Torsion hypersurfaces on abelian schemes and Betti coordinates

Abstract

In this paper we extend to arbitrary complex coefficients certain finiteness results on unlikely intersections linked to torsion in abelian surface schemes over a curve, which have been recently proved for the case of algebraic coefficients; in this way we complete the solution of Zilber–Pink conjecture for abelian surface schemes over a curve. As experience has shown also in previous cases, the extension from algebraic to complex coefficients often requires entirely new arguments, whereas simple specialization arguments fail. The outcome gives as a byproduct new finiteness results when the base of the scheme has arbitrary dimension; another consequence is a proof of an expectation of Mazur concerning the structure of the locus in the base when a given section is torsion. Finally, we show the link with an old work of Griffiths and Harris on a higher dimensional extension of a theorem of Poncelet.

Appendices

Appendix 1: Uniform Silverman’s theorem

In this Appendix we prove a uniform version of a special case of the main result in [20]. Let S be a smooth affine algebraic variety of dimension d over ${\bar{{{{\mathbb {Q}}}}}}$, $\varphi : S\rightarrow {{{\mathbb {P}}}}_{d-1}$ a morphism whose generic fibers $\varphi ^{-1}(\lambda )=: {{{\mathcal {C}}}}_\lambda$ are geometrically irreducible.

As a rule, we shall denote by $\lambda$ a ${\bar{{{{\mathbb {Q}}}}}}$-generic point of ${{{\mathbb {P}}}}_{d-1}$, while algebraic specializations of $\lambda$ shall be denoted by the letter $\theta$.

Theorem 5.1

Let $\pi :\, {{{\mathcal {A}}}}\rightarrow S$ be a family of abelian varieties over S, defined over a number field. Let $\sigma : S\rightarrow {{{\mathcal {A}}}}$ be a non-torsion section. Let $\varphi :S\rightarrow {{{\mathbb {P}}}}_{d-1}$ be a morphism as before. Then, there exist an open dense subset $S_0\subset S$, a positive real number $C_1>0$ depending only on $\sigma$ such that for all $\theta \in {{{\mathbb {P}}}}_{d-1}({\bar{{{{\mathbb {Q}}}}}})$, with the property that the restriction of $\sigma$ to ${{{\mathcal {C}}}}_\theta$ is non-torsion, every point $p\in {{{\mathcal {C}}}}_\theta \cap S_0 ({\bar{{{{\mathbb {Q}}}}}})$ such that $\sigma (p)$ is torsion satisfies

$$\begin{aligned} h(p)<C_1 (h(\theta )+1). \end{aligned}$$

Remarks

• Here the height of p is defined by any previously fixed projective embedding of S.

• If the point $\theta$ is fixed, this is a corollary of Silverman’s specialization bound applied to the family ${{{\mathcal {A}}}}\rightarrow {{{\mathcal {C}}}}_\theta$.

Proof

We use the setting of Chapter 12 in Lang’s book [8]. We note at once that by induction on $\dim S$ we may replace in the sequel S with a Zariski open dense subset.

We can compactify the scheme ${{{\mathcal {A}}}}\rightarrow S$ above a suitable dense open subset $S_0$ of S, to obtain a morphism ${{\bar{\pi }}}:{\overline{{{{\mathcal {A}}}}}}\rightarrow {{\bar{S}}}$ between projective varieties which coincides with our previous abelian scheme above $S_0$ and is flat above $S_0$.

Take an ample divisor on ${\overline{{{{\mathcal {A}}}}}}$, inducing an ample symmetric divisor on each abelian variety ${{{\mathcal {A}}}}_s:=\pi ^{-1}(s)$, for s in $S_0$. This divisor gives an embedding ${{{\mathcal {A}}}}\hookrightarrow {{{\mathbb {P}}}}_N$, hence a naive height in ${{{\mathcal {A}}}}({\bar{{{{\mathbb {Q}}}}}})$. For each $s\in S_0$, we also get en embedding of the abelian variety ${{{\mathcal {A}}}}_s$ into ${{{\mathbb {P}}}}_N$, so a (naive) height in $\pi ^{-1}(s)$ induced by that of ${{{\mathcal {A}}}}$.

