Picard groups, pull back and class groups

TL;DR

This paper investigates the relationship between the Picard group of a certain affine algebraic surface over $Q$, its pullback properties, and the class groups of quadratic fields, providing new insights into torsion line bundles and their arithmetic implications.

Contribution

It demonstrates that non-trivial torsion line bundles in the relative Picard group can be pulled back to ideal classes of quadratic fields with arbitrarily large order, answering a question by Agboola and Pappas.

Findings

Torsion line bundles can be pulled back to ideal classes in quadratic fields.

The Picard group of fibers remains stable over a Zariski open subset.

Existence of elements of odd order in class groups of imaginary quadratic fields.

Abstract

Let $S$ be a certain affine algebraic surface over $\mathbb{Q}$ such that it admits a regular map to $\mathbb{A}^2/\mathbb{Q}$. We show that any non-trivial torsion line bundle in the relative Picard group $Pic^0\left(S/\mathbb{A}^2\right)$ can be pulled back to ideal classes of quadratic fields whose order can be made sufficiently large. This gives an affirmative answer to a question raised by Agboola and Pappas, in case of affine algebraic surfaces. For a closed point $P\in \mathbb{A}^2/\mathbb{Q}$, we show that the cardinality of a subgroup of the Picard group of the fiber $S_P$ remains unchanged when $P$ varies over a Zarisky open subset in $\mathbb{A}^2$. We also show by constructing an element of odd order $n\geq 3$ in the class group of certain imaginary quadratic fields that the Picard group of $S_P$ has a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$.

Equations

$$C_d^1(X/B)=\{(D_b,b)|\mathrm{supp}(D_b)\subset X_b,\mathrm{deg}(D_b)=d\}.$$

$$\mathcal{Z}:=\{(b,D_b)|[D_b]=0\in \mathrm{Pic}(X_b)\}.$$

$$f(0)=D_b^{+}+\gamma \text{ and }f(\infty )=D_b^{-}+\gamma ,$$

$$ev:Ho{m}^v({\mathbb{P}}_k^1,C_d^1(X/B))\to C_d^1(X/B)\times C_d^1(X/B),$$

$$(b,f)\mapsto (b,f(0),f(\infty )).$$

$$(A,C,B,D)\mapsto (A+C,C,B+D,D).$$

$$f:{\mathbb{P}}^1\to C_{d+u,u,d+u,u}^1(X_b)$$

$$f(0)=(A+C,C)\text{ and }f(\infty )=(B+D,D),$$

$$f:{\mathbb{P}}_k^1\to C_{d+u,u}^1(X_b)$$

$$W_d^{u,v}=p{r}_{1,2}({\tilde{s}}^{-1}(W_{d+u}^{0,v}\times W_u^{0,v})),$$

$$\tilde{s}:B\times C_{d,d,u,u}^1(X)\to B\times C_{d+u,d+u,u,u}^1(X)$$

$$\tilde{s}(b,A,B,C,D)=(b,A+C,B+D,C,D).$$

$$g(0)=A+C,g(\infty )=B+D\text{ and }h(0)=C,h(\infty )=D$$

$$f(0)=(A+C,C)\text{ and }(f(\infty ))=(B+D,D).$$

$$f(0)=(A+C,C)\text{ and }f(\infty )=(B+D,D),$$

$$g(0)=A+C,g(\infty )=B+D$$

$$h(0)=C,h(\infty )=D.$$

$$W_d=p{r}_{1,2}({\tilde{s}}^{-1}(W_{d+u}\times W_u)).$$

$$e:U_{v,d}(X)\subset B\times {\mathrm{Hom}}^v({\mathbb{P}}_k^1,C_d^1(X))\to C_{d,d}^1(X)$$

$$(b,f)\mapsto (b,f(0),f(\infty ))$$

$${\mathcal{Z}}_d:=\{(b,D_b)|n[D_b]=0\in \mathrm{Pic}(X_b)\}$$

$$\phi :\mathrm{Pic}(X_b)\to \mathrm{Pic}(X_{b\mathbb{C}})$$

$${\theta }_d:C_d^1(X/B)\to \mathrm{Pic}(X/B),\text{ for all }d.$$

$$\mathcal{Z}=\{[D]\mid n[D]=0\}.$$

$${\oplus }_i\mathbb{Z}[P_i]\to \mathrm{Pic}(C)\to \mathrm{Pic}(U)\to 0.$$

$$\mathcal{P}:=\{(x,b)|x\in X_b\setminus U_b\}\to U.$$

$${\mathcal{Z}}_d=\{(b,z)|\mathrm{supp}(z)\subset X_b,nz=\sum _i{n}_i{P}_{ib}\}\to U.$$

$${\vartheta }_d({\mathcal{Z}}_d)=\mathcal{Z}={\cup }_i{\mathcal{Z}}_i.$$

$$nz=\sum _i{n}_i{P}_{ib}$$

$$y^2=t^2{q}^2-z^n$$

Full text

1. Introduction

Notations: Given a point $P$ in a surface defined over ${\mathbb{Q}}$, by $S_{P}$ we mean a fiber; by ${\mathbb{Q}}(S_{P})$ we mean the field of definition of $S_{P}$; by ${\mathbb{A}}^{2}$ we mean an affine plane; for a number field $K$, by $\mathcal{O}_{K}$ we mean the ring of integers of $K$.

