Comptes Rendus Mathématique

Priyankur Chaudhuri

Comptes Rendus
Mathématique
Priyankur Chaudhuri
An inductive approach to generalized abundance using nef reduction
Volume 361 (2023), p. 417-421
https://doi.org/10.5802/crmath.420
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 417-421
https://doi.org/10.5802/crmath.420
Algebra / Algèbre
An inductive approach to generalized
abundance using nef reduction
Priyankur Chaudhuri∗, a
a Department of Mathematics, University of Maryland, College Park, MD 20742, USA
E-mail: pchaudhu@umd.edu
Abstract. We use the canonical bundle formula for parabolic fibrations to give an inductive approach to the
generalized abundance conjecture using nef reduction. In particular, we observe that generalized abundance
holds for a klt pair (X ,B) if the nef dimension n(KX +B +L) = 2 and KX +B ≥ 0 or n(KX +B +L) = 3 and
κ(KX +B) > 0.
Manuscript received 4 November 2021, revised 25 January 2022 and 19 May 2022, accepted 7 September 2022.
1. Introduction
In [10], Lazić and Peternell propose the following
Conjecture 1 (Generalized Abundance). Let (X ,B) be a klt pair with KX +B pseudoeffective. If
L is a nef divisor on X such that KX +B +L is also nef, then KX +B +L ≡ M for some semiample
Q-divisor M.
They show that this conjecture holds in case dim X = 2 and for dim X = 3 if κ(KX +B) > 0
(see [10, Corollary C, D]). The main purpose of this article is to prove the following result.
Theorem 2. Let (X ,B) be an n-dimensional klt pair with KX +B ≥ 0 and let L ∈ Pic X be nef such
that KX +B +L is nef with nef dimension n(KX +B +L) = d.
Assume termination of klt flips [7, Conjecture 5-1-13] in dimension d, abundance conjecture [8,
Conjecture 3.12] in dimension ≤ d, semiampleness conjecture [10, page 2] in dimension ≤ d and
generalized non-vanishing conjecture [10, page 2] in dimension ≤ d −1.
Then KX +B +L ≡ M for some semiample divisor M.
Note that the phrase “Conjecture (∗) holds in dimension d (resp. ≤ d)” above means that (∗)
holds for all klt pairs (X ,B) such that dim X = d (resp. ≤ d).
In case L = 0, such a theorem has been proved by [1]. The main ingredients of our proof are the
canonical bundle formula for parabolic fibrations [5] and Nakayama-Zariski decomposition [12].
We have the following corollary
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
418 Priyankur Chaudhuri
Corollary 3. Generalized abundance holds (in any dimension) in the following two cases:
(1) n(KX +B +L) = 2 and KX +B ≥ 0, or
(2) n(KX +B +L) = 3 and κ(KX +B) > 0.
Note that any divisor of nef dimension 1 is already numerically equivalent to a semiample
divisor by [4, 2.4.4]. Thus generalized abundance also holds in this case.
2. Preliminaries
Definition 4 (Singularities of pairs). Let (X ,B) be a sub-pair consisting of a normal variety X and
a Q-divisor B on X such that KX +B is Q-Cartier. It is called sub-klt if there exists a log resolution
Y
µ−→ X of (X ,B) such that letting BY be defined by KY +BY =µ∗(KX +B), all coefficients of BY are
< 1 and sub-lc if all coefficients of BY are ≤ 1. If B ≥ 0, we drop the prefix sub.
Definition 5 (Nef reduction and nef dimension [4]). Let X be a normal projective variety and
let L ∈ Pic X be nef. Then there exists a dominant rational map φ : X 99K Y with connected fibers
which is proper and regular over an open subset of Y (i.e. there exists V ⊂ Y nonempty open such φ
restricts to a proper morphism φ|φ−1(V ) :φ−1(V ) →V ) where Y is also normal projective such that
(1) If F ⊂ X is a general compact fiber of φ with dimF = dim X −dimY , then L|F ≡ 0.
(2) If x ∈ X is a very general point and C ⊂ X a curve passing x such that dim(φ(C )) > 0, then
(L ·C ) > 0.
φ is called the nef reduction map of L and dimY the nef dimension n(L) of L.
Remark 6. Note that if φ : X 99K Y is proper and regular over an open subset of Y , then there
exists a resolution φ̂ : X̂ → Y of φ such that the exceptional divisor E x(X̂ /X ) is φ̂-vertical. For
example, X̂ can be chosen to be the closure of the graph of φ.
Notation 7. For a pseudoeffective divisor D on a smooth projective variety X , Pσ(D) and Nσ(D)
will denote the positive and negative parts of the Nakayama-Zariski decomposition of D (see [12,
Chapter 3] for details). κ(D) will denote the Iitaka dimension and ν(D) the numerical dimension
(see [6, Definition 2.10]) of D .
3. Inductive approach to generalized abundance
Proof of Theorem 2. We will follow some ideas of [1, Theorem 5.1] and [6, Lemma 4.4]. Let Φ be
the nef reduction map of KX +B +L. Let X̂ be the normalization of the closure of the graph of Φ.
We have an induced commutative diagram:
X̂
X Y
β
φ
Φ
Note that since Φ is proper and regular over an open subset of Y , E x(β) is φ-vertical. We will
make base changes that preserve this property. Let F ⊂ X̂ be a general fiber of φ. Then(
K X̂ +B X̂ +L X̂
)∣∣
F ≡ 0. (1)
Here L X̂ := β∗L and B X̂ is defined by K X̂ +B X̂ = β∗(KX +B). Now since K X̂ +B X̂ ≥ 0, we have
KF +BF ≥ 0 which implies KF +BF ∼Q 0. Indeed, suppose that KF +BF > 0. Then there exists a
curve C ⊂ F such that ((KF +BF )·C ) > 0. Then ((KF +BF +LF )·C ) > 0. This contradicts (1). We also
conclude that L X̂ |F ≡ 0.
Priyankur Chaudhuri 419
By [10, Lemma 3.1], there exists σ : Y
′ → Y birational from a smooth projective variety Y ′ such
that letting X
′
denote the normalization of the main component of X̂ ×Y Y ′ → Y ′ andφ′ : X ′ → Y ′
the induced morphism, there exists a nefQ-Cartier divisor LY ′ on Y
′
such that LX ′ ≡φ
′∗(LY ′ ). For
a divisor D , let D+ and D− denote its positive and negative parts respectively. Letting F
′
denote a
general fiber of φ
′
, note that (B−
X ′
)|F ′ ∼ 0. Thus(
KX ′ +BX ′
)∣∣
F ′ ∼
(
KX ′ +B+X ′
)∣∣∣
F ′
∼ 0.
Then by [5, Section 4], there exists a birational morphism Ỹ → Y ′ , X̃ birational to the main
component of X
′ ×Y ′ Ỹ where X̃ and Ỹ are both smooth projective such that letting φ̃ : X̃ → Ỹ
denote the induced morphism, there exists aQ-divisor∆ on X̃ such that φ̃ : (X̃ ,B+
X̃
+∆) → Ỹ is a klt
trivial fibration ([9, Definition 2.1]) where∆+ is exceptional over Ỹ and X
′
and φ̃∗OX̃ (bl∆−c) ∼=OỸ
for all l ∈N.
X̃ X
′
X̂ X
Ỹ Y
′
Y
π
φ̃ φ
′
β
φ
By [6, Lemma 2.15], the last condition implies that the vertical part (∆−)v is φ̃-degenerate (see [6,
Definition 2.14]). Letting BỸ and MỸ denote the induced discriminant and moduli divisors on Ỹ ,
we have
K X̃ +B+X̃ +∆
+−∆− ∼Q φ̃∗
(
KỸ +BỸ +MỸ
)
.
Since (∆−)v is φ̃-degenerate, it follows from the definition of BỸ that BỸ ≥ 0. Since K X̃ +B+X̃ =
π∗(KX ′ +B+X ′ )+E , where E ≥ 0 is π-exceptional, if F̃ ⊂ X̃ is a general fiber of φ̃,(
K X̃ +B+X̃