Let now $\sigma : S_0\rightarrow {{{\mathcal {A}}}}$ be a non-torsion section. Composing with the embedding ${{{\mathcal {A}}}}\hookrightarrow {{{\mathbb {P}}}}_N$ we obtain a map $S_0\rightarrow {{{\mathbb {P}}}}_N$. Recall the construction of a rational map $\lambda : S\rightarrow {{{\mathbb {P}}}}_{d-1}$; for the generic $\lambda \in {{{\mathbb {P}}}}_{d-1}$, we can consider the restriction of the section $\sigma$ to the curve $\mathcal{C}_\lambda$, which will be denoted by $\sigma _\lambda$. Then we can write $\sigma _\lambda :\mathcal{C}_\lambda \rightarrow \AA \hookrightarrow {{{\mathbb {P}}}}_N$ as

$$\begin{aligned} (f_{0,\lambda }:\ldots :f_{N,\lambda }):\mathcal{C}_\lambda \rightarrow {{{\mathbb {P}}}}_N. \end{aligned}$$

More generally, replacing $\sigma$ by $M\cdot \sigma$, we obtain corresponding maps $(f_{0,\lambda }^{(M)}:\ldots :f_{N,\lambda }^{(M)}): \mathcal{C}_\lambda \rightarrow {{{\mathbb {P}}}}_N$. Here each $f_{i,\lambda }^{(M)}$ is a rational function on the curve $\mathcal{C}_\lambda$. We may also suppose $f_{0,\lambda }^{(M)}=1$.

Provided $\theta$ is an algebraic point in ${{{\mathbb {P}}}}_{d-1}$ lying outside a proper Zariski closed set, we may specialize at $\lambda =\theta$ obtaining corresponding functions $f_{i,\theta }^{(M)}$ on the irreducible curve $\mathcal{C}_\theta$ [which are irreducible by Lemma 2.5 (1)].

Remark

In the proof of the theorem and of the subsequent lemmas, we shall choose a fixed M; hence we can argue for specializations $\theta$ of $\lambda$ outside some prescribed hypersurface, and then use induction with a new S, namely the pre-image by $\varphi$ of such a hypersurface.

We shall use the lemmas below, for which we introduce a definition: given a rational function $f\in {\bar{{{{\mathbb {Q}}}}}}(x)$ its height will be the height of the vector of the coefficients of the defining polynomials. This height is naturally defined also for rational functions on curves, once x is chosen and the function field of the curve is viewed as a finite extension of ${\bar{{{{\mathbb {Q}}}}}}(x)$. We shall not use sharp estimates, but only inequalities up to a constant factor.

Lemma 5.2

There is a $c_3>0$ depending only on $\sigma$ such that for large M we have $\max _i \deg f_{i,\theta }^{(M)}\ge c_3 M^2$ for all $\theta \in {{{\mathbb {P}}}}_{d-1}({\bar{{{{\mathbb {Q}}}}}})$ outside a suitable proper Zariski-closed set depending on M.

Sketch of the proof of the Lemma. The canonical height ${\hat{h}}(\sigma _{\lambda })$ is $>0$, where this height is referred to the function field ${\bar{{{{\mathbb {Q}}}}}}({{{\mathcal {C}}}}_{\lambda })$. Hence there exists $c_4>0$ with ${\hat{h}}(M\sigma _{\lambda })>c_4 M^2$ for all $M\ge 0$. By the usual average procedure of Néron-Tate used to construct the canonical height from the naive height (see §3, page 29 in Serre’s book [18]), as in the proof of Zimmer’s theorem, applied to ${{{\mathcal {A}}}}\rightarrow {{{\mathcal {C}}}}_{\lambda }$ over the function field ${\bar{{{{\mathbb {Q}}}}}}({{{\mathcal {C}}}}_{\lambda })$, we deduce that $\max _i (\deg f_{i,\lambda }^{(M)})>c_5 M^2$. However, the special degree of a rational function in a family may decrease with respect to the generic one only in a proper Zariski closed set, proving the lemma.

