Journal de Théorie des Nombres de Bordeaux

Sam Frengley

Explicit $7$-torsion in the Tate–Shafarevich groups of genus $2$ Jacobians

Jungwon Lee

Bharathwaj Palvannan

An ergodic approach towards an equidistribution result of Ferrero–Washington

Ningchuan Zhang

Congruences of Eisenstein series of level $\Gamma _1(N)$  via Dieudonné theory of formal groups

Anton Mosunov

Absolute Bound On the Number of Solutions of Certain Diophantine Equations of Thue and Thue–Mahler Type

Dinesh S. Thakur

Fermat–Wilson Supercongruences, Arithmetic Derivatives and Strange Factorizations

Alexandru Ghitza

Takuya Yamauchi

Automorphy of mod 2 Galois representations associated to certain genus 2 curves over totally real fields

Han Wu

Chang Lv

Han WU et Chang LV
Genus one half stacky curves violating the local-global principle
Tome 36, no 3 (2024), p. 835-841.
https://doi.org/10.5802/jtnb.1297
© Les auteurs, 2024.
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Journal de Théorie des Nombres
de Bordeaux 36 (2024), 835–841
Genus one half stacky curves violating the
local-global principle
par Han WU et Chang LV
Résumé. Pour un corps de nombres quelconque, nous prouvons qu’il existe
un champs algébrique de Deligne–Mumford propre, lisse, géométriquement
connexe de dimension 1 et de genre 1/2, défini sur l’anneau des entiers du
corps et violant le principe local-global pour les points entiers.
Abstract. For any number field, we prove that there exists a stacky curve
of genus 1/2 defined over the ring of its integers violating the local-global
principle for integral points.
1. Introduction
Given a number field K, let OK be the ring of its integers, and let ΩK be
the set of all its nontrivial places. Let Kv be the completion of K at v ∈ ΩK .
For a finite place v, let Ov be the valuation ring of Kv. For an archimedean
place v, let Ov = Kv. Let X be an algebraic stack of finite type over OK . If
the set X(OK) ̸= ∅, then the set X(Ov) ̸= ∅ for all v ∈ ΩK . The converse
does not always hold. We say that X violates the local-global principle for
integral points if X(Ov) ̸= ∅ for all v ∈ ΩK , whereas X(OK) = ∅. For
K = Q, Darmon and Granville [3] implicitly gave an example of a stacky
curve violating the local-global principle for integral points. In the paper [1],
Bhargava and Poonen proved that any stacky curve over OK of genus less
than 1/2 satisfies the local-global principle for integral points. For K = Q,
they gave an example of a genus-1/2 stacky curve violating the local-global
principle for integral points in loc. cit.
Our goal is to generalize their counterexample to any number field. We
will prove the following theorem.
Manuscrit reçu le 9 mars 2023, révisé le 16 août 2023, accepté le 15 septembre 2023.
2020 Mathematics Subject Classification. 11G30, 14A20, 14G25, 14H25.
Mots-clefs. stacky curves, local points, integral points, local-global principle for integral
points.
Han Wu was partially funded by Open Foundation of Hubei Key Laboratory of Applied Math-
ematics (Hubei University) and by the National Natural Science Foundation ofChina (NSFC)
Grant No. 12301014, and supported by Hubei Provincial Natural Science Foundation of China
Grant No. 2023AFB282. Chang Lv was partially funded by the National Natural Science Foun-
dation of China (NSFC) Grant No. 11701552. Chang Lv is the correspondence author.
836 Han Wu, Chang Lv
Theorem 1.1 (Theorem 5.1). For any number field K, there exists a stacky
curve of genus-1/2 over OK violating the local-global principle for integral
points.
The way to prove this theorem is to give an explicit construction of
a genus-1/2 stacky curve violating the local-global principle for integral
points. The paper is organised as follows. In Section 2, we set up the back-
ground by recalling some facts on stacky curves. Then we introduce a class
of genus-1/2 stacky curves in Section 3. In Section 4, we prove that the
stacky curves given in Section 3 have local integral points. Finally, in Sec-
tion 5, we put some restrictions on the stacky curves given in Section 3 so
that they do not have integral points. Then Theorem 5.1 holds.
2. Notation and preliminaries
2.1. Notation. Given a number field K, let OK be the ring of its integers,
and let ΩK be the set of all its nontrivial places. Let ∞rK ⊂ ΩK be the subset
of all real places, and let ΩfK ⊂ ΩK be the subset of all finite places. Let
Kv be the completion of K at v ∈ ΩK . For a finite place v, let Ov be the
valuation ring of Kv, and let Fv be the residue field. For an archimedean
place v, let Ov = Kv. We say that an element is a prime element, if the
ideal generated by this element is a prime ideal. If an element p ∈ OK is a
prime element, we denote its associated valuation by vp, and its associated
valuation ring (field) by Op (respectively Kp). Let K be an algebraic closure
of K.
2.2. Stacky curves. In this subsection, we briefly recall some facts on
stacky curves. We refer to [5], [8] and [1] for more details.
We say that X is a stacky curve over K, if X is a smooth, proper and
geometrically connected 1-dimensional Deligne–Mumford stack over K that
contains a nonempty open substack isomorphic to a scheme, cf. [8, Definiton
5.2.1]. Given a stacky curve X over a number field K, by [4, Theorem 1.1],
let Xcoarse be its coarse moduli space, which is a smooth, projective and
geometrically connected curve over K. Let π : X → Xcoarse be the coarse
space morphism. For any finite extension L/K and any closed point P ∈
Xcoarse(L), let GP be the stabilizer of X above P, which is a finite group
scheme over K. Let P ⊂ Xcoarse be the reduced finite subsheme above
which the stabilizer is nontrivial. Hence, π is an isomorphism over the
open subscheme Xcoarse\P. Motivated by the Riemann–Hurwitz formula,
the genus of X is defined by
(2.1) g(X) : = g(Xcoarse) +
1
2
∑
P ∈P
(
1 − 1deg GP
)
deg P.
Genus one half stacky curves violating the local-global principle 837
This formula is stable under base field change. It can be defined using the
geometrically closed points of P by
(2.2) g(X) : = g(Xcoarse) +
1
2
∑
P ∈P(K)
(
1 − 1deg GP
)
.
In particular, the genus is a nonnegative rational number. From this for-
mula, one deduces the following lemma.
Lemma 2.1 ([6, Lemma 6 and Proposition 8]). Let X be a stacky curve
over a number field K, then g(X) ≥ 0. If g(X) < 1, then g(Xcoarse) = 0
and X is geometrically isomorphic to P1.
It follows from the Hasse–Minkowski theorem that for a stacky curve of
genus less than one over a number field, the local-global principle for ratio-
nal points always holds. Bhargava and Poonen [6, Theorem 5] proved that
the local-global principle for integral points always holds for a stacky curve
of genus less than 1/2 over a number field. Furthermore, Christensen [2,
Theorem 13.0.6] proved that it satisfies strong approximation. Because of