)∣∣∣
F̃
= E |F̃ .
Note that E |F̃ is an exceptional divisor for the induced birational morphism π|F̃ : F̃ → F
′
.
Indeed, assume otherwise. Then there exists U
′ ⊂ Y ′ open such that letting V ′ :=φ′−1(U ′ ), codim
(π(E) ∩V ′ ) ≤ 1. But then E can’t be π-exceptional which is a contradiction. We conclude that
κ((K X̃ +B+X̃ )|F̃ ) = ν((K X̃ +B
+
X̃
)|F̃ ) = 0. Thus we can run a relative (K X̃ +B+X̃ )-MMP with ample scaling
over Ỹ to get
X̃ X̃m
Ỹ
ψ
φ̃
φ̃m
such that letting K X̃m +B+X̃m = ψ∗(K X̃ +B
+
X̃
) and ∆X̃m := ψ∗∆, we have (K X̃m +B+X̃m )|F̃m ∼Q 0, F̃m
being the general fiber of φ̃m (see the first paragraph of the proof of [6, Lemma 4.4] for details).
Consider the induced klt-trivial fibration
φ̃m :
(
X̃m ,B X̃m +∆+X̃m −∆
−
X̃m
)
→ Ỹ .
Note: the fact that φ̃m is klt-trivial can be shown by choosing a smooth resolution of indetermi-
nacy X̃
p←−W q−→ X̃m of ψ and using the fact that p∗(K X̃ +B+X̃ ) = q
∗(K X̃m +B+X̃m )+E where E ≥ 0 is
q-exceptional. By [2, Lemma 2.6], φ̃m and φ̃ induce the same discriminant and moduli divisors
on Ỹ . In the case of φ̃m , (∆−X̃m )|F̃m ∼Q 0, thus by [3, Theorem 3.3], MỸ is b-nef and b-good. Hence,
by the arguments in the proof of [3, Theorem 4.1], we can write
K X̃ +B+X̃ +∆
+−∆− ∼Q φ̃∗
(
KỸ +∆Ỹ
)
.
420 Priyankur Chaudhuri
where (Ỹ ,∆Ỹ ) is klt. We write this as
K X̃ +B+X̃ +∆
+− (∆−)h ∼Q φ̃∗
(
KỸ +∆Ỹ
)+ (∆−)v (2)
where h and v denote the horizontal and vertical parts with respect to φ̃ respectively. As observed
above, (K X̃ + B+X̃ )|F̃ is π|F̃ -exceptional and (K X̃ + B
+
X̃
)|F̃ ∼Q (∆−)h |F̃ . Thus (∆−)h |F̃ = Nσ((K X̃ +
B+
X̃
)|F̃ )(= (K X̃ +B+X̃ )|F̃ ). Now it follows from the definition of Nσ that
Nσ
((
K X̃ +B+X̃
)∣∣∣F̃ )≤ Nσ (K X̃ +B+X̃ )∣∣∣F̃ . (3)
Since F̃ is a general fiber and (∆−)h has no vertical components, it follows that
(∆−)h ≤ Nσ
(
K X̃ +B+X̃
)
(4)
Note that Pσ(K X̃ +B+X̃ ) is effective. Indeed, let µ : X̃ → X be the induced birational morphism.
Then we have K X̃ +B+X̃ =µ
∗(KX +B)+B−X̃ where B
−
X̃
is effective and µ-exceptional. Thus
κ
(
K X̃ +B+X̃
)
= κ(µ∗(KX +B))= κ(KX +B) ≥ 0.
Then κ(Pσ(K X̃ +B+X̃ )) = κ(K X̃ +B
+
X̃
) ≥ 0 by [6, Lemma 2.9]. It follows that K X̃ +B+X̃ − (∆
−)h ≥ 0.
By [12, Lemma 1.8], (4) implies that Nσ(K X̃ + B+X̃ − (∆
−)h) = Nσ(K X̃ + B+X̃ ) − (∆
−)h and thus
Pσ(K X̃ + B+X̃ − (∆
−)h) = Pσ(K X̃ + B+X̃ ). Now we take Pσ in (2). Since κ(D) = κ(Pσ(D)) for any
pseudoeffective divisor D ([6, Lemma 2.9]), the fact that ∆+ and B−
X̃
are exceptional over X , the
φ̃-degeneracy of (∆−)v and [6, Lemma 2.16] imply that
κ(KX +B) = κ
(
KỸ +∆Ỹ
)
(5)
Then (2) gives
K X̃ +B+X̃ +L X̃ +∆
+− (∆−)h ≡ φ̃∗ (KỸ +∆Ỹ +LỸ )+ (∆−)v (6)
where ∆+ is exceptional over X
′
and Ỹ and (∆−)v is φ̃-degenerate. Since L X̃ |F̃ ≡ 0, as observed
above, we have
Nσ
((
K X̃ +B+X̃ +L X̃
)∣∣∣F̃ )= Nσ ((K X̃ +B+X̃ )∣∣∣F̃ )≥ (∆−)h |F̃ implying (∆−)h ≤ Nσ (K X̃ +B+X̃ +L X̃ )
and exactly as before, we conclude that
Pσ
(
K X̃ +B+X̃ +L X̃ +∆
+− (∆−)h
)
= Pσ
(
K X̃ +B+X̃ +L X̃ +∆
+
)
.
But the latter equals
Pσ
(
µ∗ (KX +B +L)+B−X̃ +∆
+
)
= Pσ
(
µ∗(KX +B +L)
)=µ∗ (KX +B +L)
since ∆+ and B−
X̃
are exceptional over X and KX +B +L is nef. Now taking Pσ in (6) and using [6,
Lemma 2.16], we get
µ∗ (KX +B +L) = Pσ
(
φ̃∗
(
KỸ +∆Ỹ +LỸ
))
.
By [10, Theorem 5.3], there exists a smooth projective Y and w : Y → Ỹ birational such that
Pσ(w∗(KỸ +∆Ỹ + LỸ )) is numerically equivalent to a semiample divisor. Letting X denote a
desingularization of the main component of X̃ ×Ỹ Y and v : X → X̃ , φ : X → Y the induced
morphisms, we have
φ
∗
Pσ
(
w∗
(
KỸ +∆Ỹ +LỸ
))= Pσ (φ∗w∗ (KỸ +∆Ỹ +LỸ ))(
sincePσ
(
w∗
(
KỸ +∆Ỹ +LY
))
is nef; see [10, Lemma 2.5]
)
= Pσ
(
v∗φ̃∗
(
KỸ +∆Ỹ +LỸ
))= v∗Pσ (φ̃∗ (KỸ +∆Ỹ +LỸ ))(
since Pσ
(
φ̃∗
(
KỸ +∆Ỹ +LỸ
))
is nef
)= v∗µ∗(KX +B +L)
(∗)
is numerically equivalent to a semiample divisor. Thus KX +B +L is numerically equivalent to a
semiample divisor. 
Priyankur Chaudhuri 421
Remark 8. The above result can also be proved by using the MMP instead of the canonical
bundle formula. See for example the proof of [11, Theorem 3.5].
Proof of Corollary 3.
(1) If n(KX +B +L) = 2, then dim Ỹ = 2 and KỸ +∆Ỹ ≥ 0 by (5). Termination of flips for three-
fold pairs and abundance for surface and threefold pairs are classical. Semiampleness
and generalized non-vanishing conjectures hold for surface pairs by [10, Corollary C,
page 4]. Thus the case n(KX +B +L) = 2 i.e. when dim Ỹ = 2 is immediate.
(2) If n(KX +B +L) = 3, then dim Ỹ = 3 and by (5), κ(KỸ +∆Ỹ ) > 0. Then by [10, Remark 5.4],
Pσ(w∗(KỸ +∆Ỹ +LỸ )) is numerically equivalent to a semiample divisor for some bira-
tional morphism w : Y → Ỹ from a smooth projective variety Y . Then so is KX +B +L by
(∗) above. 
Acknowledgements
I thank my advisor Patrick Brosnan for several helpful conversations on the contents of this note,
Vladimir Lazić for his comments on a draft version, Brian Lehmann for answering a question and
the referee whose comments improved the readability of this paper. An email conversation with
Florin Ambro provided the impetus for this work.
References
[1] F. Ambro, “The nef dimension of log minimal models”, https://arxiv.org/abs/math/0411471, 2004.
[2] ——— , “Shokurov’s Boundary property”, J. Differ. Geom. 67 (2004), no. 2, p. 229-255.
[3] ——— , “The moduli b-divisor of an lc-trivial fibration”, Compos. Math. 141 (2005), no. 2, p. 385-403.
[4] T. Bauer, F. Campana, T. Eckl, S. Kebekus, T. Peternell, S. Rams, T. Szemberg, L. Wotzlaw, “A reduction map for nef line
bundles”, in Complex geometry (Göttingen, 2000), Springer, 2002, p. 27-36.
[5] O. Fujino, S. Mori, “A canonical bundle formula”, J. Differ. Geom. 56 (2000), no. 1, p. 167-188.
[6] Y. Gongyo, B. Lehmann, “Reduction maps and minimal model theory”, Compos. Math. 149 (2013), no. 2, p. 295-308.
[7] Y. Kawamata, K. Matsuda, K. Matsuki, “Introduction to the minimal model problem”, in Algebraic geometry, Sendai,
1985, Advanced Studies in Pure Mathematics, vol. 10, North-Holland, 1987, p. 283-360.
[8] J. Kollár, S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge
University Press, 1998, with the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese
original. ISBN: 0-521-63277-3.
[9] V. Lazić, E. Floris, “A travel guide to the canonical bundle formula”, in Birational Geometry and Moduli Spaces,
Springer INdAM Series, vol. 39, Springer, 2020, p. 37-55.
[10] V. Lazić, T. Peternell, “On Generalised Abundance, I”, Publ. Res. Inst. Math. Sci. 56 (2020), no. 2, p. 353-389.
[11] ——— , “On Generalised Abundance, II”, Peking Math. J. 3 (2020), no. 1, p. 1-46.
[12] N. Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, 2004.


