Quadratic forms and Genus Theory a link with 2-descent and an application to non-trivial specializations of ideal classes

2025-05-02 0 0 674.4KB 26 页 10玖币

侵权投诉

Quadratic forms and Genus Theory: a link with

2-descent and an application to non-trivial specializations

of ideal classes

William Dallaporta

April 2024

Keywords: binary quadratic form, Picard group, Genus Theory, 2-descent on hyperelliptic curves,

density on S-integers

MSC classes: 11E16, 14H25, 14H40 (Primary); 11R45 (Secondary)

Abstract Genus Theory is a classical feature of integral binary quadratic forms. Using the

author’s generalization of the well-known correspondence between quadratic form classes and

ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and

have coeﬃcients in any PID R. When R=K[X], we show that the Genus Theory map is the

quadratic form version of the 2-descent map on a certain hyperelliptic curve. As an application,

we make a contribution to a question of Agboola and Pappas regarding a specialization problem

of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of

non-trivial specializations has density 1.

1 Introduction

It has been well-known since the work of Gauss in his Disquisitiones Arithmeticae that, given

∆∈Z, the set

equivalence classes of primitive binary quadratic forms

ax2+bxy +cy2with a, b, c ∈Zand discriminant b2−4ac = ∆ 

can be endowed with a group structure, whose group operation is called the composition law

(see Section 2 for the deﬁnitions).

Before going further, we must take care which notion of equivalence class we use. Over Z,

the natural action of SL2(Z)on quadratic forms is usually considered. In that setting, and

when ∆is a negative integer, it is a classical fact that the above group (restricted to classes

of positive deﬁnite quadratic forms) is isomorphic to the Picard group of the quadratic Z-

algebra of discriminant ∆(see [Cox13, Theorem 7.7] for a modern exposition). There have

been numerous generalizations of this group structure and of this correspondence to other rings

than Z, possibly with a diﬀerent action (see for example [Tow80] with SL2or [Kne82] with GL2).

More recently, Wood gave a set-theoretical bijection over an arbitrary base scheme [Woo11],

and the present author derived from her work the sought group isomorphism, when 2is not a

zero divisor on the base scheme [Dal21]. In her work, Wood pointed out the importance of the

twisted action of GL2, which we denote by GLtw

2(see Deﬁnition 2.2). This is the only action

we consider through this article, except in Subsection 3.2, where we make the link with the

classical SL2action over Z.

arXiv:2210.13045v4 [math.NT] 28 Apr 2024

This general group isomorphism from [Dal21] is stated over any base scheme S. Here,

we shall consider aﬃne schemes S= Spec(R), where Ris an integral domain of characteristic

diﬀerent from 2such that every locally free R-module of ﬁnite rank is free. Under an additional

assumption (see Proposition 2.4), the set of (twisted-)equivalence classes of primitive binary

quadratic forms with coeﬃcients in Rand with discriminant ∆∈Ris a group, which we denote

by Cltw

R(∆) (Proposition 2.4). The neutral element of Cltw

R(∆) is called the principal form class.

Given a primitive binary quadratic form, it is natural to wonder if it lies in the principal

form class or not. A classical feature of quadratic forms over Zis Genus Theory, which partially

answers this question and which is the main topic of this paper. Roughly speaking, the operation

associating a class of quadratic forms to its set of values modulo its discriminant ∆yields a

group homomorphism

ψ: Cltw

Z(∆) −→ Z

⧸

∆Z×

⧸

whose kernel is called the principal genus (Theorem 3.8). Here, H0denotes the set of values

of the principal form class. In particular, a quadratic form whose class is not in the principal

genus cannot be equivalent to the principal form.

In this article, we extend Genus Theory (for the twisted action GLtw

2) to quadratic forms over

principal ideal domains. In this general setting, it is already a diﬃcult problem to determine

precisely the principal genus. A simple argument shows that it always contains the subgroup

of squares. Over Z, when the discriminant is negative, we show in Proposition 3.14 that the

converse is true (this is just an adaptation in our context of the proofs of the classical results).