these, we consider the local-global principle for integral points of genus-
1/2 stacky curves. We say that X is a stacky curve over OK , if X is a
proper algebraic stack over OK whose generic fiber is a stacky curve over
K. For any OK-algebra R, let X (R) be the set of isomorphism classes of
OK-morphisms Spec R → X .
3. A class of genus-1/2 stacky curves
Let K be a number field. Let
µ2 := Spec OK [λ]/(λ2 − 1) ⊂ Gm := Spec OK [λ, 1/λ]
be the closed subgroup scheme. Let
Z/2Z := Spec OK [λ]/(λ − 1)
⊔
Spec OK [λ]/(λ + 1).
The following lemma states that these two finite group schemes are isomor-
phic over OK [1/2].
Lemma 3.1. Given a number field field K, the natural morphism Z/2Z →
µ2 given by
OK [λ]/(λ2 − 1) −→ OK [λ]/(λ − 1) × OK [λ]/(λ + 1) : λ 7−→ (λ, λ)
is a group homomorphism. And it is an isomorphism over OK [1/2].
Proof. By a direct check of group operators of these two group schemes,
this is a group homomorphism. After base change to OK [1/2], the ring
homomorphism is an isomorphism. □
838 Han Wu, Chang Lv
Let p, q be two coprime integers in K. Let z2−px2−qy2 be a homogeneous
polynomial in OK [x, y, z] with homogeneous coordinates (x : y : z). Let
Y(p,q) := Proj OK [x, y, z]/(z2 − px2 − qy2), and let Y(p,q) be its base change
to K. We define a µ2-action on Y(p,q) by letting λ ∈ µ2 act as (x : y : z) 7→
(x : y : λz). Let [Y(p,q)/µ2] and [Y(p,q)/µ2] be the quotient stacks over OK
and K respectively.
Proposition 3.2. The quotient stack [Y(p,q)/µ2] is a Deligne–Mumford
stack over OK [1/2]. The quotient stack [Y(p,q)/µ2] is a genus-1/2 stacky
curve.
Proof. Since [Y(p,q)/(Z/2Z)] is a Deligne–Mumford stack over OK , the first
argument follows from Lemma 3.1. In particular, the quotient stack
[Y(p,q)/µ2] is a Deligne–Mumford stack. For a Deligne–Mumford stack, the
properties of being smooth, proper and geometrically connected of di-
mension one follow from these properties of Y(p,q). Let Pz=0 ⊂ Y(p,q) be
the finite K-subscheme defined by z = 0. The group µ2 acts freely on
Proj K[x, y, z]/(z2 − px2 − qy2)\Pz=0, so the stack (Proj K[x, y, z]/(z2 −
px2 − qy2)\Pz=0)/µ2 is representable by a scheme, which is an open sub-
stack of [Y(p,q)/µ2]. Since Proj K[x, y, z]/(z2 − px2 − qy2)\Pz=0 is geometri-
cally isomorphic to Gm, geometrically this action over it can be viewed as
the action from the Kummer sequence 1 → µ2 → Gm → Gm → 1. Hence
the stack (Proj K[x, y, z]/(z2 − px2 − qy2)\Pz=0)/µ2 is geometrically iso-
morphic to Gm. So [Y(p,q)/µ2] is a stacky curve and g([Y(p,q)/µ2]coarse) = 0.
As µ2 acts trivially on Pz=0 containing two geometric points, by the genus
formula (2.2), we have g([Y(p,q)/µ2]) = 1/2. □
The stacky curves that we consider in this paper, are the quotient stacks
of the form [Y(p,q)/µ2]. We denote [Y(p,q)/µ2] by X(p,q).
4. Existence of local points
In this section, we prove that the stacky curve X(p,q) has local integral
points, i.e. the set X(p,q)(Ov) ̸= ∅ for all v ∈ ΩK .
Lemma 4.1. Given a number field K, let p, q be two coprime integers
in K. Let S = ∞rK ∪ {v ∈ Ω
f
K |v(2pq) ̸= 0} be a finite set. Then the set
Y(p,q)(Ov) ̸= ∅ for all v ∈ ΩK\S.
Proof. For any finite place v ∈ ΩK , by Chevalley–Warning theorem (cf. [7,
Chapter I §2, Corollary 2]), the set Y(p,q)(Fv) ̸= ∅. For any v ∈ ΩK\S, the
scheme Y(p,q) is smooth over Ov. By the smooth lifting theorem, the set
Y(p,q)(Ov) ̸= ∅ for all v ∈ ΩK\S. □