An inductive approach to generalized abundance using nef reduction

Rémi Lodh

Comptes Rendus
Mathématique
Rémi Lodh
The connectedness of degeneracy loci in positive characteristic
Volume 361 (2023), p. 959-964
Published online: 7 September 2023
https://doi.org/10.5802/crmath.448
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 959-964
https://doi.org/10.5802/crmath.448
Algebraic geometry / Géométrie algébrique
The connectedness of degeneracy loci in
positive characteristic
Rémi Lodha
a Springer-Verlag, Tiergartenstr. 17, 69121 Heidelberg, Germany
E-mail: remi.shankar@gmail.com
Abstract. A well-known result of Fulton–Lazarsfeld ensures the connectedness of degeneracy loci under an
ampleness condition. We extend it to positive characteristic, along with the variants for degeneracy loci of
symmetric and alternating maps of even rank, due to Tu in characteristic zero. The proof uses the explicit
determination of the top étale cohomology group of an algebraic variety, a result communicated by Esnault.
2020 Mathematics Subject Classification. 14F20, 14N05, 14J60, 14F06, 14M12, 14F45, 14E15, 14G17.
Manuscript received 26 October 2021, revised 14 December 2022, accepted 23 November 2022.
1. Introduction
It is classical that a hyperplane section of a projective variety of dimension greater than 1 is
connected. This fundamental fact can be rephrased in terms of the connectedness of the zero
scheme of a global section of an ample line bundle. This latter statement admits a generalisation
to degeneracy loci, due to Fulton–Lazarsfeld [4] (see also [11, §7.2]). Recall that, for a map of
vector bundles φ : E → F , the degeneracy locus Dr (φ) is the zero scheme of ∧r+1φ : ∧r+1E →
∧r+1F .
In this note we extend the Fulton–Lazarsfeld theorem to positive characteristic, namely
Theorem 1. Let k be a field and X a proper k-scheme of dimension d. Letφ : E →F be a morphism
of vector bundles on X , of ranks e and f . If E∨⊗OX F is ample, then Dr (φ) is
(i) nonempty if d ≥ (e − r )( f − r )
(ii) geometrically connected if d > (e − r )( f − r ) and X is geometrically irreducible.
This was shown in [4] for k = C and the proof given there also works in positive characteristic
if X is smooth. For singular X in positive characteristic, Fulton–Lazarsfeld [5] deduced (i) from
deep results in intersection cohomology. More recently, Flenner–Ulrich [3] showed that the
conclusions of Theorem 1 hold in characteristic p > 0 under the stronger assumption that
E∨⊗OX F is cohomologically p-ample.
As an immediate consequence of Theorem 1, we have the following result which, surprisingly,
appears to have been open.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
960 Rémi Lodh
Corollary. Let k be a field, X a proper, geometrically irreducible k-scheme, and F an ample
vector bundle on X . If dim X > rkF , then the zero scheme of a global section of F is geometrically
connected.
There are variants of Theorem 1 for symmetric and alternating degeneracy loci, conjectured
in [5] and shown by Tu [14] in characteristic zero for r ∈ 2Z, and by Flenner–Ulrich [3] for arbitrary
r in characteristic p > 0 under their standing cohomologically p-ample assumption. Here we
extend Tu’s results to positive characteristic, obtaining
Theorem 2. Let k be a field (resp. a field of characteristic other than 2) and X a proper k-scheme
of dimension d. On X , let E be a vector bundle of rank e, L a line bundle, and ϕ : E⊗2 → L an
alternating (resp. symmetric) map with induced map φ : E → E∨⊗OX L .
If L ⊗OX ∧2E∨ is ample (resp. L ⊗OX Sym2 E∨ is ample), then, for an even integer r , Dr (φ) is
(i) nonempty if d ≥ (e−r2 ) (resp. d ≥ (e−r+12 ))
(ii) geometrically connected if d > (e−r2 ) (resp. d > (e−r+12 )) and X is geometrically irreducible.
As in Theorem 1, case (i) has been known for a long time (without restriction on r or k) thanks
to Fulton–Lazarsfeld [5]; we have only included it to emphasise that cases (i) and (ii) are proved
here in parallel. Note that the hypothesis r ∈ 2Z is not restrictive in the alternating case. The
restriction to characteristic ̸= 2 in the symmetric case is necessary in order to interpret Sym2 E as
a space of maps E∨ → E .
Apart from a single topological lemma, the proofs given in [4] and [14] can be made charac-
teristic free. So what remains is to find a suitable substitute for that lemma, and this is the orig-
inal content of this note. The required result in étale cohomology (Corollary 7) turns out to be
a straightforward consequence of de Jong’s theorem on alterations. In fact, as pointed out to us
by Esnault, one can use Gabber’s refinement of de Jong’s theorem to precisely determine the top
étale cohomology group, which easily implies the required result; see Proposition 3.
There are further generalisations of Theorems 1 and 2 that may be deduced from the results
of this paper, for instance the generalisation to n-ample vector bundles of [15] and [13]. Broadly
speaking, any generalisation based on the approach of [4] should be accessible.
Conventions
If E is a vector bundle (i.e., a locally free coherent sheaf) on X we write E∨ = HomOX (E ,OX )
for the dual bundle. If s : OX → E a section, then the zero scheme of s is the closed subscheme
Z (s) ⊂ X whose defining ideal sheaf is the image of the dual map š : E∨ → OX . If φ : E → F is a
map of vector bundles, then the zero scheme ofφ is defined to be the zero scheme of the induced
section OX → E∨⊗OX F .
Our convention for projective bundles and Grassmannians is [7], the opposite of [4, 14]. With
this convention, a vector bundle E is ample if and only if the tautological quotient line bundle
OP(E )(1) on P(E ) is ample [8, 3.2].
To simplify the notation, when k is separably closed and (n,char(k)) = 1 we will fix an
isomorphism Z/n(1) ≃Z/n, and omit to write Tate twists.
All cohomology is étale. Cohomology with compact support is denoted H ic (X ,−).
2. A result in étale cohomology
In this section we determine the top étale cohomology group of algebraic varieties and derive
some useful consequences. The main result, communicated by Esnault, is of independent inter-
est. Its proof uses Gabber’s refinement of de Jong’s alterations.
Rémi Lodh 961
Let k be an algebraically closed field, X be a separated k-scheme of finite type, d := dim X , and
n an integer prime to char(k).
Proposition 3 (Esnault). We have
H 2d (X ,Z/n) =⊕Y Z/n
where the direct sum is taken over the irreducible components Y of X of dimension d which are
proper over k. Moreover, the trace map is an isomorphism on each direct summand.
We first prove a lemma.
Lemma 4.
(a) If X is irreducible but not proper, then H 2d (X ,Z/n) = 0.
(b) If X̃ is the disjoint union of the proper irreducible components of dimension d of X , then
the canonical map H 2d (X ,Z/n) → H 2d (X̃ ,Z/n) is an isomorphism.
Proof. We may assume X is reduced.
(a). First assume X is smooth. Since X is not proper we have H 0c (X ,Z/n) = 0, hence the result
follows by Poincaré duality.
We reduce the general case to the smooth case as follows. Firstly, we may assume n = l m for a
prime l ̸= char(k). By Gabber’s theorem on alterations [9, 2.1], there exists an alterationπ : X ′ → X
with X ′ smooth, connected and quasi-projective, withπ of degree prime to n. Note that X ′ cannot
be proper, so H 2d (X ′,Z/n) = 0.
Let j : U ,→ X be a dense open such that πU := π|U is finite flat, U ′ =U ×X X ′, and j ′ : U ′ → X ′
be the inclusion. Since πU is finite flat, composition of the adjunction map
Z/n −→πU ,∗π∗UZ/n =πU ,∗Z/n (1)
with the trace map for πU is multiplication by the degree on Z/n [2, 2.9]. On the other hand, note
that j ◦πU = π ◦ j ′, so j!πU ,∗Z/n = Rπ∗ j ′!Z/n and the global sections of this complex compute
H∗(X ′, j ′!Z/n). Therefore, for all i the canonical map
H i (X , j!Z/n) −→ H i (X ′, j ′!Z/n)
obtained by applying j! to (1) has kernel annihilated by the degree of π, hence is injective by
choice of π and n. So it suffices to show that H 2d (X , j!Z/n) = H 2d (X ,Z/n) and similarly for X ′.
This follows from the exact sequence
H 2d−1(X \U ,Z/n) −→ H 2d (X , j!Z/n) −→ H 2d (X ,Z/n) −→ H 2d (X \U ,Z/n)
because U is dense in X , hence dim(X \U ) < d . Similarly for X ′.
(b). Let X̃ ′ be the disjoint union of all of the irreducible components of X and µ : X̃ ′ → X the
canonical finite map. Consider the adjunction map
Z/n −→µ∗µ∗Z/n = Rµ∗Z/n
Since the kernel and cokernel of this map are torsion sheaves supported on subschemes of
dimension at most d −1, it follows that the canonical map
H 2d (X ,Z/n) −→ H 2d (X̃ ′,Z/n)
is an isomorphism. Now apply (a) to X̃ ′. □
Proof of Proposition 3. By Lemma 4 we may assume X is proper and irreducible. Furthermore,
we may assume X to be reduced. If U ⊂ X is a smooth dense open, then dim(X \ U ) < d , so
H 2d (X ,Z/n) = H 2dc (U ,Z/n), with the latter isomorphic toZ/n via the trace by Poincaré duality [2,
3.2.6]. □
962 Rémi Lodh
Corollary 5. If X is proper and Z ⊂ X is a closed subset intersecting an irreducible component of
dimension d, the trace map H 2dZ (X ,Z/n) →Z/n is surjective.
Proof. Let X ′ ⊂ X be the union of the irreducible components meeting Z . Then dim X ′ = d by
assumption, and H 2dZ (X
′,Z/n) = H 2dZ (X ,Z/n) by excision. So we may assume X = X ′, i.e. Z meets
every irreducible component Y of X . In that case, no irreducible component of X \ Z is proper,
hence H 2d (X \ Z ,Z/n) = 0 by Proposition 3. Now we have the exact sequence
H 2dZ (X ,Z/n) −→ H 2d (X ,Z/n) −→ H 2d (X \ Z ,Z/n)
and the trace map H 2d (X ,Z/n) →Z/n is surjective by Proposition 3. □
Corollary 6. If x ∈ X is a closed point, then the trace map H 2dx (X ,Z/n) →Z/n is surjective, where
d = dimx X .
Proof. Use excision to reduce to the case X is projective and apply Corollary 5. □
In the sequel we will make use of the following.
Corollary 7. Fix a prime l ̸= char(k). Assume X is proper and let Z ⊂ X be closed.
(a) If H 2d (X \ Z ,Ql ) = 0, then Z ̸= ;.
(b) If X is irreducible and H 2d−1(X \ Z ,Ql ) = 0 = H 2d (X \ Z ,Ql ), then Z is connected.
Proof. (a) follows at once from Proposition 3. In case (b) we find
H 2dZ (X ,Ql ) = H 2d (X ,Ql )
so by Proposition 3 we get dim H 2dZ (X ,Ql ) = 1. If Z = Z1 ⊔ Z2 is the disjoint union of two closed
subspaces, then H 2dZ (X ,Ql ) = H 2dZ1 (X ,Ql )⊕ H 2dZ2 (X ,Ql ), so H 2dZi (X ,Ql ) = 0 for some i , say i = 2.
Then it follows from Corollary 5 that Z2 =;, whence Z = Z1. Thus, Z is connected. □
Remark. One can prove Corollary 7 using only de Jong’s alteration theorem [10], avoiding
Gabber’s more delicate result. This is what we had originally done, and it sufficed for Theorems 1
and 2.
3. Proofs of the theorems
We first make some reductions. Clearly, we may assume k algebraically closed. If s : OX → G is a
global section of a vector bundle, then from the exact sequence
G∨ š−→OX −→OZ (s) −→ 0
we see that for a morphism Y → X the zero scheme satisfies Z (s)×X Y = Z (s|Y ). In particular,
for Y = X red we find that Z (s|X red ) → Z (s) is a homeomorphism. Since the pullback of an ample
vector bundle by a finite morphism is ample, to show a topological property of Dr (φ) we may
assume X is reduced. And in case (i), taking Y to be an irreducible component of X of dimension
d , we may also assume X is irreducible.
3.1. The codimension of simple degeneracy loci
A key input to the arguments of [4] and [14] is the computation of the codimension of some
simple degeneracy loci (cf. [4, p. 276] and [14, p. 388]), used almost implicitly. We provide a proof
here, which also demonstrates how one arrives at the expected codimensions.
Let E be a vector space of dimension e over a field k of characteristic different from 2. Let
P := P((Sym2 E)∨) and E := E ⊗k OP . Since char(k) ̸= 2, we have (Sym2 E)∨ = Sym2(E∨), hence on
P there is a canonical map ϕ : E∨ → E (1) := E ⊗OP OP (1).
Rémi Lodh 963
Proposition 8. For 0 ≤ r ≤ e, Dr := Dr (ϕ) ⊂ P is an irreducible subscheme of codimension
(e−r+1
2
)
.
Proof. Let G = Grasse−r (E) and Q the tautological quotient on G . On P ×G we have the vector
bundle E := ker(E →Q).
Consider the closed subscheme P :=P(Sym2 E∨) ⊂ P ×G . A simple dimension count, using the
well-known dimensions of projective bundles and Grassmannians, yields dimP = dimP−(e−r+12 ).
Lemma 9. The canonical map P → P induced by the projection P ×G → P factors over Dr .
Moreover, the resulting map P → Dr is surjective and an isomorphism over a dense open subset
of (Dr )red.