Lemma 5.3

Let ${{{\mathcal {C}}}}_\lambda ,$ be as above defined over a number field k, $f_\lambda \in k({{{\mathcal {C}}}}_\lambda )$ be a rational function of degree d. Let $h: S({\bar{{{{\mathbb {Q}}}}}})\rightarrow {{{\mathbb {R}}}}$ be a height function associated to an ample divisor; let $\delta$ be the degree of such divisor on the curve ${{{\mathcal {C}}}}_\lambda$. Then there exists a function $c_5=c_5(\epsilon ,\deg f_\theta )$ such that for all $\epsilon >0$ and for all $\theta \in {{{\mathbb {P}}}}_{d-1}({\bar{{{{\mathbb {Q}}}}}})$ for which ${{{\mathcal {C}}}}_\theta$ is irreducible and $f_\theta$ is defined,

$$\begin{aligned}&(\deg f_\theta -\epsilon ) h(p) - c_5\cdot (h(f_\theta )+h(\theta )+1)\\&\quad \le \frac{h(f_\theta (p))}{\delta }\le (\deg f_\theta +\epsilon ) h(p) +c_5\cdot (h(f_\theta )+h(\theta )+1). \end{aligned}$$

Remark

A proof should follow from the classical height machine, since the precise dependence of $c_5$ is not specified. Moreover, for our purpose it is sufficient to use the left inequality with $\deg (f)/2$ instead of $\deg (f)-\epsilon$. In any case, a formal proof of the lemma as stated here follows for instance from Abouzaid’s Corollary 1.2 of [1] as follows: we take a fixed rational function x on ${{{\mathcal {C}}}}_\lambda$ and we apply Abouzaid’s result to $F(\lambda ,x,y)\in {\bar{{{{\mathbb {Q}}}}}}(\lambda )[x,y]$, the minimal polynomial such that $F(\lambda ,x,f_{\lambda })=0$. It is irreducible also as a polynomial over $\overline{{{{{\mathbb {Q}}}}}(\lambda )}$, since ${{{\mathcal {C}}}}_\lambda$ is absolutely irreducible over ${\bar{{{{\mathbb {Q}}}}}}(\lambda )$.

The lemma will be applied with $f_\lambda =f_{i,\lambda }^{(M)}$.

In the following lemma we bound the height of the coefficients of $f_{i,\lambda }^{(M)}$:

Lemma 5.4

In the above notation, there exists a function $M\mapsto c_7(M)$ (depending only on ${{{\mathcal {A}}}}\rightarrow S$ and $\sigma$), such that for all $i=0,\ldots ,N,\, \theta \in {{{\mathbb {P}}}}_{d-1}({\bar{{{{\mathbb {Q}}}}}})$ and $M\ge 1$,

$$\begin{aligned} h(f_{i,\theta }^{(M)})\le c_7(M)\cdot (1+h(\theta )) . \end{aligned}$$

(5.1)

Remark

We omit the easy proof. We could obtain $c_7(M)$ to be $c_7\cdot M^2$ but we shall not need this.

Let ${\hat{h}}_{{{{\mathcal {A}}}}_s}$ denote the Néron-Tate height in ${{{\mathcal {A}}}}_s$ associated to the above embedding (i.e. the renormalization of the naive height associated to the embedding) and $h_{{{\mathcal {A}}}}$ the naive height on ${{{\mathcal {A}}}}$, still relative to the given embedding; by restriction this gives the naive heights on ${{{\mathcal {A}}}}_s$. Then, with the above setting:

Lemma 5.5

There exist $\gamma _1,\gamma _2>0$ such that for all $s\in S$ and all $P\in \pi ^{-1}(s)$,

$$\begin{aligned} |{\hat{h}}_{{{{\mathcal {A}}}}_s}(P)-h_{{{\mathcal {A}}}}(P)|\le \gamma _1 h(s)+\gamma _2. \end{aligned}$$

This may be viewed as a ‘Uniform Zimmer’s theorem’ and essentially follows from Theorem 1.3, due to Silverman-Tate, Chapter 12 in Lang’s book [8]. Note that we need the estimate only above a Zariski-dense subset $S_0$ of S.

We now proceed to the proof of the theorem. As remarked before the above lemmas it suffices to prove the statement for all $\theta$ outside any prescribed hypersurface, because then we may argue by induction on the dimension of S, replacing S with the inverse image $\varphi ^{-1}$(hypersurface).