We then study the case when the base ring is K[X](where Kis a ﬁeld of characteristic 0),

and when the discriminant is of the form ∆ = 4fwith f∈K[X]a square-free monic polynomial

of odd degree at least 3. In this situation, the group of (twisted)-equivalence classes of quadratic

forms of discriminant 4fis isomorphic to the group of K-points of the Jacobian variety of the

hyperelliptic curve Cdeﬁned over Kby the equation Y2=f(X). This correspondence is closely

related to Mumford’s description of the Jacobian, and was already used by Gillibert in that

setting [Gil21]. We prove in Subsection 3.3 that Genus Theory over K[X]turns out to be the

quadratic form version of the 2-descent map on the Jacobian of C. More precisely, by combining

Proposition 3.19 and Theorem 3.20, we obtain

Theorem 1.1. Let Kbe a ﬁeld of characteristic 0, let f∈K[X]be a square-free monic poly-

nomial of odd degree at least 3, let L:= K[X]

⧸

⟨f(X)⟩, and let Jbe the Jacobian variety of the

hyperelliptic curve deﬁned by the aﬃne equation Y2=f(X)over K. Let us denote by Ψthe

Genus Theory homomorphism (3.15) and by λthe 2-descent map on J(K). Then the following

diagram commutes

Cltw

K[X](4f)

⧸

Cltw

K[X](4f)□J(K)

⧸

2J(K)

L×

⧸

K×L×□L×

⧸

L×□

∼

Ψλ

(1.2)

where the exponent □denotes the subgroup of squares, and pr is the natural projection. Fur-

thermore, Ψis injective; in other words, the principal genus is precisely the subgroup of squares.

The fact that our base ring is a principal ideal domain is heavily used to ﬁnd an adequate

representative of a given class of quadratic forms (Lemma 3.4). This is a property which is at

the heart of most of the technical arguments. If one wants to extend Genus Theory to quadratic

forms over more general rings than PIDs, then one must in particular extend Lemma 3.4 or

ﬁnd a way to deal without it.

An as application of Genus Theory, the last Section of our article is devoted to the following

question, which is closely related to a question raised by Agboola and Pappas [AP00].

Question 1.3. Let Kbe a number ﬁeld. Let Cbe a hyperelliptic curve over Kof genus

g≥1, with a K-rational Weierstrass point. Let us choose an aﬃne equation of Cof the

form Y2=f(X)where f∈ OK[X]is a square-free monic polynomial of odd degree 2g+ 1.

Let I∈Pic OK[X, Y ]

⧸

⟨Y2−f(X)⟩be a non-trivial ideal class. Can we ﬁnd n∈ OKsuch

that the specialization of Iat X=ngives a non-trivial ideal class Inin Pic OK(√f(n)), or at

least in Pic OK[Y]

⧸

⟨Y2−f(n)⟩?

We answer positively the second part of Question 1.3 for ideal classes Iwhich are not

squares, at least after inverting a ﬁnite number of prime ideals of OK. We further prove that

the density of non-trivial specializations is 1, for any “reasonable” density. In the case of square

ideal classes, our arguments which rely on Genus Theory cannot be extended, since squares are

already in the principal genus.

Regarding hyperelliptic curves, several results have already been established about non-

trivial specializations:

•when Cis an elliptic curve over K=Qand Ihas inﬁnite order, Soleng proved that there

exist inﬁnitely many non-trivial specializations Inin imaginary quadratic extensions of Q

whose order is unbounded as ngoes to inﬁnity [Sol94, Theorem 4.1];

•when K=Qand Ihas ﬁnite order, Gillibert and Levin used Kummer Theory and

Hilbert’s Irreducibility Theorem to show that, after inverting primes of bad reduction,

there exist inﬁnitely many non-trivial specializations Inin imaginary quadratic extensions

of Q[GL12, Corollary 3.8];

•when K=Qand Ihas inﬁnite order, Gillibert showed that there exist inﬁnitely many

negative integers nsuch that Inis a non-trivial ideal class of the order Z[pf(n)]. With

the additional assumption that the irreducible factors of fall have degree at most 3, this

leads to inﬁnitely many non-trivial ideal classes in Pic OQ(√f(n))[Gil21, Theorems 1.2

and 1.3]. Among Gillibert’s main ingredients, one can ﬁnd Wood’s correspondence with

binary quadratic forms and a generalization of Soleng’s argument.