Proof. The key observation is that P is the zero subscheme of the composition E∨ → E (1) →
Q(1). This follows easily from the exact sequence
Q∨⊗E∨ −→ Sym2 E∨ −→ Sym2 E∨ −→ 0
In particular, on P the canonical map E∨ → E (1) has rank at most r , hence the first assertion.
The surjectivity of this map is readily checked on fibres. In particular, Dr is irreducible.
Consider the open subset U := Dr \ Dr−1 of Dr . It is nonempty because symmetric e ×e matrices
of rank r exist (take the r × r identity matrix and add zeros). We claim that PU → U red is an
isomorphism. Indeed, it is easily checked that its fibres are trivial, so the map is unramified, and
by [6, 17.4.1] it suffices find a section. But over U red the cokernel of the map E∨ → E (1) is a vector
bundle of rank e − r , which defines a section. □
Thus, dimDr = dimP and, since dimP = dimP −
(e−r+1
2
)
, this concludes the proof of Proposi-
tion 8. □
The alternating case is analogous, replacing Sym by ∧ throughout, and also works in charac-
teristic 2.
For the (easier) variant of Proposition 8 required for the proof of Theorem 1 we consider
another k-vector space F of dimension f and let P = P(E∨⊗F ). On P there is a canonical map
ϕ : F → E (1), where F := F ⊗k OP .
Proposition 10. For 0 ≤ r ≤ min{e, f }, Dr (ϕ) ⊂ P is an irreducible subscheme of codimension
(e − r )( f − r ).
Proof. Define G and E as above, and let P =P(E∨⊗F ). Then P ⊂ P ×G is the zero scheme of the
composition F
ϕ→ E (1) →Q(1) and we can argue as before. The details are left to the reader. □
3.2. Proofs of Theorems 1 and 2
Fix a prime l ̸= char(k). The basic strategy of [4] and [14] for proving Theorems 1 and Theorem 2
is as follows. A Grassmannian G over X is constructed with a closed subset Y ⊂ G (denoted
Z (t ) in [14]) mapping surjectively onto Dr (φ), and it suffices to show that Y is nonempty (resp.
connected) in case (i) (resp. (ii)). By Corollary 7, this reduces to showing H j (G \ Y ,Ql ) = 0 for
j = 2dimG (resp. j ≥ 2dimG−1).
Proof of Theorem 1. The point is that the analogue of [4, Prop. 1.2] in l-adic étale cohomology
holds over an arbitrary algebraically closed field, cf. [4, Rem. 1.7]. Indeed, the constructions
are geometric and, besides the dimension count of Proposition 10, the proof only relies on the
classical finiteness and vanishing results of SGA 4. In fact, the only place in the proof of [4,
Thm. 1.1] where the hypothesis k = C is necessary is [4, Lem. 1.3], which is not used in the proof
of [4, Prop. 1.2].
From [4, Prop. 1.2] we obtain H j (G\Y ,Ql ) = 0 for j = 2dimG in case (i) (resp. for j ≥ 2dimG−1
in case (ii)), proving Theorem 1. □
964 Rémi Lodh
Proof of Theorem 2. Again most of the arguments of [14] hold in our setting with singular
cohomology replaced by l -adic étale cohomology.
In [14, §1], the assumption k = C can be replaced by k algebraically closed and char(k) ̸= 2
(cf. [1, III, §3], [12, Ch. 2]). However, the stated results of [14, §1] also hold for char(k) = 2 in the
alternating case by [1, III, §3].
Then, in [14, §2], the conclusion of [14, Prop. 2.1] should be replaced by: P(E∗)− Z (s∗) is an
affine space bundle on X −Z (s).
Now, after replacing [14, Lem. 3.3] by our Corollary 7, the results of [14, §3–4] hold with étale
cohomology (here one needs Proposition 8), without any smoothness hypothesis.
This proves Theorem 2 in the symmetric case when L is trivial. To remove this latter hypoth-
esis, follow the argument of [14, §5] from [14, Lem. 5.1] onwards. For the alternating case, apply
the argument of [14, §6]. □
Acknowledgments. I am very grateful to Hélène Esnault for helpful comments, in particular for
communicating the statement and proof of Proposition 3, which greatly improved my original
exposition. I also thank Lorenzo Ramero for commenting on an early version and this journal’s
editor for helpful advice.
References
[1] E. Artin, Geometric Algebra, Interscience Tracts in Pure and Applied Mathematics, vol. 3, Interscience Publishers,
1957.
[2] P. Deligne, “Exposé XVIII: La formule de dualité globale”, in Théorie des topos et cohomologie étale des schémas (SGA
4), tome 3 (M. Artin, A. Grothendieck, J.-L. Verdier, P. Deligne, B. Saint-Donat, eds.), Lecture Notes in Mathematics,
vol. 305, Springer, 1973.
[3] H. Flenner, B. Ulrich, “Codimension and connectedness of degeneracy loci over local rings”, Math. Z. 286 (2017),
no. 1-2, p. 723-740.
[4] W. Fulton, R. Lazarsfeld, “On the connectedness of degeneracy loci and special divisors”, Acta Math. 146 (1981),
p. 271-283.
[5] ——— , “Positive polynomials for ample vector bundles”, Ann. Math. 118 (1983), p. 35-60.
[6] A. Grothendieck, “Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de sché-
mas (Quatrième partie)”, Publ. Math., Inst. Hautes Étud. Sci. 32 (1967), p. 5-361, written in collaboration with J.
Dieudonné.
[7] A. Grothendieck, J. A. Dieudonné, Éléments de géométrie algébrique I, Grundlehren der Mathematischen Wis-
senschaften, vol. 166, Springer, 1971.
[8] R. Hartshorne, “Ample vector bundles”, Publ. Math., Inst. Hautes Étud. Sci. 29 (1966), p. 63-94.
[9] L. Illusie, M. Temkin, “Exposé X. Gabber’s modification theorem (log smooth case)”, in Travaux de Gabber sur
l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l’École Polytechnique 2006–
2008 (L. Illusie, Y. Laszlo, F. Orgogozo, eds.), Astérisque, no. 363-364, Société Mathématique de France, 2014, p. 167-
212.
[10] A. J. de Jong, “Smoothness, semi-stability and alterations”, Publ. Math., Inst. Hautes Étud. Sci. 83 (1996), p. 51-93.
[11] R. Lazarsfeld, Positivity in Algebraic Geometry II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 49,
Springer, 2004.
[12] W. Scharlau, Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften, vol. 270, Springer,
1985.
[13] S.-K. Tin, “Numerically positive polynomials for k-ample vector bundles”, Math. Ann. 294 (1992), no. 4, p. 579-590.
[14] L. W. Tu, “The connectedness of symmetric and skew-symmetric degeneracy loci: even ranks”, Trans. Am. Math. Soc.
313 (1989), no. 1, p. 381-392.
[15] ——— , “The connectedness of degeneracy loci”, in Topics in algebra. Part 2: Commutative rings and algebraic
groups, vol. 26, Banach Center Publications, no. 2, PWN-Polish Scientific Publishers, 1990, p. 235-248.