Let $S$ be an affine algebraic surface defined over ${\mathbb{Q}}$ and it admits a regular map to ${\mathbb{A}}^{2}$ over ${\mathbb{Q}}$. One of the goals of this paper is to give an affirmative answer to the following question in case of imaginary quadratic fields.

Question 1.1*.*

Let ${\mathscr{L}}$ be a non-trivial line bundle in the relative Picard group ${\rm{Pic}}^{0}(S/{\mathbb{A}}^{2})$. Is it possible to some find points $P\in{\mathbb{A}}^{2}({\mathbb{Q}})$ for which the pull back of ${\mathscr{L}}$ at $P$ is a non-trivial ideal class of the number field ${\mathbb{Q}}(S_{P})$?

This a special case of a question raised by Agboola and Pappas in [1]. Gillibert and Levin [11] affirmatively answered the question raised by Agboola and Pappas in the case of torsion line bundles on a hyperelliptic curve. Gillibert [13] also considered a similar question in case of non-trivial line bundles of degree zero on certain hyperelliptic curves which are not torsion.

Another goal of this paper is to show for a closed point $P$ in ${\mathbb{A}}^{2}/{\mathbb{Q}}$ that the cardinality of certain types of subgroups of the Picard group of the fiber $S_{P}$ remains constant when $P$ varies over a Zariski open subset in ${\mathbb{A}}^{2}$. Moreover, we show that this Picard group has a subgroup isomorphic to ${\mathbb{Z}}/n{\mathbb{Z}}$ by constructing an element of odd order $n\geq 3$ in the class group of ertain families of imaginary quadratic fields. Producing an element of a given order in a class group of an imaginary quadratic field (in fact, any number field) provides more informations about the class group than showing the divisibility of the class number of the same field. However, the divisibility properties of the class number of a number field help us to understand the structure of class group. Many families of imaginary quadratic fields with class number divisible by a given integer $n\geq 2$ are known. Most of such families are of the type $\mathbb{Q}(\sqrt{k^{2}-\lambda t^{n}})$, where $k$ and $t$ are positive integers with some restrictions, and $\lambda=1,2,4$ (for $\lambda=1$ see [2, 12, 16, 17, 22, 26]; for $\lambda=2$ see [7, 15] and for $\lambda=4$ see [6, 14, 20, 29]). Here, we consider the family $K_{k,p,n}=\mathbb{Q}(\sqrt{k^{2}-p^{n}})$, where $k,n$ are positive integers and $p$ is an odd prime such that $k^{2}<p^{n}$. We produce an ideal class of order $n$ in $K_{k,p,n}$, and then we use this ideal class to show that the Picard group of the fiber $S_{P}$ has a subgroup which is isomorphic to ${\mathbb{Z}}/n{\mathbb{Z}}$. The main idea that used to connect the two concepts of $n$-divisibility is to understand the fiber of the natural map from the relative Chow schemes to the relative Picard scheme associated to a flat family of projective schemes.

2. Mumford-Roĭtman argument on Chow schemes and relative Picard schemes

For a flat morphism $X\to B$ of projective schemes, we consider the Chow scheme, $C^{1}_{d}(X/B)$ of relative co-dimension one subschemes of $X\to B$ of degree $d$, that is

[TABLE]

Then there is a natural map $C^{1}_{d}(X_{b})\to{\rm{Pic}}(X_{b})$ associating to a $D$ in its divisor class $[D]$. In this set up, we consider the following:

[TABLE]

The proof of the following theorem is based on the idea introduced by Mumford in [21]. This idea had been elaborated by Roĭtman in [23] and Voisin in [28]. This idea had been used and presented in [3].

Theorem 2.1**.**

${\mathscr{Z}}$* is a countable union of Zariski closed subsets in $C^{1}_{d}(X/B)$.*

Proof.

Assume that the relation $D_{b}=D_{b}^{+}-D_{b}^{-}$ is rationally equivalent to zero. This means that there exists a map $f:{\mathbb{P}}^{1}\to C^{1}_{d,d}(X_{b})$ such that

[TABLE]

where $\gamma$ is a positive divisor on $X_{b}$. In other words, we have the following map:

[TABLE]

given by $f\mapsto(f(0),f(\infty))$ and image of $f$ is contained in $C^{1}_{d,d}(X_{b})$.

Let us denote $C^{1}_{d}(X/B)$ by $C^{1}_{d}(X)$ for simplicity.

We now consider the subscheme $U_{v,d}(X)$ of $B\times Hom^{v}({\mathbb{P}}^{1}_{k},C^{1}_{d}(X))$ consisting of the pairs $(b,f)$ such that image of $f$ is contained in $C^{1}_{d}(X_{b})$ (such a universal family exists, for example see [18, Theorem 1.4]). This gives a morphism from $U_{v,d}(X)$ to $B\times C^{1}_{d,d}(X)$ defined by

[TABLE]

Again, we consider the closed subscheme ${\mathscr{V}}_{d,d}$ of $B\times C^{1}_{d,d}(X)$ given by $(b,z_{1},z_{2})$, where $(z_{1},z_{2})\in C^{1}_{d,d}(X_{b})$. Suppose that the map from ${\mathscr{V}}_{d,u,d,u}$ to ${\mathscr{V}}_{d+u,d+u}$ is given by

[TABLE]

Then one writes the fiber product ${\mathscr{V}}$ of $U_{v,d}(X)$ and ${\mathscr{V}}_{d,u,d,u}$ over ${\mathscr{V}}_{d+u,d+u}$. If we consider the projection from ${\mathscr{V}}$ to $B\times C^{1}_{d,d}(X)$, then we observe that $A$ and $B$ are supported as well as rationally equivalent on $X_{b}$. Conversely, if $A$ and $B$ are supported as well as rationally equivalent on $X_{b}$, then one gets the map

[TABLE]

of some degree $v$ satisfying

[TABLE]

where $C$ and $D$ are supported on $X_{b}$. This implies that the image of the projection from ${\mathscr{V}}$ to $B\times C^{1}_{d,d}(X)$ is a quasi-projective subscheme $W_{d}^{u,v}$ consisting of the tuples $(b,A,B)$ such that $A$ and $B$ are supported on $X_{b}$, and that there exists a map

[TABLE]

such that $f(0)=(A+C,C)$ and $f(\infty)=(B+D,D)$. Here $f$ is of degree $v$, and $C,D$ are supported on $X_{b}$ and they are of co-dimension $1$ and degree $u$ cycles. This shows that $W_{d}=\cup_{u,v}W_{d}^{u,v}$. We now prove that the Zariski closure of $W_{d}^{u,v}$ is in $W_{d}$ for each $u$ and $v$. For this, we prove the following:

[TABLE]

where

[TABLE]

defined by

[TABLE]

We assume $(b,A,B,C,D)\in B\times C^{1}_{d,d,u,u}(X)$ in such a way that $\widetilde{s}(b,A,B,C,D)\in W^{0,v}_{d+u}\times W^{0,v}_{u}$. This implies that there exists an element $(b,g)\in B\times{\rm Hom}^{v}({\mathbb{P}}^{1}_{k},C^{p}_{d+u}(X))$ and an element $(b,h)\in{\rm Hom}^{v}({\mathbb{P}}^{1}_{k},C^{p}_{u}(X))$ satisfying

[TABLE]

as well as the image of $g$ and $h$ are contained in $C^{1}_{d+u}(X_{b})$ and $C^{1}_{u}(X_{b})$ respectively.

Also if $f=g\times h$ then $f\in{\rm Hom}^{v}({\mathbb{P}}^{1}_{k},C^{p}_{d+u,u}(X))$ such that the image of $f$ is contained in $C^{1}_{d+u,u}(X_{b})$ as well as it satisfies the following:

[TABLE]

This shows that $(b,A,B)\in W^{d}_{u,v}$.

On the other hand, we assume that $(b,A,B)\in W^{d}_{u,v}$. Then there exists $f\in{\rm Hom}^{v}({\mathbb{P}}^{1}_{k},C^{1}_{d+u,u}(X_{b}))$ such that

[TABLE]

and image of $f$ is contained in the Chow scheme of $X_{b}$.

We now compose $f$ with the projections to $C^{1}_{d+u}(X_{b})$ and to $C^{1}_{u}(X_{b})$ to get a map $g\in{\rm Hom}^{v}({\mathbb{P}}^{1}_{k},C^{1}_{d+u}(X))$ and a map $h\in{\rm Hom}^{v}({\mathbb{P}}^{1}_{k},C^{1}_{u}(X))$ satisfying

[TABLE]

and

[TABLE]

Also, the image of $g$ and $h$ are contained in the respective Chow schemes of the fibers $X_{b}$. Therefore, we have

[TABLE]

We are now in a position to prove that the closure of $W_{d}^{0,v}$ is contained in $W_{d}$. Let $(b,A,B)$ be a closed point in the closure of ${W_{d}^{0,v}}$. Let $W$ be an irreducible component of ${W_{d}^{0,v}}$ whose closure contains $(b,A,B)$. Assume that $U$ is an affine neighborhood of $(b,A,B)$ such that $U\cap W$ is non-empty. Then there is an irreducible curve $C$ in $U$ passing through $(b,A,B)$. Suppose that $\bar{C}$ is the Zariski closure of $C$ in $\overline{W}$. The map

[TABLE]

given by

[TABLE]

is regular and $W_{d}^{0,v}$ is its image. We now choose a curve $T$ in $U_{v,d}(X)$ such that the closure of $e(T)$ is $\bar{C}$. Let $\widetilde{T}$ be denote the normalization of the Zariski closure of $T$, and $\widetilde{T_{0}}$ be the pre-image of $T$ in this normalization. Then the regular morphism $\widetilde{T_{0}}\to T\to\bar{C}$ extends to a regular morphism from $\widetilde{T}$ to $\bar{C}$. If $(b,f)$ is a pre-image of $(b,A,B)$, then $f(0)=A,~{}f(\infty)=B$ and the image of $f$ is contained in $C^{p}_{d}(X_{b})$ by the definition of $U_{v,d}(X)$. Therefore, $A$ and $B$ are rationally equivalent. This completes the proof. ∎

As a consequence, one gets the following:

Corollary 2.1**.**

The collection

[TABLE]

is a countable union of Zariski closed subsets in the scheme $C^{1}_{d}(X/B)$.

