John R. Doyle Department of Mathematics
University of Georgia
Athens, GA 30602
Xander Faber Department of Mathematics
University of Hawaii
Honolulu, HI 96822
and  David Krumm Department of Mathematics
Claremont McKenna College
Claremont, CA 91711
###### Abstract.

To each quadratic number field and each quadratic polynomial with -coefficients, one can associate a finite directed graph whose vertices are the -rational preperiodic points for , and whose edges reflect the action of on these points. This paper has two main goals. (1) For an abstract directed graph , classify the pairs such that the isomorphism class of is realized by . We succeed completely for many graphs by applying a variety of dynamical and Diophantine techniques. (2) Give a complete description of the set of isomorphism classes of graphs that can be realized by some . A conjecture of Morton and Silverman implies that this set is finite. Based on our theoretical considerations and a wealth of empirical evidence derived from an algorithm that is developed in this paper, we speculate on a complete list of isomorphism classes of graphs that arise from quadratic polynomials over quadratic fields.

\renewbibmacro

## 1. Introduction

### 1.1. Background

Let be a number field and let be a rational function defined over , where and are coprime polynomials with coefficients in . The function naturally induces a map ; a fundamental problem in dynamics is that of describing the behavior of points under repeated iteration of the map . Thus, we consider the sequence

 P,f(P),f(f(P)),f(f(f(P))),….

For convenience we denote by the -fold composition of : is the identity map, and for all . We say that a point is preperiodic for if the orbit of under , i.e., the set , is finite. Furthermore, we say that is periodic for if it satisfies the stronger condition that for some ; in this case, the least positive integer with this property is called the period of . The set of all points that are preperiodic for is denoted by . Finally, the degree of is defined to be the number .

Using the theory of height functions, Northcott [northcott] proved that the set is finite as long as the degree of is greater than 1. This set can be given the structure of a directed graph by letting the elements be the vertices of the graph, and drawing directed edges for every such point . Thus, we obtain a finite directed graph representing the -rational preperiodic points for . It is then natural to ask how large the set can be, and what structure the associated graph can have. Drawing an analogy between preperiodic points of maps and torsion points on abelian varieties, Morton and Silverman [morton-silverman, 100] proposed the following conjecture regarding the size of the set :

###### Uniform Boundedness Conjecture (Morton-Silverman).

Fix integers and . There exists a constant such that for every number field of degree , and every rational function of degree ,

 #PrePer(f,K)≤M(n,d).

Very little is currently known about this conjecture; indeed, it has not been proved that such a constant exists, even in the simplified setting where and is a quadratic polynomial. However, Poonen [poonen_prep, Cor. 1] proposed an upper bound of 9 in this case, and moreover gave a conjecturally complete list of all possible graph structures arising in this context — see [poonen_prep, 17].

###### Theorem 1 (Poonen).

Assume that there is no quadratic polynomial over having a rational periodic point of period greater than 3. Then, for every quadratic polynomial with rational coefficients,

 #PrePer(f,Q)≤9.

Moreover, there are exactly 12 graphs that arise from as varies over all quadratic polynomials with rational coefficients.

Regarding the assumption made in Poonen’s result, it is known that there is no quadratic polynomial over having a rational periodic point of period or 5 (see [flynn-poonen-schaefer, Thm. 1] and [morton_4cycles, Thm. 4]), and assuming standard conjectures on -series of curves, the same holds for (see [stoll_6cycles, Thm. 7]). In [flynn-poonen-schaefer] Flynn, Poonen, and Schaefer conjecture that no quadratic polynomial over has a rational point of period greater than 3, a hypothesis which Hutz and Ingram have verified extensively by explicit computation (see [hutz-ingram, Prop. 1]). However, a proof of this conjecture seems distant at present.

One direction in which to continue the kind of work carried out by Poonen in studying the Uniform Boundedness Conjecture (UBC) is to consider preperiodic points for higher degree polynomials over , such as was done by Benedetto et al. [benedetto_cubics] in the case of cubics. In this paper we take a different approach and consider maps defined over number fields of degree . The case of degree is very special because there is only one number field with this degree; hence, in this case the UBC is a statement only about uniformity as one varies the rational function with -coefficients. More striking is that the conjecture predicts upper bounds even when all number fields of a fixed degree are considered. Our goal in this article is to carry out an initial study of preperiodic points for maps defined over quadratic number fields (the case of the UBC). Having fixed this value of , we will focus on the simplest family of maps to which the conjecture applies, namely quadratic polynomials. Thus, we wish to address the following questions:

1. How large can the set be as varies over all quadratic number fields and varies over all quadratic polynomials with coefficients in ?

2. What are all the possible graph structures corresponding to sets as and vary as above?

### 1.2. Outline of the paper

Our initial guesses for answers to the above questions were obtained by gathering large amounts of data, and doing this required an algorithm for computing all the preperiodic points of a given quadratic polynomial defined over a given number field. In §4 we develop an algorithm for doing this which relies heavily on a new method, due to the first and third authors [doyle-krumm], for listing elements of bounded height in number fields. Our algorithm for computing preperiodic points can in principle be applied to quadratic polynomials over any number field, but we will focus here on the case of quadratic fields. Using this algorithm we computed the set for roughly 250,000 pairs consisting of a quadratic field and a quadratic polynomial with coefficients in . Our strategy for choosing the fields and polynomials is explained in §4.4. The graph structures found by this computation are shown in Appendix B, and for each such graph we give in Appendix C an example of a pair for which the graph associated to is isomorphic to .

In order to state the more refined questions addressed in this article and our main results, we introduce some notation. From a dynamical standpoint, quadratic polynomials form a one-parameter family; more precisely, if is a number field and is a quadratic polynomial, then is equivalent, in a dynamical sense, to a unique polynomial of the form . (See the introduction to §3 for more details.) In studying the dynamical properties of quadratic polynomials, we will thus consider only polynomials of the form . We denote by the directed graph corresponding to the set , excluding the point at infinity.

The graphs arising from our computation did not all occur with the same frequency: some of them appeared only a few times, while others were extremely common. For each graph that was found we may then ask:

1. How many pairs are there for which the graph is isomorphic to ?

2. If there are only finitely many such pairs, can they be completely determined?

3. If there are infinitely many such pairs, can they be explicitly described?

Our strategy for addressing these questions is to translate them into Diophantine problems of determining the set of quadratic points on certain algebraic curves over . In essence, the idea is to attach to each graph an algebraic curve whose points parameterize instances of the graph . This philosophy of studying rational preperiodic points via algebraic curves was first taken up by Morton [morton_4cycles], and then pushed much further by Flynn-Poonen-Schaefer [flynn-poonen-schaefer], Poonen [poonen_prep], and Stoll [stoll_6cycles]. However, the Diophantine questions we need to answer in this article differ from those studied by previous authors, since they were interested primarily in finding -rational points on curves, whereas we need to determine all -rational points on a given curve , where is allowed to vary over all quadratic number fields. A survey of known theoretical results on this type of question is given in §2, where we also develop our basic computational methods for attacking the problem in practice. In §3 we construct the algebraic curves corresponding to the graphs found by our computation, and the methods of §2 are used to describe or completely determine their sets of quadratic points.

### 1.3. Main results

In our extensive computation of preperiodic points mentioned above, we obtained a total of 46 non-isomorphic graphs, and the maximum number of preperiodic points for the polynomials considered was 15 (counting the fixed point at infinity). This data may provide the correct answers to questions (1) and (2) posed in §1.1, though it is not our goal here to make this claim and attempt a proof. However, our computations do yield the following result:

###### Theorem 2.

Suppose that there exists a constant such that for every quadratic number field and quadratic polynomial with coefficients in . Then . Moreover, there are at least 46 directed graphs that arise from the set for such a field and polynomial .

For each of the 46 graphs that were found we would like to answer questions (1) - (3) stated in §1.2. This can be done easily for 12 of the graphs, namely those that appeared in Poonen’s paper [poonen_prep] — see §3.1 below. The essential tool for this is Northcott’s theorem on height bounds for the preperiodic points of a given map. For 15 of the 34 remaining graphs we were able to determine all pairs giving rise to the given graph. In most cases this was done by finding all quadratic points on the parameterizing curve of the graph, using our results concerning quadratic points on elliptic curves and on curves of genus 2 with Mordell-Weil rank 0. With the notation used in Appendix B, the labels for these graphs are

4(1), 5(1,1,)b, 5(2)a, 5(2)b, 6(2,1), 7(1,1)a, 7(1,1)b, 7(2,1,1)a,

7(2,1,1)b, 9(2,1,1), 10(1,1)a, 12(2,1,1)a, 14(2,1,1), 14(3,1,1), 14(3,2).

With the exception of graph 5(2)a — which occurs for two Galois conjugate pairs, namely — all of these graphs turned out to be unique; i.e., they occur for exactly one pair . Moreover, the parameter in all of these pairs is rational.