The connectedness of degeneracy loci in positive characteristic

Robert E. Gaunt

Siqi Li

The variance-gamma ratio distribution

Arka Mallick

Hoai-Minh Nguyen

The Caffarelli–Kohn–Nirenberg inequalities for radial functions

Jinghao Huang

Fedor Sukochev

Derivations with values in noncommutative symmetric spaces

Georgios Dalezios

Ioannis Emmanouil

Homological dimension based on a class of Gorenstein flat modules

Jean-Paul Dufour

Daniel Lehmann

Étude des $(n+1)$-tissus de courbes en dimension $n$

Cristina Ballantine

Mircea Merca

Comptes Rendus
Mathématique
Cristina Ballantine and Mircea Merca
Congruences modulo 4 for the number of 3-regular partitions
Volume 361 (2023), p. 1577-1583
https://doi.org/10.5802/crmath.512
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 1577-1583
https://doi.org/10.5802/crmath.512
Combinatorics, Number theory / Combinatoire, Théorie des nombres
Congruences modulo 4 for the number of
3-regular partitions
Cristina Ballantine ∗,a andMirceaMerca b,c
a Department of Mathematics and Computer Science, College of The Holy Cross, Worcester, MA 01610, USA
b Department of Mathematical Methods and Models, Fundamental Sciences Applied in Engineering
Research Center, University Politehnica of Bucharest, RO-060042 Bucharest, Romania
c Academy of Romanian Scientists, RO-050044, Bucharest, Romania
E-mails: cballant@holycross.edu, mircea.merca@profinfo.edu.ro
Abstract. The last decade has seen an abundance of congruences for bℓ(n), the number of ℓ-regular parti-
tions of n. Notably absent are congruences modulo 4 for b3(n). In this paper, we introduce Ramanujan type
congruences modulo 4 for b3(2n) involving some primes p congruent to 11,13,17,19,23 modulo 24.
Keywords. partitions, regular partitions, congruences.
2020 Mathematics Subject Classification. 11P83, 05A17, 11F33.
Manuscript received 14 March 2023, revised 19 April 2023, accepted 6 May 2023.
1. Introduction
A partition of a positive integer n is a weakly decreasing sequence of positive integers whose sum
is n. The positive integers in the sequence are called parts. For more on the theory of partitions,
we refer the reader to [1].
For an integer ℓ > 1, a partition is called ℓ-regular if none of its parts is divisible by ℓ. The
number of the ℓ-regular partitions of n is usually denoted by bℓ(n) and its arithmetic properties
were investigated extensively. See, for example, [3–7, 10–12, 17–20, 22]. The generating function
for bℓ(n) is given by ∞∑
n=0
bℓ(n) q
n = (q
ℓ; qℓ)∞
(q ; q)∞
.
Here and throughout q is a complex number with |q | < 1, and the symbol (a; q)∞ denotes the
infinite product
(a; q)∞ =
∞∏
n=0
(1−a qn).
In a recent paper, W. J. Keith and F. Zanello [9] discovered infinite families of Ramanujan type
congruences modulo 2 for b3(2n) involving every prime p with p ≡ 13,17,19,23 (mod 24).
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1578 Cristina Ballantine and Mircea Merca
Theorem 1 (Keith–Zanello). The sequence b3(2n) is lacunary modulo 2. If p ≡ 13,17,19,23
(mod 24) is prime, then
b3
(
2(p2n +pk −24−1))≡ 0 (mod 2)
for 1⩽ k ⩽ p −1, where 24−1 is taken modulo p2.
Motivated by the Keith–Zanello result, O. X. M. Yao [23] provided new infinite families of
Ramanujan type congruences modulo 2 for b3(2n) involving every prime p ⩾ 5.
Theorem 2 (Yao). Let p ⩾ 5 be a prime.
(1) If b3
(
p2−1
12
)
≡ 1 (mod 2), then for n,k ⩾ 0
b3
(
2p4k+4 n +2p4k+3 j + p
4k+4 −1
12
)
≡ 0 (mod 2)
where 1⩽ j ⩽ p −1 and for n,k ⩾ 0
b3
(
p4k −1
12
)
≡ 1 (mod 2).
(2) If b3
(
p2−1
12
)
≡ 0 (mod 2), then for n,k ⩾ 0 with p ∤ (24n +1)
b3
(
2p6k+2 n + p
6k+2 −1
12
)
≡ 0 (mod 2)
and for n,k ⩾ 0
b3
(
p6k −1
12
)
≡ 1 (mod 2).
Very recently, Ballantine, Merca and Radu [2] introduced new infinite Ramanujan type con-
gruences modulo 2 for b3(2n). They complement naturally the results of Keith–Zanello and Yao
and involve primes in the set
P = {p prime : ∃ j ∈ {1,4,8}, x, y ∈Z, gcd(x, y) = 1 with x2 +216y2 = j p}
whose Dirichlet density is 1/6.
Theorem 3. For every p ∈P and n ⩾ 0, we have
b3
(
2(p2 n +pα−24−1p )
)≡ 0 (mod 2),
where 0 ⩽α< p, α ̸= ⌊p/24⌋, and 24−1p is the inverse of 24 modulo p taken such that 1 ⩽−24−1p ⩽
p −1.
In this work, motivated by the results on the parity of b3(2n), we investigate Ramanujan type
congruences modulo 4 for b3(2n). We note that congruences modulo 4 for ℓ-regular partitions,
have been studied in [8] for ℓ = 4,5,9, and in [15] for ℓ = 2. However, congruences modulo 4 for
3-regular partitions are missing from the literature.
Theorem 4. For every p ∈ {43,47,59,61,67,89,137,139,157} and n ⩾ 0 we have
b3
(
2(p2 n +pα−24−1p )
)≡ 0 (mod 4), (1)
where 0⩽α< p, α ̸= ⌊p/24⌋, 24−1p is the inverse of 24 modulo p taken such that 1⩽−24−1p ⩽ p −1.
Cristina Ballantine and Mircea Merca 1579
We conjecture that there are infinitely many primes for which (1) holds. For example, we
verified numerically that, in addition to the primes in Theorem 4, the statement of the theorem
holds for
p ∈ {167,181,229,233,277,331,359,379,401,419,421,431,443,479,499,541,
569,593,599,613,643,647,691,709,719,757,761,787,809,827,829,853,
859,863,877,911,929,953,977,983,1021,1031}.
We were unable to finish the proof of Theorem 4 for these values of p due to computing time
limitations.
2. Proof of Theorem 4
2.1. Modular forms
As is the case with many proofs of congruences in the literature, we use [13, Lemma 4.5]. For
the conveniece of the reader, we first introduce all necessary notation and the statement of [13,
Lemma 4.5]. This exposition is nearly identical to that in [2].
Let Γ := SL(2,Z), and define
Γ∞ :=
{(
1 b
0 1
)
∈ Γ
}
.
For a positive integer N , we define the congruence subgroup
Γ0(N ) :=
{(
a b
c d
)
∈ Γ : c ≡ 0 (mod N )
}
.
If M is a positive integer, we write R(M) for the set of finite integer sequences r = (rδ1 ,rδ2 , . . . ,rδk ),
where 1 = δ1 < δ2 < ·· · < δk = M are the positive divisors of M . We note for the remainder of
this section we only consider positive divisors of a given integer. Given a positive integer m, we
denote by S24m the set of invertible quadratic residues modulo 24m and, for fixed 0 ⩽ t ⩽ m −1,
we define
Pm,r (t ) :=
{
t s + s −1
24
∑
δ|M
δrδ (mod m) : s ∈ S24m
}
.
Let m, M and N be positive integers. Moreover, let t be an integer such that 0⩽ t ⩽m −1 and
let r = (rδ) ∈ R(M). We set κ := gcd(1−m2,24) and write
∏
δ|M δ|rδ| =: 2s v , where s is a nonnegative
integer and v is odd. Then, we say that the tuple (n, M , N , (rδ), t ) ∈ ∆∗ if and only if all of the
following six conditions are satisfied.
(1) p | m, p prime, implies p | N ;
(2) δ | M , δ⩾ 1 such that rδ ̸= 0 implies δ | mN ;
(3) κN
∑
δ|M rδ mNδ ≡ 0 (mod 24);
(4) κN
∑
δ|M rδ ≡ 0 (mod 8);
(5) 24mgcd(κ(−24t−∑δ|M δrδ),24m)
∣∣∣N ;
(6) If 2 | m, then (4 | κN and 8 | N s) or (2 | s and 8 | N (1− v)).
Finally, for γ= (a bc d ) ∈ Γ, and m and r = (rδ) ∈ R(M) as above, we define
pm,r (γ) := min
d∈{0,...,m−1}
1
24
∑
δ|M
rδ
gcd2(δ(a +κdc),mc)
δm
and for a = (aδ) ∈ R(N ), we define
p∗a (γ) :=
1
24
∑
δ|N
aδ
gcd2(δ,c)
δ
.
1580 Cristina Ballantine and Mircea Merca
We use the notation ∞∑
n=0
c(n) qn ≡
∞∑
n=0
d(n) qn (mod u)
to mean that c(n) ≡ d(n) (mod u) for all n ⩾ 0. Similarly,
∞∑
n=0
c(n) qn ≡ 0 (mod u)
means c(n) ≡ 0 (mod u) for all n ⩾ 0.
Lemma 5 ([13, Lemma 4.5]). Let u be a positive integer, (m, M , N , t ,r = (rδ)) ∈ ∆∗, a = (aδ) ∈
R(N ). Let {γ1, . . . ,γn} ⊂ Γ be a complete set of representatives of the double cosets in Γ0(N )\Γ/Γ∞.
Assume that pm,r (γi )+p∗a (γi )⩾ 0 for all 0⩽ i ⩽ n. Let tmin := mint ′∈Pm,r (t ) t ′ and
ν := 1
24
((∑
δ|N
aδ+
∑
δ|M
rδ
)
[Γ : Γ0(N )]−
∑
δ|N
δaδ
)
− 1
24m
∑
δ|M
δrδ−
tmin
m
.
Suppose ∏
δ|M
∞∏
n=1
(1−qδn)rδ =
∞∑
n=0
cr (n)q
n .
If
⌊ν⌋∑
n=0
cr (mn + t ′)qn ≡ 0 (mod u), for all t ′ ∈ Pm,r (t ),
then ∞∑
n=0
cr (mn + t ′)qn ≡ 0 (mod u), for all t ′ ∈ Pm,r (t ).
2.2. Proof of Theorem 4
As customary, we use the notation
fi :=
∞∏
k=1
(1−qki ).
From [21], identity (2.18), we have
∞∑
n=0
b3(2n) q
n = f2 f3 f8 f
2
12
f 21 f4 f6 f24
. (2)
Moreover, since for i ≥ 1, f 22i ≡ f 4i (mod 4), we have
f2i
fi
= f
2
2i
f 4i
f 3i
f2i
≡ f
3
i
f2i
(mod 4). (3)
Using (2) and (3), we obtain
∞∑
n=0
b3(2n) q
n ≡ f
2
1 f3 f
3
4 f
3
6
f2 f8 f24
(mod 4).
To use the Lemma 5, we write
∞∑
n=0
c(n)qn := f
2
1 f3 f
3
4 f
3
6
f2 f8 f24
= ∏
δ|M
∞∏
n=1
(1−qδn)rδ .
Thus, with the notation of Lemma 5, we take u = 4, m = p2, M = 24, and
(r1,r2,r3,r4,r6,r8,r12,r24) = (2,−1,1,3,3,−1,0,−1).
Cristina Ballantine and Mircea Merca 1581
We have κ= 24 and we calculate∑
δ|M
rδ
δ
= 35
12
,
∑
δ|M
rδ = 6,
∑
δ|M
δrδ = 1.
We take N = 24p. It is straightforward to verify that for any t = pα−24−1p conditions (1)–(6) are
satisfied.
Since
[Γ : Γ0(N )] = N
∏
x|N
(1+x−1),
where the product is taken after all prime divisors of N , we have
[Γ : Γ0(N )] = 48(p +1).
In general, it is nontrivial to find a complete set of representatives for the double cosets
in Γ0(N ) \ Γ/Γ∞. If N is square free, it is shown in [14, Lemma 2.6] that a complete set of
representatives for Γ0(N )\Γ/Γ∞ is given by
AN =
{(
1 0
δ 1
)
: δ | N , δ⩾ 1
}
.
This result has been extended to N such that N /2 is square free in [16, Lemma 4.3]. While for
N = 24p, neither N nor N /2 is square free, when m = p2,κ= 24, and (r1,r2,r3,r4,r6,r8,r12,r24) =
(2,−1,1,3,3,−1,0,−1) we can avoid finding a complete set of representatives all together.
Since for any integers i , j ≥ 1 we have
gcd( j (a +κdc),mc)⩽ gcd(i j (a +κdc),mc)⩽ i gcd( j (a +κdc),mc),
an easy calculation shows that for each γ= (a bc d ) ∈ Γ, we have∑
δ|M
rδ
gcd2(δ(a +κdc),mc)
δm
≥ 0,
and thus pm,r (γ) ≥ 0. Hence, we can use a := (aδ)δ|N with aδ = 0 for each δ | N to calculate ⌊ν⌋. It
is clear from the definition of ν in Lemma 5 that
⌊ν⌋ = 12(p +1)−1.
Let
Rp := {pα−24−1p : 0⩽α< p, α ̸= ⌊p/24⌋}.
For each p, we used Mathematicatm to write the set Rp as
Rp = Pm,r (−24−1p )∪Pm,r (Ap −24−1p )
for a minimal A. For example, when p = 43, we have A = 2 and the Mathematicatm calculation
gives
Pm,r (−24−1p ) = {34,163,206,292,378,421,593,851,894,937,1023,1195,
1238,1281,1324,1367,1453,1496,1539,1668,1754}
and
Pm,r (2 ·43−24−1p ) = {120,249,335,464,507,550,636,679,722,765,808,980,
1066,1109,1152,1410,1582,1625,1711,1797,1840}.
If α= ⌊43/24⌋ = 1 we have Pm,r (p −24−1p ) = Pm,r (77) = {77}.
For p ∈ {43,47,139,157} we obtained A = 2 and for p ∈ {59,61,67,89,137} we obtained A = 1.
Moreover, if t∗ = p⌊p/24⌋−24−1p , then Pm,r (t∗) = {t∗}.
Finally, in each case, we verified that for each t ∈Rp we have
c(p2n + t ) ≡ 0 (mod 4) for 0⩽ n ⩽ 12(p +1)−1.
1582 Cristina Ballantine and Mircea Merca
Then, [13, Lemma 4.5] implies that, for each
p ∈U := {43,47,59,61,67,89,137,139,157}
we have
c(p2n + t ) ≡ 0 (mod 4) for all n ⩾ 0 and t ∈Rp .
Our calculations show that for each prime p ∈U , we have that
c(p⌊p/24⌋−24−1p ) ̸≡ 0 (mod 4)
and thus the requirement that α ̸= ⌊p/24⌋ in the statement of Theorem 4 is necessary.
3. Concluding remarks
Several Ramanujan type congruences modulo 4 for b3(2n) involving some primes p with
p ≡ 11,13,17,19,23 (mod 24) have been proved in this paper using modular forms and [13,
Lemma 4.5].
As mentioned in the introduction, we verified the statement of Theorem 4 numerically up to
108 for many more primes equivalent to 11,13,17,19,23 modulo 24. In fact, our computations
suggest that (1) is also true for primes p = 1033 and p = 1153 which are congruent to 1 modulo
24. We did not encounter any primes congruent to 5 or 7 modulo 24 for which (1) holds. We leave
it as an open problem to characterize an infinite family of primes for which the statement of
Theorem 4 holds and to prove the theorem for all these primes.
References
[1] G. E. Andrews, The Theory of Partitions, Cambridge Mathematical Library, Cambridge University Press, 1998.
[2] C. Ballantine, M. Merca, C.-S. Radu, “Parity of 3-regular partition numbers and Diophantine equations”, 2022,
https://arxiv.org/abs/2212.09810.
[3] R. Carlson, J. J. Webb, “Infinite families of infinite families of congruences for k-regular partitions”, Ramanujan J. 33
(2014), no. 3, p. 329-337.
[4] S.-P. Cui, N. S. S. Gu, “Arithmetic properties of ℓ-regular partitions”, Adv. Appl. Math. 51 (2013), no. 4, p. 507-523.
[5] B. Dandurand, D. Penniston, “l-Divisibility of l-regular partition functions”, Ramanujan J. 19 (2009), no. 1, p. 63-70.
[6] D. Furcy, D. Penniston, “Congruences for ℓ-regular partition functions modulo 3”, Ramanujan J. 27 (2012), no. 1,
p. 101-108.
[7] M. D. Hirschhorn, J. A. Sellers, “Elementary proofs of parity results for 5-regular partitions”, Bull. Aust. Math. Soc. 81
(2010), no. 1, p. 58-63.
[8] W. J. Keith, “Congruences for m-regular partitions modulo 4”, Integers 15A (2015), article no. A11 (12 pages).
[9] W. J. Keith, F. Zanello, “Parity of the coefficients of certain eta-quotients”, J. Number Theory 235 (2022), p. 275-304.
[10] J. Lovejoy, D. Penniston, “3-regular partitions and a modular K 3 surface”, in q-series with applications to combi-
natorics, number theory, and physics, Contemporary Mathematics, vol. 291, American Mathematical Society, 2001,
p. 177-182.
[11] D. Penniston, “The pa -regular partition function modulo p j ”, J. Number Theory 94 (2002), no. 2, p. 320-325.
[12] ——— , “Arithmetic of ℓ-regular partition functions”, Int. J. Number Theory 4 (2008), no. 2, p. 295-302.
[13] S. Radu, “An algorithmic approach to Ramanujan’s congruences”, Ramanujan J. 20 (2009), no. 2, p. 215-251.
[14] S. Radu, J. A. Sellers, “Congruence properties modulo 5 and 7 for the pod function”, Int. J. Number Theory 7 (2011),
no. 8, p. 2249-2259.
[15] Ø. Rødseth, “Congruence properties of the partition functions q(n) and q0(n)”, Årbok Univ. Bergen, Mat.-Naturv. Ser.
1969, no. 13.
[16] L. Wang, “Arithmetic properties of (k,ℓ)-regular bipartitions”, Bull. Aust. Math. Soc. 95 (2017), no. 3, p. 353-364.
[17] ——— , “Congruences for 5-regular partitions modulo powers of 5”, Ramanujan J. 44 (2017), no. 2, p. 343-358.
[18] ——— , “Arithmetic properties of 7-regular partitions”, Ramanujan J. 47 (2018), no. 1, p. 99-115.
[19] J. J. Webb, “Arithmetic of the 13-regular partition function modulo 3”, Ramanujan J. 25 (2011), p. 49-56.
[20] E. X. W. Xia, “Congruences for some l-regular partitions modulo l ”, J. Number Theory 152 (2015), p. 105-117.
Cristina Ballantine and Mircea Merca 1583
[21] E. X. W. Xia, O. X. M. Yao, “New Ramanujan-like congruences modulo powers of 2 and 3 for overpartitions”, J. Number
Theory 133 (2013), no. 6, p. 1932-1949.
[22] ——— , “Parity results for 9-regular partitions”, Ramanujan J. 34 (2014), no. 1, p. 109-117.
[23] O. X. M. Yao, “New parity results for 3-regular partitions”, Quaest. Math. 46 (2023), no. 3, p. 465-471.