3. Mumford-Roitman arguments and monodromy representation

The main idea of this section, is due to Voisin [27, Chapter 3]. We consider the case when $X$ is a projective algebraic surface over $\bar{\mathbb{Q}}$ fibered over a projective algebraic curve over $\bar{\mathbb{Q}}$. Then the fibers of the map $X\to B$ are projective algebraic curves. Moreover, the fiber $X_{b}$ is smooth for a general $b\in B$. We now consider the complexification of the above family. That is, we consider the map $X_{{\mathbb{C}}}\to B_{{\mathbb{C}}}$, where $X_{{\mathbb{C}}}=X\times_{\bar{\mathbb{Q}}}{\mathbb{C}}$. Then there is a map

[TABLE]

given by pull back. The set of all images of the torsions on ${\rm{Pic}}(X_{b})$ under $\varphi$ is a subgroup of the group of torsions in ${\rm{Pic}}(X_{b{\mathbb{C}}})$. We now consider the map

[TABLE]

Then $\cup_{d}\theta_{d}({\mathscr{Z}}_{d})\subset{\rm{Pic}}(X/B)$, where ${\mathscr{Z}}_{d}$ is as defined in Corollary 2.1. Let ${\mathscr{Z}}=\cup_{i}{\mathscr{Z}}_{i}$, then

[TABLE]

We consider the map ${\mathscr{Z}}\to B$. Then for any $b\in B$ there exists a $D$ supported on $X_{b}$ such that $n[D]=0$ on ${\rm{Pic}}(X/B)$. If $B_{{\mathbb{C}}}$ is connected then there exists a ${\mathscr{Z}}_{i}$ such that the map from ${\mathscr{Z}}_{i{\mathbb{C}}}$ to $B_{{\mathbb{C}}}$ is onto. Therefore, if $U$ is the Zariski open set in $B$ consisting of $b$ such that $X_{b}$ is smooth then one gets a surjective map $\mathfrak{f}:{\mathscr{Z}}_{i{\mathbb{C}},U}\to U_{{\mathbb{C}}}$. By removing a finite collection points from $U_{{\mathbb{C}}}$, one can assume that the map $\mathfrak{f}$ is smooth and proper in the sense of underlying smooth manifolds. Hence, $\mathfrak{f}$ is a fibration in the sense of differential topology. It is easy to see that $\mathfrak{f}$ has finite fibers since these fibers contain torsion points on ${\rm{Pic}}(X_{b,{\mathbb{C}}})$. In fact, the torsion points are inside ${\rm{Pic}}^{0}(X_{b{\mathbb{C}}})$, which is an abelian variety isomorphic to $J(X_{b,{\mathbb{C}}})$, the Jacobian variety of $X_{b{\mathbb{C}}}$.

We now consider the first cohomology $H^{1}(J(X_{b{\mathbb{C}}}),{\mathbb{Q}})$ which is also isomorphic to $H^{1}(X_{b,{\mathbb{C}}},{\mathbb{Q}})$. We also consider the cohomology classes in $H^{1}(J(X_{b{\mathbb{C}}}),{\mathbb{Q}})$ of the elements in the fibers of $\mathfrak{f}$, and let $V_{b}$ be its ${\mathbb{Q}}$-span. Since the this map is smooth and proper, so that the cohomology classes in $V_{b}$ gives rise to a $\pi_{1}(U,b)$ module which is a $\pi_{1}(U,b)$ submodule of $H^{1}(X_{b,{\mathbb{C}}},{\mathbb{Q}})$. Therefore, the cohomology classes of the torsion points on the fibers of ${\mathscr{Z}}_{i{\mathbb{C}},U}\to U_{{\mathbb{C}}}$ give rise to a locally constant sheaf ${\mathscr{L}}$ over $U$ by the equivalence between locally constant sheaves and $\pi_{1}(U,b)$-module representations. Hence, the dimension of the vector space $V_{b}$ over ${\mathbb{Q}}$ remains constant as $U$ is connected. Thus, the cardinality of the finite group of torsions coming from ${\mathscr{Z}}_{i{\mathbb{C}},U}\to U_{{\mathbb{C}}}$ in ${\rm{Pic}}(X_{b{\mathbb{C}}})$ remains constant. This gives the following result.

Theorem 3.1**.**

The cardinality of the subgroup of torsions in $X_{b}$ coming from the fibration ${\mathscr{Z}}_{i{\mathbb{C}},U}\to U_{{\mathbb{C}}}$ for each $b\in U$ remains constant and they vary in a family.

We now consider a smooth projective curve $C$ over an algebraically closed field $K\subset{\mathbb{C}}$ in the projective plane ${\mathbb{P}}^{2}$ over $K$. Let $U$ be an affine piece of $C$. That is, $U$ is $C$ minus finitely many points, viz. $P_{1},\cdots,P_{m}$. Consider the following localization exact sequence of Picard groups

[TABLE]

Then the set of all torsion points in ${\rm{Pic}}(U)$ gives rise to elements of ${\rm{Pic}}(C)$ of the form $nz$ such that $nz=\sum_{i}n_{i}P_{i}$, where $P_{1},\cdots,P_{m}$ are the finite number of points that are deleted. As before, we consider a fibration of smooth projective schemes $X\to B$ over $\bar{\mathbb{Q}}$, where $X$ is a surface embedded in ${\mathbb{P}}^{3}$ such that each fiber $X_{b}$ is contained in a projective plane ${\mathbb{P}}^{2}$ over $\bar{\mathbb{Q}}$ and $B$ is an algebraic curve. Suppose that the degree of the algebraic curve $X_{b}$ remains constant over a Zariski open set $U$ in $B$. For an affine piece $U_{b}$ of the algebraic curve $X_{b}$, we consider the following:

[TABLE]

By the above assumption, this a finite-to-one map from ${\mathscr{P}}$ to $U$ and the degree of this map is constant. For a given any $b\in U$, let us suppose the fiber ${\mathscr{P}}_{b}$ contains the points $P_{1b},\cdots,P_{mb}$. We define the set:

[TABLE]

Then as a conseqeunce of Theorem 2.1 one gets the following result.

Corollary 3.1**.**

${\mathscr{Z}}_{d}$* is a countable union of Zariski closed subsets in the ambient relative Chow scheme $C^{1}_{d}(X_{U}/U)$, where $X_{U}\to U$ is the pullback of the family $X\to B$ to $U$.*

We consider the map $\vartheta_{d}:C^{1}_{d}(X_{U}/U)\to{\rm{Pic}}(X_{U}/U)$ and the union

[TABLE]

Since $U$ is connected, so that there exists at least one $i$ such that ${\mathscr{Z}}_{i{\mathbb{C}}}\to U_{{\mathbb{C}}}$ is surjective. Therefore as in Theorem 3.1, it gives rise to a local system. More precisely, for each $b\in U$ if one considers the $\pi_{1}(U_{{\mathbb{C}}},b)$ module $V_{b}$ consisting of cohomology classes of the elements in the fibers of the map ${\mathscr{Z}}_{i{\mathbb{C}}}\to U_{{\mathbb{C}}}$ in $H^{1}(X_{b{\mathbb{C}}},{\mathbb{Q}})$, then the dimension of this vector space $V_{b}$ remains constant over $U$. Therefore, we have the following:

Corollary 3.2**.**

The cardinality of the set of $z$ in ${\mathscr{Z}}_{ib}$ for $b\in U$ such that

[TABLE]

for points $P_{ib}\in X_{b}$ is constant as $b$ varies over $U$.

These points on ${\rm{Pic}}(X_{b})$ are correspondind to the torsion elements in ${\rm{Pic}}(U_{b})$, where $U_{b}$ is the open complement of $X_{b}$ obtained from $X_{b}$ by deleting the points $P_{1b},\cdots,P_{mb}$.

4. Torsion in the Picard group of algebraic surfaces

In this section, we will show that certain algebraic surfaces have $n$-torsion elements in the Picard group. We begin section with the algebraic surface defined by

[TABLE]

over ${\mathbb{Q}}$. Its co-ordinate ring is given by

[TABLE]

We now consider the maximal ideal $(t-m,z-\ell)$, for some algebraic numbers $m,\ell$, in the polynomial ring ${\mathbb{Q}}[t,z]$. We also consider the map

[TABLE]

which is defined by

[TABLE]

and the map ${\mathbb{Q}}[t,z]\to{\mathbb{Q}}$ which is given by

[TABLE]

Then the tensor product

[TABLE]

is given by ${\mathbb{Q}}[y]/(y^{2}-m^{2}q^{2}+\ell^{n})$. Further, if the polynomial $p(y):=y^{2}-m^{2}q^{2}+\ell^{n}$ is irreducible over ${\mathbb{Q}}$, then the above co-ordinate ring is isomorphic to $L$, where $L$ is the imaginary quadratic extension of ${\mathbb{Q}}$ given by adjoining a root of $p(y)$. Therefore if we consider the family

[TABLE]

then the fibers are the ring of integers of ${\mathbb{Q}}(\sqrt{m^{2}q^{2}-\ell^{n}})$.

Let us consider an affine surface $S$ fibered over $\AA^{2}_{{\mathbb{Q}}}$ as mentioned in the beginning of this section. Then if the pullback of the fibration over $\AA^{2}_{{\mathbb{Z}}}$, that is $S_{{\mathbb{Z}}}\to\AA^{2}_{{\mathbb{Z}}}$ then this is the family of ring of integers ${\mathscr{O}}(\sqrt{m^{2}q^{2}-\ell^{n}})$. Consider the Zariski closure of $S$ in ${\mathbb{P}}^{3}_{{\mathbb{Q}}}$ and the Zariski closure of the family $S_{{\mathbb{Z}}}\to\AA^{2}_{{\mathbb{Z}}}$ in ${\mathbb{P}}^{3}_{{\mathbb{Z}}}$. We denote it by $\bar{S}_{{\mathbb{Z}}}$. We also consider the Chow scheme

[TABLE]

and the subset

[TABLE]

where $P_{1b},\cdots,P_{mb}$ are the points in the complement of $S_{{\mathbb{Z}},b}$ inside the Zariski closure $\bar{S}_{{\mathbb{Z}},b}$. Then by Theorem 2.1, we get the following result.

Proposition 4.1**.**

The set ${\mathscr{Z}}_{d}$ is a countable union of Zariski closed subsets in the Chow scheme.

Applying the same argument as in Corollary 3.2, we see that there exists an irreducible Zariski closed subset ${\mathscr{Z}}_{i}$ inside the relative Picard scheme ${\rm{Pic}}(\bar{S_{{\mathbb{Z}}U}}\to U)$, where $U$ is Zariski open in $\AA^{2}_{{\mathbb{Z}}}$, such that the complexification of ${\mathscr{Z}}_{i,{\mathbb{C}}}$ maps dominantly onto $U_{{\mathbb{C}}}$ as well as the number of points in the fiber of this map is constant. Therefore, one gets the following:

Theorem 4.1**.**

The cardinality of a certain subgroup of ${\rm{Pic}}(S_{{\mathbb{Z}},b})$ remains constant as $b$ varies over $U$. However, the Picard group ${\rm{Pic}}(S_{{\mathbb{Z}},b})$ is nothing but the class group of the quadratic field ${\mathbb{Q}}(\sqrt{m^{2}q^{2}-\ell^{n}})$ for some fixed integers $m$ and $\ell$.

This concludes that given an element of order $n$ in ${\rm{Pic}}(S_{{\mathbb{Z}},b})$, one can find an element of the same order in ${\rm{Pic}}(S_{{\mathbb{Z}},b^{\prime}})$ for some $b^{\prime}\in U$ which is different from $b$.

5. Class group of $\mathbb{Q}(\sqrt{k^{2}-p^{n}})$

In this section, we will construct a subgroup in the class group of ${\mathbb{Q}}(\sqrt{m^{2}q^{2}-\ell^{2}})$ which is isomorphic to ${\mathbb{Z}}/n{\mathbb{Z}}$. More precisely, we prove:

Theorem 5.1**.**

Let $k\geq 1$ be an integer, $n\geq 3$ an odd integer and $p$ an odd integer such that $\gcd(k,p)=1$ and $k^{2}<p^{n}$. Let $-d~{}(<-3)$ be the square-free part of $k^{2}-p^{n}$. Then the class group of $K_{k,p,n}=\mathbb{Q}(\sqrt{-d})$ has an element of order $n$ if the following conditions hold:

• (i)

$k\not\equiv\pm 1\pmod{d}$.

• (ii)

$t^{q}\not\equiv\pm k\pmod{d}$* for any proper divisor $t$ of $k$ and any prime divisor $q$ of $n$.*

• (iii)

$p^{n/3}\neq(k^{\prime 3}+2k)/3$* when $d\equiv 3\pmod{4}$, $3\mid n$ and $k^{\prime}\mid k$ with odd positive integer $k^{\prime}$.*

Remark 5.1*.*

By putting $k=q$ an odd prime in Theorem 5.1, we obtain [5, Theorem 1.1].

Remark 5.2*.*

Theorem 5.1 gives [5, Theorem 4.1] when $k=1$.

We begin with the following crucial proposition on the solutions in positive integers $x,n$ of the following equation,

[TABLE]

where $d\geq 2$ and $k\geq 1$ are fixed integers such that $\gcd(k,d)=1$ and $p$ is odd prime number.

Proposition 5.2**.**

The equation (5.1) has at most one solution in positive integer $x$ and $n$, except $(x,n)=(11,5)$ which occurs only when $d=2$ and $k=1$.

In order to prove this proposition, we need to recall the following result of Bugeaud and Shorey [4]. Before stating this result, one needs to introduce some definitions and notations.

Let $F_{\ell}$ be denote the $\ell$-th term in the Fibonacci sequence defined by $F_{\ell+2}=F_{\ell}+F_{\ell+1}$ for $\ell\geq 0$ with the initials $F_{0}=0$ and $F_{1}=1$. Analogously, $L_{\ell}$ denotes the $\ell$-th term in the Lucas sequence defined by $L_{\ell+2}=L_{\ell}+L_{\ell+1}$ for $\ell\geq 0$ with the initials $L_{0}=2$ and $L_{1}=1$. For $\lambda\in\{1,\sqrt{2},2\}$, we define the subsets $\mathcal{F},\ \mathcal{G}_{\lambda},~{}\mathcal{H}_{\lambda}\subset\mathbb{N}\times\mathbb{N}\times\mathbb{N}$ by

[TABLE]

except when $\lambda=2$, in which case the condition “odd” on the prime $p$ should be removed from the definitions of $\mathcal{G}_{\lambda}$ and $\mathcal{H}_{\lambda}$.

Theorem I** (Bugeaud and Shorey, [4]).**

Given $\lambda\in\{1,\sqrt{2},2\}$, a prime $p$ and positive co-prime integers $D_{1}$ and $D_{2}$, the number of positive integer solutions $(x,y)$ of the equation

[TABLE]

is at most one except for

[TABLE]

and $(D_{1},D_{2},p)\in\mathcal{F}\cup\mathcal{G_{\lambda}}\cup\mathcal{H_{\lambda}}$.

We also recall the following result of Ljunggren [19] which will be needed to prove Proposition 5.2.

Theorem II**.**

For an odd integer $n$, the only solution $(x,y,n)$ to the equation

[TABLE]

in positive integers with $y>1$ is $(11,3,5)$.

We also need the following result of Cohn [8] which talks about appearance of squares in the Lucas sequence.

Theorem III**.**

The only perfect squares appearing in the Lucas sequence are $L_{1}=1$ and $L_{3}=4$.

Proof of Proposition 5.2.

Here, $(\lambda,D_{1},D_{2},p)=(1,d,k^{2},p)$. Thus if $(1,d,k^{2},p)\in\mathcal{S}$, then $d=2,k=1,p=3$, and hence (5.1) gives $2x^{2}+1=3^{n}$. Therefore by putting $y=3$ in (II) we see that $(x,n)=(11,5)$ is the only solution in positive integer of this equation.

We now assume that $(d,k^{2},p)\in\mathcal{F}$. Then $L_{\ell+\varepsilon}=k^{2}$, and thus by Theorem III only possible values of $\ell$ are $0,1,2$ and $4$. Again utilizing $F_{\ell}=p$, one gets $\ell=4$ which corresponds to $p=3,~{}k=2$ and $\varepsilon=-1$. Therefore by using $F_{\ell-2\varepsilon}=d$, we get $d=8$. This shows that $2\mid 3$ by (5.1) since $k=2$ and $p=3$. Hence $(d,k^{2},p)\not\in\mathcal{F}$.

Again, if $(d,k^{2},p)\in\mathcal{G}_{1}$ then $4p^{r}-1=k^{2}$ for some integer $r\geq 1$. This shows that $k^{2}\equiv 3\pmod{4}$ which is not true.

Finally, if $(d,k^{2},p)\in\mathcal{H}_{1}$ then there are positive integers $r$ and $s$ such that

[TABLE]

and

[TABLE]

One can conclude by reading (5.4) modulo $3$ that $3ds^{2}-k^{2}=1$ is not possible, and thus 5.4) becomes,

[TABLE]

This equation together with (5.5) imply

[TABLE]

This further implies

[TABLE]

This leads to the following possibilities:

[TABLE]

or

[TABLE]

The first pair of equations imply $2=3p^{r}-1$ which is not possible. Again, the last pair of equations implies $k=2,~{}p=5$ and $r=1$. Thus (5.5) gives $d=1$ which is a contradiction. Hence by Theorem I, we complete the proof. ∎

We now recall the following result [5, Proposition 2.1] which will be needed in the proof of the next result.

Theorem IV**.**

Let $d\equiv 5\pmod{8}$ be an integer and $q$ a prime. For any odd integers $a$ and $b$, the following holds:

[TABLE]

We now prove the following result which is the main ingredient in the proof of Theorem 5.1.

Proposition 5.3**.**

Let $k,p,n,d$ be as in Theorem 5.1 and let $m$ be the positive integer such that $k^{2}-p^{n}=-m^{2}d$. Then $\alpha=k+m\sqrt{-d}$ is not an $q^{th}$ power of an element in $\mathcal{O}_{K_{k,p,n}}$ for any prime divisor $q$ of $n$.

Proof.

Let $q$ be a prime number such that $q\mid n$. Then $q$ is odd since $n$ is odd.

We first consider the case when $d\equiv 1,2\pmod{4}$. Suppose $\alpha=\left(a+b\sqrt{-d}\right)^{\ell}$ for some integers $a$ and $b$. That is,

[TABLE]

We now compare the real parts to get,

[TABLE]

This shows that $a\mid k$ and thus $a=\pm 1,\pm t$ or $\pm k$ for some proper divisor $t$ of $k$.

If $a=\pm 1$, then (5.6) implies

[TABLE]

Reading this modulo $d$, we get $k\equiv\pm\pmod{d}$. This contradicts to (i).

When $a=\pm t$, then (5.6) becomes

[TABLE]

As in previous case, reading this modulo $d$ one gets $t^{q}\equiv\pm k\pmod{d}$. This contradicts to (ii).

We now consider the remaining case when $a=\pm k$. In this case, we have $k+m\sqrt{-d}=\left(\pm k+b\sqrt{-d}\right)^{q}$. We take the norm on both sides to get,

[TABLE]

This implies

[TABLE]

Also we have,

[TABLE]

Since $q$ is a prime divisor of $n$, so that (5.7) and (5.8) together show that $(|b|,n/q)$ and $(m,n)$ are two distinct solutions of (5.1) in positive integers $x$ and $y$. This contradicts to Proposition 5.2

We now consider the case when $d\equiv 3\pmod{4}$. If $\alpha$ is an $q^{th}$ power of some element in $\mathcal{O}_{K_{k,p,n}}$, then there are some rational integers $a,b$ with same parity such that

[TABLE]

If both $a$ and $b$ are even then one can proceed as in the case $d\equiv 1,2\pmod{4}$, and gets a contradiction to Proposition 5.2. Therefore one assumes that both $a$ and $b$ are odd. Taking norm on both sides, one gets

[TABLE]

Since $a,b$ and $p$ are odd, so that reading (5.9) modulo $8$, we obtain $d\equiv 3\pmod{8}$. As $k+m\sqrt{-d}=\left(\frac{a+b\sqrt{d}}{2}\right)^{q}\in\mathbb{Z}[\sqrt{-d}]$ and $-d\equiv 5\pmod{8}$, so that by Theorem IV we obtain $q=3$. Therefore,

[TABLE]

We compare the real parts to get,

[TABLE]

Since $a$ is odd, so that $a=\pm 1,~{}\pm k^{\prime}$ for some odd divisor $k^{\prime}$ (other than $\pm 1$) of $k$.