Of the remaining 19 graphs, there are 4 for which we were not able to determine all pairs giving rise to the graph, but instead proved an upper bound on the number of all such pairs that could possibly exist. This was done by reducing the problem of determining all quadratic points on the parameterizing curve to a problem of finding all rational points on certain hyperelliptic curves. Table 1 below summarizes our results for these graphs. The first column gives the label of the graph under consideration, the second column gives the number of known pairs corresponding to this graph structure, and the third column gives an upper bound for the number of such pairs.

In order to complete our analysis of these four graphs we would need to determine all rational points on the hyperelliptic curves defined by the following equations:

 w2=x7+3x6+x5−3x4+x3+3x2−3x+1, s2=−6x7−6x6+34x5+22x4−18x3+38x2−10x+10, w2=x12+2x11−13x10−26x9+67x8+124x7+26x6−44x5+179x4−62x3−5x2+6x+1, w2=(x2+1)(x2−2x−1)(x6−3x4−16x3+3x2−1).

Of the 15 graphs that remain to be considered, there are 9 for which we showed that the graph occurs infinitely many times over quadratic fields. This is achieved by using results from Diophantine geometry giving asymptotics for counting functions on the set of rational points on a curve.

###### Theorem 3.

For each of the graphs

8(1,1)a, 8(1,1)b, 8(2)a, 8(2)b, 8(4), 10(2,1,1)a, 10(2,1,1)b

there exist infinitely many pairs consisting of a real (resp. imaginary) quadratic field and an element for which contains a graph of this type. The same holds for the graphs 10(3,1,1) and 10(3,2), but these occur only over real quadratic fields.

###### Remark.

Showing the existence of infinitely many pairs for which not only contains a graph of a given type but in fact is itself of this type is a more difficult problem. This kind of result was achieved in the article [faber] for several of the graphs with .

For the six graphs that remain, our methods did not yield a satisfactory upper bound on the number of possible instances — see §1.4 below for more information.

Finally, we point out some results in this article which may be of independent interest. First, our description of quadratic points on elliptic curves in §2.1, though completely elementary, has been useful in practice for quickly generating many quadratic points on a given elliptic curve, as well as for proving several of our results stated above. The formula in §2.2 for the number of “non-obvious” quadratic points on a curve of genus 2 does not seem to be explicitly stated in the literature, nor is its application (in Theorem 4) to the study of quadratic points on the modular curves of genus 2. The techniques used here to determine all quadratic points on certain curves also appear to be new. As an example of our methods we mention our study of the graph 12(2,1,1)a in §3.14, which illustrates our approach to finding all quadratic points on a curve having maps and , where and are elliptic curves of rank 0. A second example is our study of the graph 12(4,2) in §3.17, in which we bound the number of quadratic points on a curve having a map , where is a curve of genus 2 and Mordell-Weil rank 0, and a map , where is a curve of genus 3 and Mordell-Weil rank 1.

### 1.4. Future work

This paper leaves open several questions that we intend to address in subsequent articles. First of all, there are a few graphs in Appendix B for which we are at present not able to carry out a good analysis of the quadratic points on the corresponding curves; precisely, these are the graphs labeled

10(1,1)b, 10(2), 10(3)a, 10(3)b, 12(3), and 12(6).

In most cases the difficulties do not seem insurmountable, but the methods required to study these graphs may be rather different from the ones used in this paper. Second, for some of the graphs analyzed in §3 we have only partially determined the quadratic points on the parameterizing curve, the obstruction being a problem of finding all rational points on certain hyperelliptic curves (listed above). We expect that the method of Chabauty and Coleman can be successfully applied to determine all rational points on these curves, thus completing our study of the corresponding graphs; this analysis will appear in a sequel to the present paper.

The next open question concerns 5-cycles. All evidence currently available suggests that there does not exist a quadratic polynomial defined over a quadratic field such that has a -rational point of period 5. Such a polynomial did not show up in our computations, nor was it found in a related search carried out by Hutz and Ingram [hutz-ingram]. We therefore set the following goal for future research:

Either find an example of a 5-cycle over a quadratic field, or show that it does not exist.

As a result of their own extensive search for periodic points with large period defined over quadratic fields, Hutz and Ingram [hutz-ingram, Prop. 2] provide evidence supporting the conjecture that 6 is the longest cycle length that can appear in this setting. Moreover, they found exactly one example of a 6-cycle over a quadratic field, which is the same one found during our computations and the same one that had been found earlier by Flynn, Poonen, and Schaefer [flynn-poonen-schaefer, 461]; namely the example given in Appendix C under the label 12(6). While the question of proving that 6 is the longest possible cycle length may be too ambitious, we do aim to study the following question:

Determine all instances of a 6-cycle over a quadratic field.

We remark that it follows from known results in Diophantine geometry (see Theorem 1 below) applied to the curve parameterizing 6-cycles 111The geometry of this curve, and its set of rational points, were studied by Stoll in [stoll_6cycles]. In particular, it is shown that the curve is not hyperelliptic and not bielliptic. that there are only finitely many pairs , with quadratic, giving rise to a 6-cycle.

As a final goal, we wish to prove a theorem analogous to Poonen’s result (Theorem 1), but in the context of quadratic fields. In view of the above discussion, we propose the following:

###### Speculation 4.

Assume that there is no quadratic polynomial over a quadratic field having a periodic point of period or . Then, for every quadratic polynomial with coefficients in a quadratic field ,

 #PrePer(f,K)≤15.

Moreover, there are exactly 46 graphs that arise from as varies over all quadratic polynomials with coefficients in a quadratic field .

Using techniques similar to those applied in §3, substantial progress towards proving this result — or one very similar to it — has already been made by the first author and will appear in a later paper.

### Acknowledgements

Part of the research for this article was undertaken during an NSF-sponsored VIGRE research seminar at the University of Georgia, supervised by the second author and Robert Rumely. The second author was partially supported by an NSF postdoctoral research fellowship. We would like to acknowledge the contributions of the other members of our research group: Adrian Brunyate, Allan Lacy, Alex Rice, Nathan Walters, and Steven Winburn. We are especially grateful to the Brown University Center for Computation and Visualization for providing access to their high performance computing cluster, and to the Institute for Computational and Experimental Research in Mathematics for facilitating this access. Finally, we thank Dino Lorenzini and Robert Rumely for many comments and suggestions; Anna Chorniy for her help in preparing the figures shown in Appendix B; and the anonymous referee for pointing out some gaps in our exposition in §3.

## 2. Quadratic points on algebraic curves

The material in this section forms the basis for our analysis of preperiodic graph structures in §3. As mentioned in the introduction, questions concerning the sets of preperiodic points for quadratic polynomials over quadratic number fields can be translated into questions about the sets of quadratic points on certain algebraic curves defined over . We will therefore require some basic facts about quadratic points on curves, from a theoretical as well as computational perspective.

Let be a number field, and fix an algebraic closure of . Let be a smooth, projective, geometrically connected curve defined over . We say that a point is quadratic over if , where denotes the field of definition of , i.e., the residue field of at . The set of all quadratic points on will be denoted by . It may well happen that : for instance, it follows from a theorem of Clark [clark, Cor. 4] that there are infinitely many curves of genus 1 over with this property. The set may also be nonempty and finite; several examples of this will be seen in §3. Finally, the set may be infinite. Suppose, for instance, that is hyperelliptic, so that it admits a morphism of degree 2 defined over . By pulling back -rational points on we obtain — by Hilbert’s irreducibility theorem [fried-jarden, Chap. 12] — infinitely many quadratic points on . Similarly, suppose is bielliptic, so that it admits a morphism of degree 2 to an elliptic curve . If the group has positive rank, then the same argument as above shows that has infinitely many quadratic points. Using Faltings’ theorem [faltings] concerning rational points on subvarieties of abelian varieties (formerly a conjecture of Lang), Harris and Silverman [harris-silverman, Cor. 3] showed that these are the only two types of curves that can have infinitely many quadratic points.

###### Theorem 1 (Harris-Silverman).

Let be a number field and let be a curve. If is neither bielliptic nor hyperelliptic, then the set is finite.

Thus, we have simple geometric criteria for deciding whether a given curve has finitely many or infinitely many quadratic points. However, we are not only interested here in abstract finiteness statements, but also in the practical question of explicitly determining all quadratic points on a curve, given a specific model for it. If the given curve has infinitely many quadratic points, then we will require an explicit description of all such points, and if it has only a finite number of quadratic points, then we require that they be determined. For the purposes of this paper we will mostly need to address these questions in the case of elliptic and hyperelliptic curves, or curves with a map of degree 2 to such a curve.

### 2.1. Case of elliptic curves

The following result gives an explicit description of all non-obvious quadratic points on an elliptic curve.

###### Lemma 2.

Let be an elliptic curve defined by an equation of the form

 y2=ax3+bx2+cx+d,

where and . Suppose that is a quadratic point with . Then there exist a point and an element such that and

 x2+ax0−v2+bax+ax20+v2x0+bx0−2y0v+ca=0.
###### Proof.

Since , we can write for some polynomial of degree at most 1. Note that is a root of the polynomial

 F(t):=at3+bt2+ct+d−p(t)2,

so must factor as , where is the minimal polynomial of and . Since , then . Letting we can write for some ; in particular, . Carrying out the division we obtain

 m(t)=t2+ax0−v2+bat+ax20+v2x0+bx0−2y0v+ca.\qed
###### Remark.

The description of quadratic points on given above has the following geometric interpretation: suppose is quadratic over , let be the field of definition of , and let be the nontrivial element of . We can then consider the point , where denotes the Galois conjugate of . If is the point at infinity on , then the line through and is vertical, so that and hence ; this gives rise to obvious quadratic points on . If is not the point at infinity, then it is an affine point in , say for some elements . The points , and are collinear, and the line containing them has slope in , say equal to . We then have , and this gives rise to the formula in Lemma 2.

### 2.2. Curves of genus 2

Suppose now that is a curve of genus 2. Fix an affine model for , where has degree 5 or 6, and let be the hyperelliptic involution on .

###### Lemma 3.

Suppose that has genus 2 and . Let be the Jacobian variety of .

1. The set of non-obvious quadratic points for the model is finite if and only if is finite.

2. Suppose that is finite, and let denote the number of non-obvious quadratic points for the given model. Then there is a relation

 q=2j−2+w−c2,

where ,  and is the number of points in that are fixed by .

###### Proof.

Fix a point and let be the embedding taking to 0. Let Sym denote the symmetric square of . Points in correspond to unordered pairs , where . The embedding induces a morphism taking to . We will need a few facts concerning the fibers of this morphism; see the article of Milne [milne] for the necessary background material. There is a copy of inside whose points correspond to pairs of the form . The image of under is a single point , and restricts to an isomorphism . In particular, there is a bijection

 (2.1) U(k)=S(k)∖P1(k)⟷J(k)∖{∗}.

Points in correspond to pairs of the form where either and are both in , or they are quadratic over and ; in particular, points in correspond to pairs where either or is an obvious quadratic point. Finally, the points of are either pairs with a non-obvious quadratic point, or pairs with but .

Hence, there are three essentially distinct ways of producing points in : first, we can take points and in and obtain a point . Second, we can take an obvious quadratic point and obtain . Finally, we can take a non-obvious quadratic point and obtain .

Let and denote, respectively, the set of obvious and non-obvious quadratic points on . We then have maps and , and a map defined as above. The proof of the lemma will be a careful analysis of the images of these three maps.

We have . Removing the points of from both sides we obtain

 (2.2) U(k)=(im(φ)∖P1(k))⊔im(ψn).

To prove part (1), suppose first that is finite. We know by Faltings’ theorem that is finite, so it follows from (2.2) that is finite. By (2.1) we conclude that is finite. Conversely, assume that is finite. Then is finite by (2.1), so im is finite by (2.2). But is 2-to-1 onto its image, so we conclude that is finite. This completes the proof of part (1).

To prove part (2), suppose that is finite and let , so that . By (2.1) and (2.2) we have

 (2.3) q/2=#U(k)−#(im(φ)∖P1(k))=j−1−#(im(φ)∖P1(k)).

By simple combinatorial arguments we see that

 #im(φ)=c+c(c−1)2and#(P1(k)∩im(φ))=w+c−w2.

Therefore,

 #(im(φ)∖P1(k))=c+c(c−1)2−w−c−w2=c2−w2.

By (2.3) we then have

 j−1=q2+c2−w2,

and part (2) follows immediately. ∎

### 2.3. Application to the modular curves X1(n) of genus 2

For later reference we record here some consequences of Lemma 3 in the particular case that and is one of the three modular curves of genus 2. We will fix models for these curves to be used throughout this section. The following equations are given in [poonen_prep, 15], [washington, 774], and [rabarison, 39], respectively:

 X1(13): y2=x6+2x5+x4+2x3+6x2+4x+1; X1(16): y2=−x(x2+1)(x2−2x−1); X1(18): y2=x6+2x5+5x4+10x3+10x2+4x+1.
###### Theorem 4.

1. Every quadratic point on is obvious.

2. The only non-obvious quadratic points on are the following four:

 (√−1,0),(−√−1,0),(1+√2,0),(1−√2,0).
3. The only non-obvious quadratic points on are the following four:

 (ω,ω−1),(ω2,ω2−1),(ω,1−ω),(ω2,1−ω2),

where is a primitive cube root of unity.

###### Proof.

It is known that the Jacobians for have only finitely many rational points (see [mazur_tate_13, §4] for the case of , [kenku, Thm. 1] for , and [kubert, Thm. IV.5.7] for ). Hence, Lemma 3 implies that the corresponding curves have only finitely many non-obvious quadratic points. The various quantities appearing in the lemma are either known or can be easily computed. Indeed, Ogg [ogg, 226] showed that for and

 #J1(13)(Q)=19,#J1(16)(Q)=20,#J1(18)(Q)=21.

The number in Lemma 3 is determined by the number of rational roots of the polynomial .

Applying Lemma 3 to the curve , we have , and hence . Therefore, all quadratic points on are obvious.

Similarly, with we have , and hence . We have already listed four non-obvious quadratic points, so these are all.

Finally, with we have , and hence . Therefore, has exactly four non-obvious quadratic points. Since we have already listed four such points, these must be all. ∎

## 3. Classification of preperiodic graph structures

For each graph appearing in the list of 46 graphs in Appendix B, it is our goal in this section to describe — as explicitly as possible — all the quadratic fields and quadratic polynomials such that the graph corresponding to the set is isomorphic to . There are several graphs in the list for which this description can be achieved without too much work: see §3.1 below for the case of graphs arising from quadratic polynomials over , and §5 for other graphs with special properties. Together, these two sections will cover all graphs in the appendix up to and including the one labeled 7(2,1,1)b, and also 8(2,1,1), 8(3), and 9(2,1,1); in this section we will focus on studying the remaining graphs. Our general approach is to construct a curve parameterizing occurrences of a given graph, then apply results from §2 to study the quadratic points on this curve, and from there obtain the desired description. As mentioned in §1.4, there are a few graphs for which this approach does not yet yield the type of result we are looking for; hence, we will exclude these graphs from consideration in this section.

For the purpose of studying preperiodic points of quadratic polynomials over a number field , it suffices to consider only polynomials of the form

 fc(z):=z2+c

with . Indeed, for every quadratic polynomial there is a unique linear polynomial and a unique such that . One sees easily that the graph representing the set of preperiodic points is unchanged upon passing from to ; hence, for our purposes we may restrict attention to the one-parameter family . For a quadratic polynomial with coefficients in , we denote by the directed graph corresponding to the set of -rational preperiodic points for , excluding the fixed point at infinity. If and are positive integers, a point of type for is an element which enters an -cycle after iterations of the map .

### 3.1. Graphs occurring over Q

Given a rational number and a quadratic field , we may consider the two sets and . For all but finitely many quadratic fields these sets will be equal, since Northcott’s theorem [silverman_dynamics, Thm. 3.12] implies that there are only finitely many quadratic elements of that are preperiodic for . Therefore, every graph appearing in Poonen’s paper [poonen_prep] — that is, every graph of the form with — will also occur as a graph for some quadratic field (in fact, for all but finitely many such ). These graphs all appear in Appendix B and are labeled

0, 2(1), 3(1,1), 3(2), 4(1,1), 4(2), 5(1,1)a, 6(1,1), 6(2), 6(3), 8(2,1,1), and 8(3).

For every graph in the above list, Poonen provided an explicit parameterization of the rational numbers for which . This essentially achieves, for each of the graphs above, our stated goal of describing the pairs giving rise to a given graph. There still remains the following question, which will not be further discussed here: given , how can one determine the quadratic fields for which and moreover, what are all the graphs that can arise in this way? From the data in Appendix C we see that many of the graphs shown in Appendix B are induced by a rational number .

### 3.2. Preliminaries

We collect here a few results that will be used repeatedly throughout this section.

Let be a smooth, projective, geometrically integral curve defined over ; let denote the genus of , and assume that . Let be the rank of the group , where denotes the Jacobian variety of . By Faltings’ theorem we know that is a finite set; the following three results can be used to obtain explicit upper bounds on the size of this set under the assumption that .

###### Theorem 1 (Coleman).

Suppose that and let be a prime of good reduction for . Let be a model of with good reduction. Then

 #X(Q)≤#X(Fp)+2g−2.
###### Proof.

See the proof of Corollary 4a in [coleman] and the remark following the corollary. ∎

###### Theorem 2 (Lorenzini-Tucker).

Assume that . Let be a prime of good reduction for , and let be a model of with good reduction. If is a positive integer such that and , then

 #X(Q)≤#X(Fp)+(p−1p−d)(2g−2).
###### Proof.

This follows from [lorenzini-tucker, Thm. 1.1]

###### Theorem 3 (Stoll).

Suppose that and let be a prime of good reduction for . Let be a model of with good reduction. Then

 #X(Q)≤#X(Fp)+2r.
###### Proof.

This is a consequence of [stoll_bound, Cor. 6.7].∎

The next result is crucial for obtaining equations for the parameterizing curves of the graphs to be considered in this section. Different versions of the various parts of this result have appeared elsewhere (for instance, [poonen_prep] and [walde-russo]), but not exactly in the form we will need. Hence, we include here the precise statements we require for our purposes.

###### Proposition 4.

Let be a number field and let with .

1. If has a fixed point , then there is an element such that

 p=x+1/2,c=1/4−x2.

Moreover, the point is also fixed by .

2. If has a point of period 2, then there is a nonzero element such that

 p=x−1/2,c=−3/4−x2.

Moreover, the orbit of under consists of the points and .

3. If has a point of period 3, then there is an element such that and

 p=x3+2x2+x+12x(x+1),c=−x6+2x5+4x4+8x3+9x2+4x+14x2(x+1)2.

Moreover, the orbit of under consists of the points and

 f(p) =x3−x−12x(x+1), f2(p) =−x3+2x2+3x+12x(x+1).
4. If has a point of period 4, then there are elements with such that

 y2=F16(x):=−x(x2+1)(x2−2x−1)

and

 p=x−12(x+1)+y2x(x−1),c=(x2−4x−1)(x4+x3+2x2−x+1)4x(x−1)2(x+1)2.

Moreover, the orbit of under consists of the points and

 f(p) =−x+12(x−1)+y2x(x+1), f2(p) =x−12(x+1)−y2x(x−1), f3(p) =−x+12(x−1)−y2x(x+1).
###### Proof.

1. The equation can be rewritten as . Letting we then have , and the result follows.

2. Since and , then we have the equation

 p2+p+c+1=f2(p)−pf(p)−p=0.

Letting , this equation becomes , and hence . Expressing and in terms of we obtain . We must have since and are distinct.

3. See the proof of [walde-russo, Thm. 3].

4. The existence of and and the expressions for and in terms of and can be obtained from the discussion in [morton_4cycles, 91-93]; the expressions for the elements of the orbit of are obtained by a straightforward calculation from the expressions for and . Finally, we must have since otherwise would have period smaller than 4: indeed, note that if . ∎

###### Remark.

Note that part (2) of Proposition 4 implies that can have at most two points of period 2 in , so that the graph can have at most one 2-cycle. This fact will be needed in the analysis of some of the graphs below.

The following lemma will allow us to show that certain preperiodic graph structures occur infinitely many times over quadratic fields.

###### Lemma 5.

Let have nonzero discriminant and degree . For every rational number , define a field by

 Kr:=Q(√p(r)).

Then, for every interval of positive length, the set contains infinitely many quadratic fields. In particular, if the polynomial function induced by takes both positive and negative values, then contains infinitely many real (resp. imaginary) quadratic fields.

###### Proof.

For the proof of the statement that contains infinitely many quadratic fields, see Appendix A. The second part follows from this statement by choosing an interval where and an interval where . ∎

### Use of computational software

In preparing this article we have made extensive use of both the Magma [magma] and Sage [sage] computer algebra systems. Our data-gathering computations explained in §4.4 were carried out by implementing the algorithms of §4.3 in Sage; these methods rely on the algorithm [doyle-krumm] for listing elements of bounded height in number fields, which is also implemented in Sage. In Magma, we have made use of some rather sophisticated tools; in particular, we apply the RankBound function, which implements Stoll’s algorithm [stoll_2descent] of 2-descent for bounding the rank of the group of rational points on the Jacobian of a hyperelliptic curve over . In addition, we frequently use the CurveQuotient function relying on Magma’s invariant theory functionality to determine the quotient of a curve by an automorphism. Whenever this function is used in our paper, one can easily check by hand that the output is correct. Finally, for the analysis of the graph 14(3,1,1) in §3.19 we require the Chabauty function, which implements a method due to Bruin and Stoll [bruin-stoll, §4.4] combining the method of Chabauty and Coleman with a Mordell-Weil sieve in order to determine the set of rational points on a curve of genus 2 with Jacobian of rank 1.