Congruences modulo $4$ for the number of $3$-regular partitions

Xuan Tho Nguyen

Comptes Rendus
Mathématique
Nguyen Xuan Tho
Rational points on a certain genus 2 curve
Volume 361 (2023), p. 1071-1073
Published online: 7 September 2023
https://doi.org/10.5802/crmath.471
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 1071-1073
https://doi.org/10.5802/crmath.471
Number theory / Théorie des nombres
Rational points on a certain genus 2 curve
Nguyen Xuan Thoa
a Hanoi University of Science and Technology
E-mail: tho.nguyenxuan1@hust.edu.vn
Abstract. We give a correct proof to the fact that all rational points on the curve
y2 = (x2 +1)(x2 +3)(x2 +7)
are ±∞ and (±1, ±8). This corrects previous works of Cohen [3] and Duquesne [4, 5].
2020 Mathematics Subject Classification. 14G05, 14H99.
Funding. This paper is funded by the Ministry of Education and Training of Vietnam under the project B2022-
CTT-03, 2022-2023.
Manuscript received 26 October 2022, accepted 31 January 2023.
1. Introduction
The goal of this paper is to prove the following theorems
Theorem 1. All rational points on the curve
C1 : y
2 = (x2 +1)(x2 +3)(x2 +7) (1)
are ±∞ and (±1, ±8).
The available proofs of Theorems 1 in [3] and [4] are unfortunately incorrect. The curve (1)
appeared in the work of Flynn and Wetherell [6]. However, Flynn and Wetherell also pointed out
that their method failed in determining all rational points on the curve (1). Duquesne later gave
an incorrect proof of Theorem 1 in his thesis [4, pp. 64–69]. Duquesne also reported Theorem 1
in [5, Theorem 4] and refered to his thesis for the proof. Furthermore, Duquesne’s proof was
reproduced in Cohen’s book [3, Theorem 13.3.10, pp. 459–462]. Duquesne argued as the following:
sinceQ(i ) is a unique factorization domain and
(x + i )(x2 +3)(x − i )(x2 +7) = y2,
it follows that {
αy21 = (x + i )(x2 +3),
αy22 = (x − i )(x2 +7),
whereα can be taken as a divisor of the resultant of (x+i )(x2+3) and (x−i )(x2+7). The problem is
that polynomials (x+i )(x2+3) and (x−i )(x2+7) have odd (three) degrees and so the denominator
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1072 Nguyen Xuan Tho
of x needs to be considered. Specifically, write x = X /Z , where X , Z ∈Z and gcd(X , Z ) = 1, so (1)
takes the form
Y 2 = (X 2 +Z 2)(X 2 +3Z 2)(X 2 +7Z 2).
Certainly, {
βY 21 = (X 2 +3Z 2)(X + i Z ),
βY 22 = (X 2 +7Z 2)(X − i Z ),
(2)
where β ∈Q(i ). Now
ResultantX ((X
2 +3Z 2)(X + i Z ), (X 2 +7Z 2)(X − i Z )) =−27 ·3 · i ·Z 9,
so that βmay be taken as a squarefree divisor of 2 ·3 ·Z in Z[i ]. Certainly gcd(β, Z ) = 1, otherwise
gcd(X , Z ) > 1, so β|2 ·3. Just as for α. However, the system (2) on dividing by Z 3 corresponds to
β
Z
· y21 = (x2 +3)(x + i ),
β
Z
· y22 = (x2 +7)(x − i ),
so that the original α has to depend on (the square-free part of) Z .
For a specific example, consider the curve
y2 = (x2 +1)(x2 +15)(x2 +18),
where the resultant of (x + i )(x2 +15) and (x − i )(x2 +18) equals −22 ·32 ·7 ·17i . The curve has a
rational point with x = 3/5, at which point
(x + i )(x2 +15) = (1− i )15(4+ i ) · 3
125
≡ (1− i )(4+ i ) ·3 ·5 (mod Z[i ]2).
The resultant technique will work when applied to polynomials of even degree.
In the next sections, we will prove Theorem 1. The main tools are the elliptic curve Chabauty
method in combination with Bruin and Stoll’s fake-2 Selmer set [2], which have been imple-
mented in MAGMA [1]. See Stoll [7, 8] for concrete examples. The MAGMA codes for the com-
putation of the 2-fake Selmer sets for each curve C1 and C2 are given. The MAGMA codes for the
elliptic curve Chabauty routine are available at https://www.overleaf.com/read/mgsfpypvvbrb
2. A proof of Theorem 1
Step 1. We compute the fake-2 Selmer set Sel(2)fake(C1):
MAGMA codes:
P<x> := PolynomialRing ( Rationals ( ) ) ;
C1 := HyperellipticCurve ( ( x ^2+1)*( x ^2+3)*( x ^2+7)) ;
Sel , mSel := TwoCoverDescent (C1 ) ;
# Sel ;
A<th> := Domain( mSel ) ;
Sel eq { mSel ( x0 − th ) : x0 in
{ th +1 ,1 , −1}} ;
Output:
3
true
Nguyen Xuan Tho 1073
The last line shows that for every rational point (x, y) ∈C (Q), there exists a ∈Q such that
x −α ∈ aQ(α)2 ∀α ∈ {i ,p−3,p−7}, (3)
or
x −α
1−α ∈ aQ(α)
2 ∀α ∈ {i ,p−3,p−7}, (4)
or
x −α
−1−α ∈ aQ(α)
2 ∀α ∈ {i ,p−3,p−7}. (5)
Step 2. We use the elliptic curve Chabauty method:
Case (3). Then
E1 : z
2 = (x − i )(x −p−7)(x2 +3),
where z ∈ K = Q(i ,p−7). The curve E1 has rank 2. The elliptic curve Chabauty routine in
MAGMA [1] works at p = 5 and shows that there are no points (x, z) in E1(K ) with the rational
x-coordinate.
Case (4). Then
E2 : z
2 = 2(1− i )(1−p−3)(x − i )(x −p−3)(x2 +7),
where z ∈ K = Q(i ,p−3). The curve E2 has rank 2. The elliptic curve Chabauty routine works at
p = 5 with the auxiliary prime 13, and shows that every point (x, z) ∈ E2(K ) with x ∈ Q satisfies
x =±1. Hence in (1), we have x =±1 and y =±8.
Case (5). By taking the complex conjugate and mapping x to −x, we have Case (4). Hence x =±1
and y =±8.
Acknowledgment
The author would like to thanks Professor Andrew Bremner for his help during the preparation of
this paper.
References
[1] W. Bosma, J. Cannon, C. Playoust, “The MAGMA algebra system. I. The user language”, J. Symb. Comput. 24 (1997),
no. 3-4, p. 235-265.
[2] N. Bruin, M. Stoll, “Two cover descent on hyperelliptic curves”, Math. Comput. 78 (2009), no. 268, p. 2347-2370.
[3] H. Cohen, Number Theory. Volume II: Analytic and Mordern Tools, Graduate Texts in Mathematics, vol. 240, Springer,
2007.
[4] S. Duquesne, “Calculs Effectifs des Points Entiers et Rationnels sur les Courbes”, PhD Thesis, Université Bordeaux I,
2001.
[5] ——— , “Points rationnels et méthode de Chabauty elliptique”, J. Théor. Nombres Bordeaux 15 (2003), no. 1, p. 99-113.
[6] E. V. Flynn, J. L. Wetherell, “Finding Rational Points on Bielliptic Genus 2 Curves”, Manuscr. Math. 100 (1999), no. 4,
p. 519-533.
[7] M. Stoll, “Slides of the talk Rational Diophantine quintuples and diagonal genus 5 curves”, https://www.mathe2.
uni-bayreuth.de/stoll/talks/Manchester-2017-print.pdf.
[8] ——— , “Rational Diophantine quintuples and diagonal genus 5 curves”, Acta Arith. 190 (2019), no. 3, p. 239-261.


Rational points on a certain genus 2 curve

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