Remark 4.2. Consider the quotient morphism: Y(p,q) → X(p,q). Then this
lemma implies that the set X(p,q)(Ov) ̸= ∅ for all v ∈ ΩK\S.
Genus one half stacky curves violating the local-global principle 839
In order to prove that the stacky curves X(p,q) has local integral points.
We need to check that the set X(p,q)(Ov) ̸= ∅ for all v ∈ S.
Let R be a principal ideal domain over OK . We analyze the set X(p,q)(R)
first.
By definition of the quotient stack, a morphism Spec R → X(p,q) is
given by a µ2-torsor T over R equipped with a µ2-equivariant morphism
T → Y(p,q). The torsors are classified by H1fppf(R, µ2), which is isomorphic
to R×/R×2, since H1fppf(R,Gm) = Pic R = 0. Explicitly, if t ∈ R×, the cor-
responding µ2-torsor is Tt := Spec R[u]/(u2 − t) and the µ2-action on Tt is
given by letting λ ∈ µ2 act as u 7→ λu. Let Y(p,q)t := Proj R[x, y, z′]/(tz′2 −
px2 −qy2) be the twist of Y(p,q) by t. Consider the µ2-torsor Y(p,q)t ×Spec R Tt
over Y(p,q)t. Define a morphism Y(p,q)t ×Spec R Tt → Y(p,q) given by
OK [x, y, z]/(z2 − px2 − qy2) −→ R[x, y, z′, u]/(tz′2 − px2 − qy2, u2 − t)
(x, y, z) 7−→ (x, y, uz′).
It is a µ2-equivariant morphism. This gives a morphism πt : Y(p,q)t → X(p,q).
To give a µ2-equivariant morphism Tt → Y(p,q) is the same as giving a triple
(a1, a2, a3) ∈ R3, and the µ2-equivariant morphism is given by
OK [x, y, z]/(z2 − px2 − qy2) −→ R[u]/(u2 − t)
(x, y, z) 7−→ (a1, a2, a3u).
And the triple (a1, a2, a3) gives a morphism Spec R → Y(p,q)t defined by
R[x, y, z′]/(tz′2 − px2 − qy2) −→ R
(x, y, z′) 7−→ (a1, a2, a3).
Hence, to give a µ2-equivariant morphism Tt → Y(p,q) is the same as giving
a morphism Spec R → Y(p,q)t. Thus we obtain
(4.1) X(p,q)(R) =
∐
t∈R×/R×2
πt(Y(p,q)t(R)).
With this preparation, we have the following proposition.
Proposition 4.3. Given a number field K, let p, q be two coprime integers
in K. Then the set X(p,q)(Ov) ̸= ∅ for all v ∈ ΩK .
Proof. By Lemma 4.1, we need to check that the set X(p,q)(Ov) ̸= ∅ for all
v ∈ S.
Suppose that v ∈ ∞rK or v ∤ q. Then q ∈ O×v . Since qz2 − px2 − qy2 = 0
has a nontrivial solution (x : y : z) = (0 : 1 : 1), we have Y(p,q)q(Ov) ̸= ∅.
Hence, the set X(p,q)(Ov) ̸= ∅.
Similarly, suppose that v ∤ p. Then sets Y(p,q)p(Ov) ̸=∅ and X(p,q)(Ov) ̸=∅.
Since p and q are two coprime integers, the set X(p,q)(Ov) ̸= ∅ for all
v ∈ ΩK . □
840 Han Wu, Chang Lv
5. Genus 1/2-stacky curves violating the local-global principle
for integral points
Given a number field K, we put some restrictions on the choice of inte-
gers p, q so that the stacky curve X(p,q) has no integral points, i.e. the set
X(p,q)(OK) = ∅. We choose p, q in the following way.
5.1. Choosing prime elements. Given a number field K, since the ideal
class group of K is finite, we take a positive integer N such that OK [1/N ] is
a principal ideal domain. By Dirichlet’s unit theorem, the group OK [1/N ]×
is a finitely generated abelian group. We assume that it is generated by
{ai} for i = 1, . . . , n. By Čebotarev’s density theorem and global class field
theory applied to a ray class field, we can find a pair of two different odd
prime elements (p, q) such that
(1) ai ∈ K×2p for all i = 1, . . . , n,
(2) q /∈ K×2p .
We refer to [9, Section 4.2.2] and [10, Proposition 5.3] for more details.
Then we have the following theorem.
Theorem 5.1. Let K be a number field. Let a positive integer N and a pair
