \ {0}, the fractional ideal I w (x) is non-zero and satisfies Similarly, for every (n + 1)-tuple (b 0 , . . . , b n ) of fractional ideals of O K , not all zero, the fractional ideal I w (b 0 , . . . , b n ) is non-zero and satisfies Let Ω K,∞ denote the set of Archimedean places of K, and for each v ∈ Ω K,∞ , let | | v : K → R ≥0 be the corresponding normalized absolute value. The Archimedean size on K n+1 \ {0} is the function Definition 3.4. The size function on P(w)(K) is the function S w,K : It is straightforward to check that S w,K ([x]) does not depend on the choice of the representative x.  Remark 3.7. Definition 3.4 is a special case of the notion of height for rational points on algebraic stacks defined by Ellenberg, Satriano and Zureick-Brown [9]. Namely, as explained in [9, Section 3.3], we have where ht L is the height function corresponding to the tautological line bundle L on P(w). The work of Ellenberg, Satriano and Zureick-Brown was recently used by Boggess and Sankar [2] to count elliptic curves over Q with a rational n-isogeny for n ∈ {2, 3, 4, 5, 6, 8, 9}, as mentioned in the introduction.
Theorem 3.8. Let n be a non-negative integer, let w = (w 0 , . . . , w n ) be an (n + 1)-tuple of positive integers, and let K be a number field. Let r 1 , r 2 , d K , h K , R K , µ K and ζ K be the number of real places, number of non-real complex places, discriminant, class number, regulator, number of roots of unity and Dedekind ζ-function of K, respectively. We write Then we have Proof. This was proved by Deng [7, Theorem (A)] in the case where P(w) is well-formed, i.e. each n elements from w are coprime. However, the proof works in general with only minor changes: in the paragraph before [7, Proposition 4.2], the statement that the group of roots of unity acts effectively has to be replaced by the statement that all orbits of points with all coordinates non-zero contain µ w K points, and the factor w (denoting the number of roots of unity) in [7, Proposition 4.2, Proposition 4.5, Corollary 4.6 and Theorem (A)] has to be replaced by µ w K . Remark 3.9. Theorem 3.8 also follows from recent results of Darda [5] on counting rational points on weighted projective spaces with respect to more general height functions; see in particular [5, Remark 7.3.2.5].
In the remainder of this article, we will only consider weighted projective lines, i.e. one-dimensional weighted projective spaces where the weight is given by a pair (w 0 , w 1 ).

Morphisms between weighted projective lines
Let u = (u 0 , u 1 ), w = (w 0 , w 1 ) be two pairs of positive integers. In this section, we classify the morphisms of stacks from P(w) to P(u) over a field. These morphisms form a groupoid, but for simplicity we will only be interested in the set of isomorphism classes of this groupoid, or in other words the set of morphisms from P(w) to P(u). We also prove some facts about such morphisms generalizing the corresponding facts about morphisms P 1 → P 1 .
Lemma 4.1. Let K be a field, and let u = (u 0 , u 1 ), w = (w 0 , w 1 ) be two pairs of positive integers. We consider R = K[x 0 , x 1 ] as a graded K-algebra where x 0 and x 1 are homogeneous of degrees w 0 and w 1 , respectively. Let P u,w (K) be the set of pairs (f 0 , f 1 ) ∈ R × R with the following properties: (1) There exists e = e(f 0 , f 1 ) ∈ Z ≥0 for which f 0 and f 1 are homogeneous of degrees eu 0 and eu 1 , respectively.
Proof. We apply Lemma A.2 to the following data over K: • a : G × X → X is the weight w action, given on points by a(g, x) = (g w 0 x 0 , g w 1 x 1 ), • b : H × Y → Y is the weight u action, given on points by b(h, y) = (h u 0 y 0 , h u 1 y 1 ). (Note that the lemma applies because the Picard group of X is trivial.) We first determine the morphisms h : G × X → H satisfying the "cocycle condition" (3) of Lemma A.2. A morphism h : G × X → H is given by a monomial of the form h(g, x) = λg e with λ ∈ K × and e ∈ Z, and h satisfies Given h as above, we now determine the morphisms f : It is straightforward to check that condition (4) translates to the condition that f j is homogeneous of degree eu j for j = 0, 1. In particular, morphisms f : only exist if e ≥ 0; moreover, e and therefore h are uniquely determined by f . Finally, the group H(X) is canonically isomorphic to K × , and if (f, h) is a pair as above where f is defined by (f 0 , f 1 ), and c ∈ H(X), then we have The lemma therefore follows from Lemma A.2.
Definition 4.2. Let K be a field, let u, w be two pairs of positive integers, and let φ : P(w) K → P(u) K be a morphism. The reduced degree of φ, denoted by deg red φ, is the unique integer e ≥ 0 satisfying Lemma 4.1(i) for some (hence every) pair (f 0 , f 1 ) giving rise to φ via the bijection of Lemma 4.1.
Proof. We write R + = Rx 0 + Rx 1 , S + = Sf 0 + Sf 1 and I = Rf 0 + Rf 1 = RS + . By condition (ii) of Lemma 4.1 and the fact that f 0 and f 1 are nonconstant, we have √ I = R + . Hence for m sufficiently large, we have R m + ⊆ I, so the graded K-algebra R/I is a quotient of R/R m + and is therefore finitedimensional over K. Choose homogeneous elements g 1 , . . . , g r ∈ R such that their images in R/I are a K-basis of R/I. In particular, the g i generate R/I = R/RS + over S, so we have Hence the Z ≥0 -graded S-module M = R/(Sg 1 + · · · + Sg r ) satisfies S + M = M . It follows from a variant of Nakayama's lemma (see for example Eisenbud [8,Exercise 4.6]) that M = 0 and hence R = Sg 1 + · · · + Sg r . Lemma 4.5. Let K be a field, let u, w be two pairs of positive integers, and let φ : P(w) K → P(u) K be a non-constant representable morphism. Then φ is finite.
Proof. Since P(w) K and P(u) K are Deligne-Mumford stacks and φ is representable, we may choose a Cartesian diagram where S and T are algebraic spaces and the vertical maps are étale coverings. Then φ ′ is proper because φ is proper, and is locally quasi-finite because φ ′ has relative dimension 0 [17, tag 04NV]. In particular, φ ′ is representable in schemes [17, tag 0418] and is finite [17, tag 0A4X]. It follows that φ is finite.  Proof. By Lemma 4.5, the morphism φ is finite and in particular integral. Furthermore, P(w) K is normal because K[x 0 , x 1 ] is integrally closed. This proves the claim.

Some results on scaling ideals
Let K be a number field. We prove two elementary results about scaling ideals.
Lemma 5.1. Let w = (w 0 , w 1 ) be a pair of positive integers. We consider K[x 0 , x 1 ] as a graded K-algebra by assigning weight w i to x i . Let f ∈ K[x 0 , x 1 ] be homogeneous of degree d. Let a(f ) be the fractional ideal generated by the coefficients of f . Then for all z ∈ K 2 , we have Proof. We abbreviate m = I w (z), so we have z 0 ∈ m w 0 and z 1 ∈ m w 1 . We write where the sum ranges over all pairs (k 0 , k 1 ) of non-negative integers such that k 0 w 0 + k 1 w 1 = d, and a k 0 ,k 1 ∈ K. We now compute which proves the claim.
Lemma 5.2. Let z ∈ K, and let be a monic polynomial such that h(z) = 0. Suppose b 1 , . . . , b d are fractional ideals of K such that c i ∈ b i for all i. Then we have Proof. If all the b i are zero, then z vanishes and the claim is trivial. Now assume not all of the b i are zero. We write Then for all a ∈ a we have By assumption, each a i c i lies in a i b i and hence in O K . This shows that az is integral over O K . Thus we have az ⊆ O K and hence z ∈ a −1 .

Behaviour of size functions under morphisms
Let K be a number field. Let w = (w 0 , w 1 ) and u = (u 0 , u 1 ) be two pairs of positive integers, and let φ : P(w) K → P(u) K be a non-constant morphism. Our goal in this section will be to study how the size of a point in P(w)(K) relates to the size of its image under φ.
By Lemma 4.1, the morphism φ is defined by a pair of non-constant homogeneous polynomials f 0 , f 1 ∈ K[x 0 , x 1 ] of degrees eu 0 and eu 1 , respectively, where e is the reduced degree of φ. For i ∈ {0, 1}, let a i be the fractional ideal generated by the coefficients of f i . Lemma 6.1. For all z ∈ K 2 , we have Proof. We abbreviate m = I w (z). Since f i is homogeneous of degree eu i , Lemma 5.1 gives

It follows that
I u (f (z)) ⊆ I u (a 0 m eu 0 , a 1 m eu 1 ) = I u (a 0 , a 1 )m e , which proves the claim.
For i ∈ {0, 1}, we write the rational number w i /e in reduced form as w i e = ν i δ i with ν i , δ i coprime positive integers.
By Lemma 4.4, there are integers d i > 0 and polynomials g i,j ∈ K[y 0 , y 1 ] (for i = 0, 1 and j = 1, . . . , d i ) satisfying (1) x After taking homogeneous components of degree d i w i , we may and do assume that each g i,j (f 0 , f 1 ) is homogeneous of degree jw 1 . After dividing by a power of x i if necessary, we may and do also assume g i,d i = 0. We write In particular, if g i,j = 0, then e divides jw i , so j is a multiple of the denominator of w i /e; in other words, there is a positive integer l with j = lδ i . Since we have ensured that g i,d i is non-zero, we obtain in particular a positive integer m i with d i = m i δ i , and all j for which g i,j does not vanish are of the form j = lδ i with 1 ≤ l ≤ m i . We can therefore rewrite (1) as and note that For i ∈ {0, 1} and 1 ≤ l ≤ m i , we write c i,l for the fractional ideal generated by the coefficients of g i,lδ i , i.e.
Since g i,lδ j is homogeneous of degree lν i , Lemma 5.1 gives Applying Lemma 5.2, we obtain Theorem 6.4. Let K be a number field, let u, w be two pairs of positive integers, and let φ : P(w) K → P(u) K be a non-constant morphism. Let e be the reduced degree of φ (see Definition 4.2), and for i = 0, 1 write w i /e = ν i /δ i with ν i , δ i coprime positive integers. Then for all z ∈ P(w)(K), we have S u (φ(z)) ≪ S w (z) e and S u (φ(z)) ≫ S (ν 0 ,ν 1 ) (z δ 0 0 , z δ 1 1 ), where the implied constants depend only on K, u, w and φ.
Proof. Lemma 4.1 gives us homogeneous polynomials f 0 , f 1 ∈ K[x 0 , x 1 ] such that φ is defined by (f 0 , f 1 ). For every Archimedean place v of K, the set P(w)(K v ) of points of P(w) over the completion K v of K at v is in a natural way a compact topological space. We consider the function Using the definitions of the size functions and the q v , we compute Let a i , d i (i = 0, 1) be the fractional ideals defined earlier. By Lemma 6.1, we have I w (z) e I u (f (z)) −1 ⊇ I u (a 0 , a 1 ) −1 , and hence N I w (z) e I u (f (z)) −1 ≤ N(I u (a 0 , a 1 )) −1 . By Corollary 6.3, we have d 1 )). Finally, for each v ∈ Ω K,∞ , the function q v : P(w)(K v ) → R >0 is bounded by compactness. From this the theorem follows. Corollary 6.5. In the setting of Theorem 6.4, suppose e = 1 or w = (1, 1) holds. Then for all z ∈ P(w)(K), we have where the implied constants depend only on K, u, w and φ.