We can now proceed to the main task of this paper, namely to study the preperiodic graph structures appearing in Appendix B. We will consider the graphs one at a time, following the order in which they are listed in the appendix; however, the graphs discussed in §3.1 and §5 will not be considered henceforth in this section. The format for our discussion of each graph is roughly the same for all graphs: first, a parameterizing curve is constructed and an explicit map is given which shows how to use points on to obtain instances of . Next, a theorem is proved which describes the set of quadratic points on ; finally, we use this theorem to deduce information about the instances of defined over quadratic fields. When there are infinitely many examples of a particular graph occurring over quadratic fields, we will also be interested in deciding whether it occurs over both real and imaginary fields, or only one type of quadratic field.

### 3.3. Graph 8(1,1)a

###### Lemma 6.

Let be the affine curve of genus 1 defined by the equation

 y2=−(x2−3)(x2+1).

Consider the rational map given by

 a=−2xx2−1,b=yx2−1,c=−2(x2+1)(x2−1)2.

For every number field , the map induces a bijection from the set to the set of all triples such that and are points of type for the map satisfying .

###### Proof.

Fix a number field and suppose that satisfies . Defining as in the lemma, it is straightforward to verify that is a point of type for the map ; that is fixed by ; and that the following relations hold:

 (3.1) f2c(b)−fc(b)=2(x2+1)x2−1,f2c(b)−f2c(a)=x2+3x2−1.

It follows from these relations that if , then is of type and . Hence, gives a well-defined map.

To see that is surjective, suppose that are such that and are points of type for the map satisfying . The argument given in [poonen_prep, 19] then shows that there exists a point with such that . Furthermore, the relations (3.1) imply that necessarily . To see that is injective, one can verify that if , then

 x=−aa2+c,y=2ba2+c.\qed
###### Remark.

As shown in [poonen_prep, 19], the curve is birational over to the elliptic curve 24a4 in Cremona’s tables [cremona].

###### Proposition 7.

There are infinitely many real (resp. imaginary) quadratic fields containing an element for which admits a subgraph of type 8(1,1)a.

###### Proof.

Let . Applying Lemma 5 to the polynomial we obtain infinitely many real (resp. imaginary) quadratic fields of the form with . For every such field there is a point which necessarily satisfies ; hence, by Lemma 6 there is an element such that has points of type with . In order to conclude that contains a subgraph of type 8(1,1)a we need the additional condition that , so that the points and each have two distinct preimages. We have

 ab2=2r(r2−3)(r2+1)(r2−1)3,

so the condition will be satisfied as long as . ∎

### 3.4. Graph 8(1,1)b

###### Lemma 8.

Let be the affine curve of genus 1 defined by the equation

 y2=2(x3+x2−x+1).

Consider the rational map given by

 p=yx2−1,c=−2(x2+1)(x2−1)2.

For every number field , the map induces a bijection from the set to the set of all pairs such that is a point of type for the map .

###### Proof.

Fix a number field and suppose that satisfies . Defining and as in the lemma, it is straightforward to verify that is a point of type for the map . Hence, gives a well-defined map.

To see that is surjective, suppose that are such that is a point of type for . Then an argument given in [poonen_prep, 22] shows that there exists a point with such that . To see that is injective, one can verify that if , then

 x=fc(p)f2c(p),y=p(x2−1).\qed
###### Remark.

As shown in [poonen_prep, 23], the curve is birational over to the elliptic curve 11a3 in Cremona’s tables [cremona], which is the modular curve .

###### Proposition 9.

There are infinitely many real (resp. imaginary) quadratic fields containing an element for which admits a subgraph of type 8(1,1)b.

###### Proof.

Let . Applying Lemma 5 to the polynomial we obtain infinitely many real (resp. imaginary) quadratic fields of the form with . For every such field there is a point which necessarily satisfies ; hence, by Lemma 8 there is an element such that has a point of type . In order to conclude that contains a subgraph of type 8(1,1)b we need the additional condition that . Indeed, the condition ensures that and each have two distinct preimages, while the condition guarantees that has two distinct fixed points, and each fixed point has a preimage different from itself. Now, one can check that

 p2⋅fc(p)⋅c(4c−1)=8r(r3+r2−r+1)(r2+1)(r2+3)2(r2−1)7,

so we will have as long as . ∎

### 3.5. Graph 8(2)a

###### Lemma 10.

Let be the affine curve of genus 1 defined by the equation

 y2=2(x4+2x3−2x+1).

Consider the rational map given by

 a=−x2+1x2−1,b=yx2−1,c=−x4+2x3