Let M be a sufficiently large integer to justify the estimate which shall follow, and let us argue supposing tacitly that $\theta$ does not belong to the above mentioned hypersurface, so that we can apply Lemma 5.2 to $\theta$.

Let then $\theta$ be such that ${{{\mathcal {C}}}}_\theta$ is non-torsion relative to $\sigma$. Suppose $p\in {{{\mathcal {C}}}}_\theta ({\bar{{{{\mathbb {Q}}}}}})$ is a point such that $\sigma (p)$ is torsion.

Then ${\hat{h}}_{{{{\mathcal {A}}}}_p}(\sigma (p))=0$ and also ${\hat{h}}_{{{{\mathcal {A}}}}_p}(M\sigma (p))=0$. By the last lemma,

$$\begin{aligned} |h_{{{{\mathcal {A}}}}}(M\sigma (p)|\le \gamma _1 h(p) +\gamma _2. \end{aligned}$$

However $M\sigma (p)=(1:f_{1,\theta }^{(M)}(p):\ldots :f_{N,\theta }^{(M)}(p))$. By Lemma 5.4 (left inequality), Lemma 5.3 and Lemma 5.5 we obtain

$$\begin{aligned} h_{{{{\mathcal {A}}}}}(M\sigma (p))\ge c_8 M^2 h(p) - c_9(M)\cdot (1+h(\theta )). \end{aligned}$$

for a function $c_9$, which can be suppose to be increasing in M.

Let us choose an index i providing the maximum of the degree for $f_{i,\lambda }^{(M)}$. Comparing upper and lower bounds for $h(M\sigma (p))$ and expressing in term of $f_{i,\lambda }^{(M)}(p)$ we obtain

$$\begin{aligned} h(p) (c_8 M^2-\gamma _1) \le \gamma _2 + c_9(M)(1+h(\theta )). \end{aligned}$$

Choose M such that $c_8 M^2> 2\gamma _1$ and we obtain for a suitable constant $C_1$, $h(p)<C_1(h(\theta )+1)$.

Appendix 2: Remarks on the split case

In this “Appendix”, we shall deal with a very special case of our theorems when the family is not simple. This case has been treated in the paper [15], by methods somewhat complicated. Hence we believe it is not entirely free of interest to present here an alternative approach more in line with the methods used in this paper for the case of simple families.

For these reasons, we limit ourself to a very special situation, giving moreover only a sketch of the argument.

Let S be a surface with algebraically independent regular functions $\lambda ,\mu$, so that we have a generically surjective map $(\lambda ,\mu ):S\rightarrow {{{\mathbb {A}}}}^2$.

Our abelian family will be ${{{\mathcal {A}}}}_{s}: E_{\lambda (s)}\times E_{\mu (s)}$, where $E_t$ is the Legendre curve:

$$\begin{aligned} E_t: y^2=x(x-1)(x-t) \end{aligned}$$

We tacitly assume that $\lambda ,\mu$ do not assume in S the value 0, 1. Now the section $\sigma :S\rightarrow {{{\mathcal {A}}}}$ corresponds to a pair of sections $P=P_s\in E_{\lambda (s)}, Q=Q_s\in E_{\mu (s)}$, where the coordinates of P, Q (which may be viewed as points defined over a fucntion field) are rational functions of s. We suppose that none of P, Q is torsion, which yields that they are linearly independent over $\mathrm {End}({{{\mathcal {A}}}}/S)$, since $E_\lambda , E_\mu$ are not identically isogenous.

As for the general case treated in the paper, we can define the Betti coordinates of the section, for which we adopt the same notation. Now our proof considers the two cases, according to the rank of the differential of the Betti map. We notice at once that the generic rank can be two or four, since by Proposition 2.1 it must be even, and it can be zero only for a torsion section.

Case I The generic rank of the differential of the Betti map is 2.

Note that, writing the Betti map as $s\mapsto (a_1(s),a_2(s),a_3(s),a_4(s))$, where $(a_1,a_2)$ is the Betti map corresponding to the first elliptic curve, $a_1,a_2$ (resp. $a_3,a_4$) must be analytically independent, for otherwise the fibers of the Betti map would have dimension ${\le }1$, hence 0 which we have excluded. Then locally either $a_3,a_4$ are functions of $a_1,a_2$ or vice-versa.