If $a=1$ then (5.10) gives $8k=1-3b^{2}d$. This is not possible as $d$ and $k$ are positive integers.

For $a=-1$, (5.9) and (5.10) become,

[TABLE]

This further implies

[TABLE]

For $a=\pm k^{\prime}$, (5.9) and (5.10) become,

[TABLE]

This further implies

[TABLE]

This again violates our assumption. Thus, we complete the proof. ∎

We are now in a position to present the proof of Theorem 5.1.

Proof of Theorem 5.1.

Let $m$ be the positive integer such that $k^{2}-p^{n}=m^{2}d$ and $\alpha=k+m\sqrt{-d}$. We note that $\gcd(\alpha,~{}\bar{\alpha})=1$ and $N(\alpha)=\alpha\bar{\alpha}=p^{n}$. Therefore, we can write $(\alpha)=\mathfrak{a}^{n}$ for some ideal $\mathfrak{a}$ in $\mathcal{O}_{K_{k,p,n}}$. We assume that $\mathfrak{A}$ is the ideal class containing $\mathfrak{a}$ in the class group of $K_{k,p,n}$. If the order of $\mathfrak{A}$ is less than $n$, then we obtain an odd prime divisor $q$ of $n$ and an element $\beta\in\mathcal{O}_{K_{k,p,n}}$ satisfying $(\alpha)=(\beta)^{q}$. Since $d>3$ is square-free, so that the only units in $\mathcal{O}_{K_{k,p,n}}$ are $\pm 1$, and thus these units can be absorbed into the $q$-th power of $\beta$. Hence we obatin $\alpha=\beta^{q}$ which contradicts to Proposition 5.3. Therefore, the order of $\mathfrak{A}$ is $n$ which completes the proof. ∎

For an odd positive integer $n$ not divisible by $3$ and a fixed psoitive integer $k$, the conditions (i) and (ii) in Theorem 5.1 hold very often. This can be proved using Siegel’s theorem on integral points on affine curves. More precisely, we prove the following result to show the infinititude of the imaginary quadratic fields of the form $K_{k,p,n}$ whose class group has an element of order $n$. We apply a celebrated theorem of Siegel [24] to prove this result.

Theorem 5.4**.**

Let $n\geq 5$ be an odd integer not divisible by $3$. For each positive integer $k$, the class group of $K_{k,p,n}$ has an element of order $n$ for infinitely many odd primes $p$.

Remark 5.3*.*

This is a generalization of [5, Theorem 1.2] which can be obtained by considering $k$ as an odd prime in Theorem 5.4. One can prove this theorem by a similar argument as in [5, Theorem 1.2]. However for the shake of completeness, we give a proof of it.

Proof.

Let $n\geq 2$ be an odd integer not divisible by $3$, and $k$ an arbitrary positive integer. Let $p$ be an odd prime such that $\gcd(p,k)=1$. Then by Theorem 5.1, the class group of $K_{k,p,n}$ has an element of order $n$ except for $t^{q}\equiv\pm k\pmod{d}$, where $t(\neq k)$ is a divisor of $k$ and $q$ is a prime divisor of $n$. For $t^{q}\equiv\pm k\pmod{d}$, one gets $d\leq k+t^{q}$. For any positive integer $D$,

[TABLE]

is irreducible algebraic curve of genus bigger than [math] (see [25]). Thus by Siegel’s theorem [24], it follows that there are only finitely many integral points $(x,y)$ on the curve (5.11). Therefore for each positive integer $d$, there are at most finitely many primes $p$ such that

[TABLE]

As $K_{k,p,n}=\mathbb{Q}(\sqrt{d})$, it follows that there are infinitely many fields of the form $K_{k,p,n}$ for each positive integer $k$. Furthermore from $q^{2}-p^{n}=m^{2}d$, it follows that $d>k+t^{q}$ for sufficiently large $k$. Therefore by Theorem 5.1, the class group of $K_{k,p,n}$ has an element of order $n$ for sufficiently large $k$. ∎

6. Concluding result

In Theorem 5.1, we have proved that there exists an element of order $n$ in the class group of $\mathbb{Q}(\sqrt{k^{2}-p^{n}})$. By Theorem 4.1, it follows that if we consider the family $\mathbb{Q}(\sqrt{m^{2}q^{2}-\ell^{n}})$ then there exists a Zariski open set $U$ in the base of the family such that for all $b\in U$, the order of the $n$-torsion subgroup of ${\rm{Pic}}(S_{\mathbb{Z},b})$ is invariant. Therefore we have:

Theorem 6.1**.**

${\rm{Pic}}(S_{{\mathbb{Z}},b})$* has a subgroups isomorhic to ${\mathbb{Z}}/n{\mathbb{Z}}$ for a given odd integer $n\geq 3$.*

Acknowledgements. A part of this paper was completed while A. Hoque was visiting Professor Jianya Liu at Shandong University. He is thankful to Professor Liu and the University for hosting him during this project. The authors thank Professor Dipendra Prasad and Professor Jean Gillibert for their comments and their encouragement. The authors would also like to thank Professor Kalyan Chakraborty for his valuable comments to improve the presentation of this paper. K. Banerjee is supported by DAE, Govt. of India. A. Hoque is supported by SERB N-PDF (PDF/2017/001958), Govt. of India.