Remark 6.7. By Remark 4.3, the assumption e = 1 or w = (1, 1) implies that every morphism satisfying the conditions of Corollary 6.5 is representable. However, the conclusion of Corollary 6.5 no longer holds when "e = 1 or w = (1, 1)" is weakened to "φ is representable". For example, take u = (1, 3) and w = (1, 3), and consider the morphism 0 , x 2 1 ), which has e = 2 and is therefore representable. For all primes p, taking x = (p, p 2 ) ∈ P(1, 3)(Q), we get On the other hand, for all primes p, taking x = (1, p) ∈ P(1, 3)(Q), we get This shows that the ratio between S u (φ(x)) and any fixed power of S w (x) is unbounded as x varies.

Points of bounded size on modular curves
Let Y (1) be the moduli stack over Q of elliptic curves. There is an open immersion ι : Y (1) ֒→ P(4, 6) Q defined as follows: given an elliptic curve E over a Q-scheme S, then Zariski locally on S we can choose a non-zero differential ω and define ι(E) = (c 4 (E, ω), c 6 (E, ω)), where c 4 and c 6 are defined in the usual way. A different choice of ω gives the same point of P(4, 6) Q , so ι is well defined.
Definition 7.1. Let K be a number field. Using the morphism ι, we define the size function Remark 7.2. If E is given in short Weierstrass form as This shows that if E is an elliptic curve over Q, then the ratio between S Q (E) 12 and the height of E h(E) = max{|a| 3 , |b| 2 }, as defined in [11], is bounded from above and below by a constant. Now let n be a positive integer, and let G be a subgroup of GL 2 (Z/nZ). Let K G be the subfield of the cyclotomic field Q(ζ n ) fixed by G, where G acts on Q(ζ n ) by (g, ζ n ) → ζ det g n . Let Y G be the moduli stack of elliptic curves with G-level structure, viewed as an algebraic stack over K G . There is a canonical morphism of stacks Let K be a finite extension of K G . We define E G,K = {elliptic curves admitting a G-level structure over K}/ ∼ = and Lemma 7.3. Let n be a positive integer, let G be a subgroup of GL 2 (Z/nZ), and let w be a pair of positive integers. The following are equivalent: (1) There is a commutative diagram of algebraic stacks over K G , where ι G is an open immersion and φ is representable.
(2) The integral closure of X(1) = P (4,6) in the function field of Y G is isomorphic to P(w). (3) The moduli space of generalized elliptic curves with G-level structure is isomorphic to P(w).
Proof. The equivalence of (ii) and (iii) follows from the fact that the integral closure from (ii) is canonically isomorphic to the moduli space of generalized elliptic curves with G-level structure [6, IV, Théorème 6.7(ii)].
The implication (ii) =⇒ (i) follows from the fact that the integral closure of X(1) in the function field of Y G fits in a commutative diagram as above.
Remark 7.4. If G is a group satisfying the equivalent conditions of Lemma 7.3, then the coarse moduli space of X G is isomorphic to P 1 . The converse does not hold. For example, taking G to be the group of upper-triangular matrices in GL 2 (Z/3Z) gives the modular curve X G = X 0 (3). The coarse moduli space of X 0 (3) is isomorphic to P 1 , but X 0 (3) itself (being the quotient of X 1 (3) by the trivial action of {±1}) is isomorphic to P(2) × P (1, 3), which is not a weighted projective line. One way to see this is to note that the Picard group of P(2) × P(1, 3) contains a subgroup of order 2 (coming from the factor P(2)), whereas the Picard group of a weighted projective line is infinite cyclic [10,Example 7.27].