of two different odd prime elements (p, q) be chosen as in Subsection 5.1.
Let X(p,q) be the stacky curve defined in Section 3. Then X(p,q) is a stacky
curve of genus-1/2 over OK violating the local-global principle for integral
points.
Proof. By Proposition 3.2, the genus of X(p,q) is 1/2. By Proposition 4.3,
the set X(p,q)(Ov) ̸= ∅ for any v ∈ ΩK .
Next, we prove that the set X(p,q)(OK [1/N ]) = ∅. Since the ring OK [1/N ]
is a principal ideal domain, in order to prove that X(p,q)(OK [1/N ]) = ∅, by
the equality of sets (4.1), it will be sufficient to prove that for any t ∈
OK [1/N ]×, the set Y(p,q)t(OK [1/N ]) = ∅. Since OK [1/N ]× is generated by
{ai} for i = 1, . . . , n, and by the chosen condition of Subsection 5.1 that ai ∈
K×2p , we have Y(p,q)t is isomorphic to Y(p,q) over Kp for all t ∈ OK [1/N ]×.
By the choice of elements q, the set Y(p,q)(Kp) = ∅. So X(p,q)(OK [1/N ]) = ∅,
which implies that X(p,q)(OK) = ∅.
So the stacky curve X(p,q) is of genus-1/2, and violates the local-global
principle for integral points. □
Remark 5.2. This theorem implies that the chosen stacky curve X(p,q)
violates strong approximation in the sense of [2]. In addition, this stacky
curve X(p,q) is smooth over OK [ 12pq ].
Genus one half stacky curves violating the local-global principle 841
Acknowledgements. The authors would like to thank Dasheng Wei and
Weizhe Zheng for fruitful discussions, and the referees for valuable sugges-
tions. The work was partially done during their visit to the Morningside
Center of Mathematics, Chinese Academy of Sciences. They thank the Cen-
ter for its hospitality.
References
[1] M. Bhargava & B. Poonen, “The local-global principle for integral points on stacky
curves”, J. Algebr. Geom. 31 (2022), no. 4, p. 773-782.
[2] A. Christensen, “A topology on points on stacks”, PhD Thesis, Massachusetts Institute of
Technology (USA), 2020.
[3] H. Darmon & A. Granville, “On the equations zm = F (x, y) and Axp + Byq = Czr”,
Bull. Lond. Math. Soc. 27 (1995), no. 6, p. 513-543.
[4] S. Keel & S. Mori, “Quotients by groupoids”, Ann. Math. 145 (1995), no. 1, p. 193-213.
[5] M. Olsson, Algebraic spaces and stacks, Colloquium Publications, vol. 62, American Math-
ematical Society, 2016, xi+298 pages.
[6] B. Poonen & J. F. Voloch, “The Brauer–Manin obstruction for subvarieties of abelian
varieties over function fields”, Ann. Math. 171 (2010), no. 1, p. 511-532.
[7] J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, vol. 7, Springer, 1973,
viii+115 pages.
[8] J. Voight & D. Zureick-Brown, The canonical ring of a stacky curve, Memoirs of
the American Mathematical Society, vol. 1362, American Mathematical Society, 2022,
v+144 pages.
[9] H. Wu, “Non-invariance of the Brauer–Manin obstruction for surfaces”, 2021, https://
arxiv.org/abs/2103.01784v2.
[10] ——— , “On genus one curves violating the local-global principle”, J. Number Theory 242
(2023), p. 235-243.
Han Wu
Laboratory of Applied Mathematics
Faculty of Mathematics and Statistics, Hubei University
No. 368, Friendship Avenue
Wuchang District, Wuhan, Hubei, 430062, P.R. China
E-mail: wuhan90@mail.ustc.edu.cn
URL: https://mci.hubu.edu.cn/info/1059/9625.htm
Chang Lv
State Key Laboratory of Cyberspace Security Defense
Institute of Information Engineering
Chinese Academy of Sciences
Beijing 100093, P.R. China
E-mail: lvchang@amss.ac.cn
URL: https://c-lv.github.io/


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