Let us assume that K=Qfor the time being. Given a non-trivial ideal class Iof

Z[X, Y ]

⧸

⟨Y2−f(X)⟩, can we ﬁnd non-trivial specializations of Iin real quadratic extensions

of Q? Soleng’s argument relies on properties which are speciﬁc to negative discriminants, and

do not generalize to the positive case. This leads us to consider a diﬀerent approach.

Numerical experiments show that, depending on the ideal class Iwe start from, there

may exist congruence classes of n∈Zleading to non-trivial specializations (see Example 4.1),

whatever the sign of the discriminant.

As in the work of Gillibert, we use Wood’s bijection between invertible ideal classes of

quadratic algebras and equivalence classes of primitive binary quadratic forms as described

in [Dal21, Corollary 3.25]. In this setting, the ideal class Iwe start from corresponds to

the equivalence class of a primitive quadratic form q(x, y) = ax2+bxy +cy2with discriminant

b2−4ac = 4f, where a, b, c ∈ OK[X]. Question 1.3 now asks whether one can ﬁnd n∈ OKsuch

that the specialized quadratic form a(n)x2+b(n)xy +c(n)y2is not equivalent to the principal

form x2−f(n)y2, that is, the class of quadratic forms corresponding to the trivial ideal class

in Wood’s bijection.

The presentation of Genus Theory in this article requires us to work over a principal ideal

domain. As OKmay not be a PID, we slightly modify it by inverting ﬁnitely many prime ideals.

Thus, we will work over OK,Sinstead of OK, where OK,Sis the ring of S-integers of K. Despite

the fact that OK,S[X]is not a PID, by making a suitable choice of S, one can relate classes

of quadratic forms over OK,S[X]with classes of quadratic forms over K[X](Proposition 4.2).

Notice that in the terminology of divisors, this operation is the restriction to the generic ﬁbre.

We then prove that, given a class qof quadratic forms over OK,S[X]which is not in the

principal genus when viewed over K[X], there exist inﬁnitely many n∈ OK,Ssuch that the

specialized class qnof quadratic forms is not in the principal genus. We achieve this in Theo-

rem 4.14. Together with Theorem 4.26 and Remark 4.3, a complete version of the main result

is the following.

Theorem 1.4. Let Kbe a number ﬁeld. Let f∈ OK[X]be a square-free monic polynomial of

odd degree at least 3. Let Sbe a ﬁnite set of nonzero prime ideals of OKsuch that OK,Sis a

PID. Let I∈Pic OK,S[X, Y ]

⧸

⟨Y2−f(X)⟩be a non-trivial ideal class.

Assume that the ideal class generated by Iin Pic K[X, Y ]

⧸

⟨Y2−f(X)⟩is not a square.

Then the set of n∈ OK,Ssuch that the specialization of Iat X=ngives a non-trivial ideal

class Inof OK,S[Y]

⧸

⟨Y2−f(n)⟩has density 1, for any density on OK,Sas in Deﬁnition 4.19.

In particular, there are inﬁnitely many n∈ OK,Ssuch that Inis non-trivial.

If we choose Swhich contains the prime ideals dividing 2 disc(f)and such that OK,Sis

a PID, then Theorem 1.4 gives a partial answer to a question raised by Agboola and Pap-

pas [AP00]. More precisely, following [Gil21, §2.1], there exists a smooth projective model

W −→ Spec(OK,S)of Csuch that, set-theoretically,

W= Spec OK,S[X, Y ]

⧸

Y2−f(X)∪ {∞}

together with an isomorphism Pic OK,S[X, Y ]

⧸

⟨Y2−f(X)⟩≃Pic0(W), where {∞} is the

scheme-theoretic closure of the point at inﬁnity of C. The result of Theorem 1.4 implies that a

degree 0line bundle on Wwhich is not a square has inﬁnitely many non-trivial specializations

over suitable quadratic OK,S-orders.

Notations. All through this paper, the rings we consider are commutative and endowed with

a multiplicative identity denoted by 1. If Ris a ring, R×denotes its group of units. Given

r1, . . . , rm∈R, the ideal generated by r1, . . . , rmis denoted by ⟨r1, . . . , rm⟩. If Gis a group,

then G□denotes its subgroup of squares (thinking of the group law multiplicatively).

Given two integers a < b, we denote by Ja, bKthe set of integers nsuch that a≤n≤b.