Remark 7.5. The equivalent conditions of Lemma 7.3 hold if the graded K G -algebra of modular forms for G is generated by two homogeneous elements. Over C, the groups for which this happens were classified by Bannai, Koike, Munemasa and Sekiguchi [1]. Theorem 7.6. Let n be a positive integer, and let G be a subgroup of GL 2 (Z/nZ). Let K G be the fixed field of the action of G on Q(ζ n ) given by (g, ζ n ) → ζ det g n . Assume that G satisfies the equivalent conditions of Lemma 7.3 for some (w 0 , w 1 ), and let e(G) be the reduced degree of the canonical morphism X G → X(1) (see Definition 4.2). Furthermore, assume e(G) = 1 or w = (1, 1) holds. Then for every finite extension K of K G , we have Proof. Using the commutative diagram of Lemma 7.3 and noting that for counting purposes we may ignore the cusps (cf. [7, Remark 6.2]), we obtain N G,K (X) ≍ #{z ∈ P(w)(K) | S (4,6) (φ(z)) 12 ≤ X}.
This proves the claim.

Examples
The groups corresponding to the 15 torsion groups from Mazur's theorem satisfy the conditions of Lemma 7.3. In Table 1, we list these groups and a few more satisfying these conditions. For positive integers m | n we write G(m, n) = g ∈ GL 2 (Z/nZ) g = * * 0 1 and g ≡ * 0 0 1 (mod m) .
We also put G 1 (n) = G (1, n) and For each group G we give its inverse image Γ under the canonical group homomorphism SL 2 (Z) → GL 2 (Z/nZ), the index of Γ in SL 2 (Z), the weights of the corresponding weighted projective line, and the values e(G) and d(G).
The first 12 groups can also be found in [ [SL 2 (Z) : Γ].

Future work
In work in progress of I. Manterola Ayala and the first author (see [12]), results will be proved that make it possible to count points of a moduli stack of the form P(w) directly with respect to the pull-back of the size function from X(1), rather than first relating this pull-back to the standard size function on P(w). This approach requires extending the work of Deng [7], but is conceptually simpler than the approach we have taken here.
It would be interesting to obtain a result similar to Theorem 7.6 for moduli stacks of elliptic curves that are of the form P(2) × P(1, 1). An example of such a moduli stack is X 0 (6), so such a result would enable one to count elliptic curves with a 6-isogeny over any number field.

Appendix A. Morphisms between quotient stacks
In this appendix we assume some knowledge of stacks. We place ourselves in the following situation. Let S be a scheme, let G and H be two group schemes over S, and let m G : G × S G → G and m H : H × S H → H be the group operations. Let X and Y be two S-schemes, let a : G × S X → X be a left action of G on X, and let b : H × S Y → Y be a left action of H on Y . Let p 2 : G × S X → X be the second projection, and let p 2,3 : G × S G × S X → G × S X be the projection onto the second and third factors.
We To state the next lemma, we recall the following. Given a left action of a group Γ on a set Z, the quotient groupoid Γ \\ Z is the following groupoid: the set of objects is Z, the morphisms z → z ′ are the elements γ ∈ Γ with γz = z ′ , and composition of morphisms is the group operation in Γ. The set of isomorphism classes of Γ \\ Z is just the quotient set Γ\Z.
Lemma A.2. In the above situation, assume in addition that all H-torsors on X are trivial. Let Z be the set of pairs (f : X → Y, h : G × S X → H) of morphisms of S-schemes such that for all S-schemes T , all x ∈ X(T ) and all g, g ′ ∈ G(T ) we have (3) h(g ′ g, x) = h(g ′ , gx)h(g, x) and (4) f (a(g, x)) = b(h(g, x), f (x)).
Let the group H(X) act on Z by where f ′ and h ′ are defined on points as follows: for all S-schemes T , all x ∈ X(T ) and all g ∈ G(T ) we have and h ′ (g, x) = c(a(g, x))h(g, x)c(x) −1 .
Then the groupoid of morphisms [G\X] → [H\Y ] is canonically equivalent to the quotient groupoid H(X) \\ Z. In particular, there is a canonical bijection between the set of isomorphism classes of such morphisms and the quotient set H(X)\Z.