We let U be the maximal open set where ${\mathfrak {b}}$ has rank exactly two. It is the complement of the set where the rank is zero; on this latter set the Betti coordinates are locally constant, so this set is a countable union of fibers for the Betti maps; as observed in the preliminaries, the fibers of the Betti maps are complex-analytic curves. So U is the complement of an analytic curve.

As in the opening section of the paper, U is the union of finitely many simply connected open sets $U_\alpha$.

Lemma 6.1

For every $U_\alpha$ there exists a non empty open set $V_\alpha \subset U_\alpha$ and a ’transverse’ algebraic curve $Z_\alpha$ defined over ${\bar{{{{\mathbb {Q}}}}}}$ that meets in $V_\alpha$ all the level curves which meet $V_\alpha$.

By ‘transverse’ we mean transverse in the usual sense, i.e. non-tangent, to the level curves of the Betti map.

Proof

As above, by restricting $U_\alpha$ we can suppose that two coordinates of the Betti map, say $a_3,a_4$, are functions of $a_1,a_2$ (see also the notation above). The map $s\mapsto (a_1(s),a_2(s))$ is injective with differential of maximal rank (so $a_1,a_2$ become coordinates in the image). The level curves above $a_1,a_2$ in ($U_\alpha$) are level curves for $(a_1,a_2,a_3,a_4)$, therefore are complex-analytic; also, they are smooth because the rank of the differential is constant in $U_\alpha$.

Take an algebraic point p in $U_\alpha$; it is a smooth point in the corresponding level curve. We now take an algebraic curve $Z_\alpha$, defined over ${\bar{{{{\mathbb {Q}}}}}}$, passing through p and not tangent to the level curve. Shrinking the neighborhood of p, by the implicit function theorem again, we obtain the assertion. $\square$

Let us now fix the neighborhood $V_\alpha$ as in the lemma and let $Z_\alpha$ be the algebraic curve appearing in the lemma. By the construction of the lemma, the Betti map restricted to $Z_\alpha$ still has rank 2 As remarked above, we may suppose that the projection to the first two coordinates of the Betti map has still rank 2. Then we may find on $Z_\alpha$ a z where the Betti coordinates $a_1,a_2$ are rational of any given large enough denominator and this point z will be necessarily algebraic. This means that the section P takes a torsion value $P_z$ on $E_{\lambda (z)}$, whose order is the said denominator. The point z defines a level curve $X_z$ (the level curve containing z). This is a torsion curve for P, hence it is algebraic. Since $a_3,a_4$ are constant on $X_z$, by (a very special case of) Manin’s theorem of the kernel they must be rational. Then we obtain that for every point $z\in Z_\alpha$ which is torsion for the section P, this point is also torsion for the section Q. In this way we obtain infinitely many torsion values of the section $\sigma$ restricted to Z. By the algebraic case of [12], this is impossible since the restriction $\sigma _{Z_\alpha }$ is non torsion unless P, Q are linearly dependent on $Z_\alpha$. However by Silverman’s specialization theorem we may choose the curve $Z_\alpha$ in the lemma, with tangent vector of sufficiently large height so that this does not happen (or else, it suffices to exclude that $E_\lambda ,E_\mu$ become isogenous along $Z_\alpha$; this can be achieved e.g. by taking $Z_\alpha$ to be a line in the plane $\lambda ,\mu$, since only finitely many lines might correspond to isogenous families).

Case II The generic rank of the differential is 4. Let us fix a transcendental $\lambda _0$ and consider the curve $V=V_{\lambda _0}\subset S$, defined by $\lambda =\lambda _0$. There exists a non-empty open subset (in the complex-topology) $A\subset E_{\lambda _0}$ such that $P^{-1}(A)\cap V_{\lambda _0}$ does not contain critical points of the jacobian. Let X be a torsion curve of high order N. It intersects $V_{\lambda _0}$ in a point $x\in S$ because $\lambda _0$ is transcendental. Since $\lambda _0$ is transcendental, we may view x as a ${\bar{{{{\mathbb {Q}}}}}}$-generic point of X. By Galois, if the order of torsion N is sufficiently large, there is a conjugate of x inside our open set $P^{-1}(A)$, so the open set $P^{-1}(A)$ would contain a level curve.

Appendix 3: An issue of Mazur

Let us recall the last problem considered in the introduction. We take for the base S a finite cover of the moduli space $M_2$ of (complex) curves of genus 2, which is a quasi projective algebraic variety of dimension three. The fiber ${{{\mathcal {A}}}}_x$ over a point $x\in M_2$ (representing a curve of genus 2) will be the jacobian of that curve, which is a principally polarized abelian surface. We thus obtain an abelian scheme of relative dimension 2 over a three-dimensional base.

Take now a non-torsion section $\sigma : S\rightarrow {{{\mathcal {A}}}}$. We shall prove

Theorem 7.1

There exist torsion curves for $\sigma$ of arbitrarily high torsion order, and the union of these curves is topologically dense in S, and thus Zariski-dense.

Similarly to the proof of the Main theorem, the present proof involves the rank of the Betti map, and we can split it into two cases according. Since this rank is even by Proposition 2.1, and since it cannot be zero (by Manin’s theorem, because $\sigma$ is not torsion) we have only two cases to consider.

Case I The generic rank of the differential of the Betti map is 4 (maximal rank).

In this case the Betti map is locally surjective, so in particular it takes some rational value in every open set of S. Also, on a suitable open dense subset of S, the rank is exactly 4 and there the fibers of the Betti map are complex subvarieties of dimension 1 (use Proposition 1.2 above). The fibers of the rational points, being torsion curves are algebraic curves. This proves our assertion in this case.

Case II The generic rank of the differential of the Betti map is 2 (degenerate Betti map).

We shall in fact exclude this case, thus completing the proof of Theorem 7.1.

In this Case II, all non-empty fibers of the Betti map, which are complex varieties, will have complex dimension at least 2 (use again Proposition 1.2 above). On the other hand, by Manin’s theorem, this dimension cannot be 3 (for otherwise the Betti map would be constant, hence a rational constant by Manin’s theorem, so $\sigma$ would be torsion), so it will be exactly 2. If now the Betti map takes rational values with infinitely many denominators, we can apply our Main theorem obtaining a contradiction. To prove that this would be in fact the case, we shall restrict our analysis to some special surfaces inside $M_2$, namely those corresponding to split-jacobians.

Let us first prove that there are infinitely many such surfaces. Given a (smooth, projective) genus-1 curve E and two points $P_1,P_2$ on it, one can find, for every $d\ge 3$, a cover $X\rightarrow E$ of degree d, ramified only over $P_1,P_2$, where X has genus 2: it suffices to apply Riemann’s existence theorem with the suitable combinatorial data, which in this case correspond to having exactly one ramified point over each $P_i$, $i=1,2$, with ramification index 2. Now, such curves X depend, up to isomorphisms, on two parameters: the choice of the elliptic curve E (one-dimensional moduli space) and the choice of the point $P_2-P_1$ (where the operation is made according to the group law on E). Hence we have a surface on the moduli space, and correspondingly a surface on S. Also, it is easy to see that if we take pairwise coprime values of the degree d, the corresponding surfaces are indeed distinct (here we use the fact that the Jacobian of a generic curve X corresponding to a point on this surface is isogenous to a product of two non-isogenous elliptic curves, for otherwise the moduli space of such jacobians would be one-dimensional).

Now, by our Main theorem applied to S, only finitely many such surfaces can be torsion surfaces.

Take one such surface which is not torsion; by our results in “Appendix 2”, there are infinitely many torsion values for $\sigma$ on such a surface. This concludes the proof that Case II cannot occur.

As pointed out by the referee, in the situation of “Appendix 3” the base, being a finite cover of the moduli space ${{{\mathcal {A}}}}_{2,1}$, admits an explicit compactification, where the points at infinity can be described in modular terms. This may perhaps be used to obtain a simpler proof, by ‘algebraically’ constructing the relevant curves ${{{\mathcal {C}}}}_\theta$ avoiding the torsion hypersurfaces at infinity.