If Tis a scheme, we denote by Pic(T)the Picard group of T. When Tis Noetherian and

reduced, we shall identify Pic(T)with the group of Cartier divisors modulo linear equivalence.

When Ris a domain, we write by abuse of notations Pic(R)instead of Pic(Spec(R)), which we

also identify with the group of invertible fractional ideals modulo principal ones.

We refer the reader to Deﬁnition 2.2 and Proposition 2.4 for the meaning of GLtw

2and of

Cltw

R(∆) respectively.

Acknowledgements. This work is part of my PhD at the Institut de Mathématiques de

Toulouse. I am grateful to Sander Mack-Crane, Ignazio Longhi, Florent Jouve and Yuri Bilu

for helpful discussions about densities over rings of S-integers. I also address special thanks

to Jean Gillibert and Marc Perret who made this work possible, who regularly gave precious

advice, and who were particularly encouraging all along this work. The ﬁnal writing of this

paper owes a lot to the meticulous proofreading of Christian Wuthrich and of the anonymous

referee, whom I warmly thank.

2 Binary quadratic forms and Picard groups of quadratic

algebras

The kind of rings Rwe consider in this paper are mostly of the form R′or R′[X]with R′a

principal ideal domain of characteristic diﬀerent from 2. An important property they share is

the fact that every locally free R-module of ﬁnite rank must be free, according to [Ses58].

Deﬁnition 2.1. Let Rbe a ring such that every locally free R-module of ﬁnite rank is free.

A (binary) quadratic form qover Ris a homogeneous degree 2polynomial in R[x, y]. It

is of the form ax2+bxy +cy2for some a, b, c ∈R, and is denoted by [a, b, c]. It is called

primitive if the ideal generated by a, b, c in Ris the unit ideal. Its discriminant is the quantity

∆ := b2−4ac ∈R.

Deﬁnition 2.2. Let M=α β

γ δ∈GL2(R). Following [Woo11], we deﬁne the twisted action

of GL2(R)over the set of quadratic forms via M·q:= 1

det(M)(q◦M), that is

α β

γ δ·q(x, y) := 1

αδ −βγ q(αx +βy, γx +δy)∀x, y ∈R.

We denote this action by GLtw

2. Two quadratic forms qand q′are GLtw

2-equivalent (equivalent

for short) if there exists M∈GL2(R)such that q′=M·q. In the following, a class of quadratic

forms [a, b, c]refers to the equivalence class of the quadratic form [a, b, c].

Remark 2.3.With the same notations, if q= [a, b, c], then

α β

γ δ·[a, b, c] = 1

αδ −βγ [aα2+bαγ +cγ2, b(αδ +βγ) + 2(aαβ +cγδ), aβ2+bβδ +cδ2]

αδ −βγ [q(α, γ), q(α+β, γ +δ)−q(α, γ)−q(β, δ), q(β, δ)].

We recall the main link between quadratic forms and ideals in our setting ([Dal21, Corol-

lary 3.25]).

Proposition 2.4. Let Rbe an integral domain of characteristic diﬀerent from 2such that

every locally free R-module of ﬁnite rank is free. Let ∆∈Rbe such that the equation

∆≡x2(mod 4R)has a unique solution xmodulo 2R, and let πbe any lift of xto R. De-

note by Cltw

R(∆) the set of GLtw

2-equivalence classes of primitive quadratic forms over Rwith

discriminant ∆. Then we have a bijection

Cltw

R(∆) 1: 1

←→ Pic 

R[ω]

⧸

ω2+πω −∆−π2

4



[a, b, c]with a̸= 07−→ ⟨ω+π−b

2, a⟩

.(2.5)

This allows us to endow Cltw

R(∆) with a group structure, whose operation ∗is called the com-

position law.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

QuadraticformsandGenusTheory:alinkwith2-descentandanapplicationtonon-trivialspecializationsofidealclassesWilliamDallaportaApril2024Keywords:binaryquadraticform,Picardgroup,GenusTheory,2-descentonhyperellipticcurves,densityonS-integersMSCclasses:11E16,14H25,14H40(Primary);11R45(Secondary)AbstractGenu...

展开>> 收起<<

Quadratic forms and Genus Theory a link with 2-descent and an application to non-trivial specializations of ideal classes.pdf

共26页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Quadratic forms and Genus Theory a link with 2-descent and an application to non-trivial specializations of ideal classes

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: