An Algorithm For Checking Injectivity Of Specialization Maps From Elliptic Surfaces

Let $C$ be a (complete nonsingular) curve defined over a number field $k$ with function field $k(C)$. Let $E/k(C)$ be an elliptic curve defined by the Weierstrass equation

For any $t_{0}\in C(k)$ such that the discriminant $4A(t)^{3}+27B(t)^{2}$ of $E$ does not vanish or have a pole at $t_{0}$, we define the elliptic curve $E_{0}/k$ using the Weierstrass equation $y^{2}=x^{3}+A(t_{0})x+B(t_{0})$. The specialization map at $t_{0}$ is the map

which takes the point $(x(t),y(t))\in E(k(C))$ to $(x(t_{0}),y(t_{0}))\in E_{t_{0}}(k)$. It is in fact a group homomorphism; that is, it respects the standard chord-and-tangent group laws on the domain and codomain. It is natural to ask what information can be extracted about the relationship between $E(k(C))$ and $E_{t_{0}}(k)$ through this homomorphism.

In 1952, Néron proved a theorem regarding specialization of which the following result is a special case.

Thirty years later, Silverman [11] improved on Néron’s result by proving that all but finitely many maps $\sigma_{t_{0}}$ are injective by showing that the set of $t_{0}\in k$ for which injectivity fails is a set of bounded height.

While Silverman’s result is effective, it might not give a practical way to find this finite set. So the aim of this article is to examine the injectivity of specialization homomorphisms in a different way. In particular, given an elliptic curve $E/k(t)$ given by a Weierstrass equation as above, how might one go about effectively determining the subset $\Sigma$ of $k$ containing those $t_{0}$’s for which the corresponding specialization maps are injective? We will focus on the case $k=\mathbb{Q}$ for concreteness. Gusić and Tadić [2] gave a method based on 2-descent which addresses this problem for curves with nontrivial $\mathbb{Q}(t)$-rational 2-torsion, but as they remark in their article the method does not immediately generalize to curves with trivial $\mathbb{Q}(t)$-rational 2-torsion.

We consider a different method to find $t_{0}$’s for which the specialization map is injective. More specifically, we will discuss an approach to answering the following two questions.

1. 1.

   Given some $t_{0}\in\mathbb{Q}$, how can we effectively determine whether or not the specialization map at $t_{0}$ is injective?

2. 2.

   What is the set $\Sigma$ of $t_{0}\in\mathbb{Q}$ such that the specialization map at $t_{0}$ is injective?

We will approach these questions from the perspective of irreducibility. We review the classical Néron specialization theorem to relate the above questions to the question of irreducibility of polynomials after specialization, and we give an algorithm that can be used to find a Hilbert set which intersects $\Sigma$ in an infinite set $S$. The approach taken here was inspired by a mathoverflow post by Silverman [8]. The algorithm is effective in certain cases, and we carry out the algorithm on some examples. In doing so, the elliptic curve packages in Sage [15] and Magma [1] were indispensible. They will be named wherever they were used.

We begin by reviewing the basics on division polynomials and Hilbert sets that are used in the proof of Néron’s specialization theorem.

Let $K$ be a field of characteristic zero. Given a $K$-rational point on some elliptic curve $E$, finding a point $Q$ with $nQ=P$ requires solving polynomial equations over $K$. The polynomials which appear in this way are called division polynomials. Fix $A,B\in K$ with $4A^{3}+27B^{2}\neq 0$ and define

that is, $\mathcal{O}$ is the coordinate ring of the elliptic curve

defined over $K$. Define the following sequence of polynomials in $\mathcal{O}$.

We have the following standard facts about the polynomials above, which we state without proof.

• 1.

  The polynomials $\psi_{2n+1}$, $\psi_{2n}/y$ and $\psi_{2n}^{2}$ depend only on $x$.

• 2.

  For $n\geq 1$, the set of roots of $\psi_{2n+1}$ is the set of $x$-coordinates of the nonzero $(2n+1)$-torsion points.

• 3.

  For $n\geq 2$, the set of roots of $\psi_{2n}/y$ is the set of $x$-coordinates of the nonzero $(2n)$-torsion points which are not 2-torsion. Additionally, the set of roots of $x^{3}+Ax+B$ is the set of $x$-coordinates of nonzero 2-torsion points. Since $y^{2}=x^{3}+Ax+B$, we see that the set of roots of $\psi_{2n}^{2}$ is the set of $x$-coordinates of all nonzero $(2n)$-torsion points. However, it is worth noting that this polynomial is not separable.

• 4.

  Combining the previous two statements, for $n\geq 2$ we have that the set of roots of $\psi_{n}^{2}$ is the set of $x$-coordinates of the nonzero $n$-torsion points.

• 5.

  If we additionally define

  $$\phi_{n}=x\psi_{n}^{2}-\psi_{n+1}\psi_{n-1},\qquad\omega_{n}=\frac{\psi_{n+2}\psi^{2}_{n-1}-\psi_{n-2}\psi^{2}_{n+1}}{4y},$$

  then for any $Q=(x_{Q},y_{Q})\in E(K)\setminus E[n](K)$ we have that

  $$nQ=\left(\frac{\phi_{n}(x_{Q})}{\psi^{2}_{n}(x_{Q})},\frac{\omega_{n}(x_{Q},y_{Q})}{\psi_{n}^{3}(x_{Q},y_{Q})}\right).$$

  (1)

  Note that, if $Q\in E[n](K)$, by the above discussion we must have $\psi^{2}_{n}(x_{Q})=0$.

• 6.

  The polynomials $\psi_{n}^{2}$ and $\phi_{n}$ have degrees $n^{2}-1$ and $n^{2}$, respectively.

Using Equation (1), the point $P\neq O$ is divisible by $n$ in $E(K)$ if and only if there exists a point $Q=(x_{Q},y_{Q})\in E(K)$ such that

Focusing on the equation for $x_{P}$, we must have that

Thus $x$-coordinates of points $Q$ with $nQ=P$ satisfy the polynomial

We call $d_{n,P}(x)$ the $n$-division polynomial of the point $P$.

We now mention some basic facts regarding Hilbert sets. For more details about Hilbert sets and Hilbert’s irreducibility theorem in greater generality, see Lang [3].

Notice that the intersection of any two Hilbert sets is a Hilbert set - it corresponds to the union of the sets of polynomials defining the two Hilbert sets. Additionally, note that the preceding definition varies slightly from that in [3], where a Hilbert set is also allowed to intersect finitely many Zariski open sets of $\mathbb{A}^{1}_{\bar{\mathbb{Q}}}$ - that is, they might differ from a Hilbert set as above by finitely many points. Although we will have a use for such a set later, we prefer to use the term “Hilbert set” for sets precisely corresponding to irreducible specializations.

Recall that the natural density of a subset $T$ of $\mathbb{N}$ is the limit

Néron’s specialization theorem (for elliptic curves) is the result of applying Hilbert’s irreducibility theorem to division polynomials. Following Serre [7, Chapter 11], we start by recalling a completely group-theoretic fact.

Fix an elliptic curve $E$ over $\mathbb{Q}(t)$, and set $\phi=\sigma_{t_{0}}$ the specialization homomorphism for a fixed $t_{0}\in\mathbb{Q}$, $M=E(\mathbb{Q}(t))$, $N=E_{t_{0}}(\mathbb{Q})$ and a positive integer $n\geq 2$. Then conditions 1 and 4 above are always true; indeed, condition 1 is the function field version of the Mordell-Weil Theorem [12, Chapter III §6], and condition 4 follows from basic results on formal groups of elliptic curves and their relationship to reduction mod $p$ found in Silverman [13, Chapter VII §3]. It is true that conditions 2 and 3 hold inside of a Hilbert set, but instead of proving this fact directly we will replace $E(\mathbb{Q}(t))$ with specific subgroups (which will include $E(\mathbb{Q}(t))$ itself) and prove a more general statement.

First, notice that condition 2 is equivalent to the following statement.

Now suppose $M<E(\mathbb{Q}(t))$ and $\phi=\sigma_{t_{0}}|_{M}$. If we have some $a\in M$ with $\phi(a)$ divisible by $n$, then even if we are able to conclude that $a$ is divisible by $n$ in $E(\mathbb{Q}(t))$, say $a=nb$ for some $b\in E(\mathbb{Q}(t))$, in order to verify (3) we would be required to check that $b\in M$. Unfortunately, this is a difficult problem. To avoid this, we will restrict ourselves to subgroups $M$ in which whenever $a\in M$ has the property that $a=nb$ for some $b\in E(\mathbb{Q}(t))$, we always know that $b\in M$. That is, condition 2 of Proposition 7 is satisfied for the inclusion $M\to E(\mathbb{Q}(t))$. Of course, this isn’t always true; for example, if a subgroup $G<E(\mathbb{Q}(t))$ contains an element of infinite order then $M=nG$ always fails to have this property (recall that $G$ is finitely generated). Regarding condition 3, since specialization is always injective on torsion, note that if $M[n]$ is a proper subgroup of $E[n](\mathbb{Q}(t))$, then $\phi(M[n])$ is also always a proper subgroup of $E_{t_{0}}[n](\mathbb{Q})$. Hence we also make the minor additional assumption that $M[n]=E[n](\mathbb{Q}(t))$. Thus we will assume all hypotheses of Proposition 7 for the inclusion $M\to E(\mathbb{Q}(t))$.

Now suppose that we have $M<E(\mathbb{Q}(t))$ with the inclusion satisfying the hypotheses of Proposition 7, $N=E_{t_{0}}(\mathbb{Q})$ and $\phi=\sigma_{t_{0}}$. In our goal of finding a Hilbert set on which conditions 2 and 3 of Proposition 7 hold for these very specific $M,N,\phi$, one might hope that we could discard either condition 2 or 3 and still maintain the conclusion of Proposition 7. Unfortunately neither condition implies the other, as the following two examples illustrate. We first look at an example where condition 2 holds but condition 3 does not.

Next, we consider an example where condition 3 holds but condition 2 does not.

Thus we will need to check both of the conditions as part of the following proof of Néron’s specialization theorem for subgroups. For a more general version of this theorem, see [3].

We conclude the section with a summary of how the previous proof yields an algorithm that can often be used to check when a specialization map is injective.

In this section, we discuss some explicit examples of utilizing Algorithm 13 with the modification mentioned in Remark 14. We consider two examples, one with a full Mordell-Weil group of rank 2 and another with a subgroup of rank 2.

We first consider

Our goal is to find an infinite set of rational numbers for which the corresponding specialization maps (on all of $E(\mathbb{Q}(t))$) are injective. First, we need generators of $E(\mathbb{Q}(t))$ in order to use Algorithm 13. Set $P=(t,t)$ and $Q=(0,t)$. One can check (using Magma) that the determinant of the canonical height matrix of $P$ and $Q$ is nonzero, and thus $P$ and $Q$ are linearly independent in $E(\mathbb{Q}(t))$. In addition, using Magma’s implementation of Tate’s algorithm [12, Chapter IV §9] and combining the resulting information with the Shioda-Tate formula [9], the Mordell-Weil rank of $E/{}\mkern 1.0mu\overline{\mkern-1.0mu\mathbb{Q}}(t)$ is 2. Hence the rank of $E(\mathbb{Q}(t))$ is 2. In order to show that $P$ and $Q$ generate $E(\mathbb{Q}(t))$, we will use specialization in a way that is motivated by (but different from) the method outlined in [14]. First, we show that $E(\mathbb{Q}(t))$ has trivial torsion. Since specialization is injective on torsion, it suffices to show that a single specialization has trivial torsion. To see this, consider the following example, which (at the same time) highlights Remark 15 and shows how injectivity of the specialization map for an individual $t_{0}\in\mathbb{Q}$ can often be checked directly using a computer algebra system such as Sage.

It remains to show that $P$ and $Q$ generate $E(\mathbb{Q}(t))$. Consider the specialization at $t_{0}=5$ as in Example 17. Sage yields that $E_{5}$ has Mordell-Weil group $\mathbb{Z}^{2}$ over $\mathbb{Q}$ with generators $(-1,7)$ and $(0,5)$. Since $(-1,7)+(0,5)=(5,5),$ we may instead use $(5,5)=P_{5}$ and $(0,5)=Q_{5}$ as generators. Fix the bases $\{P,Q\},\{P_{5},Q_{5}\}$ for $M,E_{5}(\mathbb{Q})$, respectively. After fixing some basis for $E(\mathbb{Q}(t))$ (which has 2 elements), we let the matrix $A$ represent the inclusion $M\to E(\mathbb{Q}(t))$ and the matrix $B$ represent the specialization map $\sigma_{5}$. We then have a sequence

The composition $BA$ is the specialization map $\sigma_{5}|_{M}$. Since this maps generators of $M$ to generators of $E_{5}(\mathbb{Q})$, $BA$ is the identity matrix. Hence $A$ is invertible, so the inclusion $M\to E(\mathbb{Q}(t))$ is surjective. Hence $E(\mathbb{Q}(t))\cong\mathbb{Z}^{2}$ with generators $P=(t,t)$ and $Q=(0,t)$.

Before moving forward with using Algorithm 13 to find injective specialization maps, it is important to notice that success of this method for a fixed $n$ is not equivalent to injectivity of the specialization map. We can’t hope for this to be true since it succeeds on (most of) a Hilbert set and Hilbert sets often have infinite complements, whereas Silverman’s specialization theorem states that the specialization map fails to be injective for only finitely many rational numbers. The next example illustrates the failure of this equivalence.

We now carry out Algorithm 13 for $n=2$. As in §2, we find the the polynomials in steps 3 and 4 to be

where $g(t,x)=\psi_{2}^{2}/4$. Notice that all four polynomials are irreducible over $\mathbb{Q}[t,x]$. As in Remark 14, we need to find $t_{0}$’s for which the specialized polynomials have no roots in $\mathbb{Q}$. Equivalently, we need to find $t_{0}$’s for which the curves defined by the polynomials have no rational points of the form $(t_{0},x_{0})$. Set

Using Sage, all of the curves are rational over $\mathbb{Q}$ and have rational points, hence they have infinitely many rational points. Because of this, we will restrict to $t_{0}\in\mathbb{N}$ and examine the case of specializing at natural numbers, which similarly reduces to looking at integral points on the curves (since each polynomial is monic in $x$). We first prove, using elementary methods, that the only obstruction to success of the method for $t_{0}>2$ comes from $C_{P}$. We begin with an algebraic lemma which will make analyzing $C_{P+Q}$ easy.