On the Parshin–Arakelov theorem and integral sections on elliptic surfaces

Abstract

It is well-known that the Parshin–Arakelov theorem implies the Mordell conjecture over complex function fields by a covering construction of Parshin. Via a similar map in the context of integral points on elliptic curves over function fields, we explain how to obtain a short geometric proof of a uniform version of Siegel’s theorem. Our technique also allows us to establish a uniform quantitative result on the set-theoretic intersection of curves with the singular divisor in the compact moduli space of stable curves.

Appendices

Appendix A.

For ease of reading, we reformulate several known results necessary for our article.

1.1 A.1 de Franchis theorems

We collect several finiteness results concerning ramified coverings of curves. Let C be a smooth, geometrically connected, projective curve of genus q over a field k. Let $F=k(C)$ and $m\in {\mathbb {N}}$. Each subfield $k\subsetneq F'\subset F$ corresponds to an k-isomorphism class of a smooth, geometrically connected, projective curve $C'$ such that the corresponding non constant map $f :C\rightarrow C'$ verifies $f^*(k(C'))=F'$.

An immediate consequence of a result of Tamme-Kani is the following effective version of the de Franchis Theorem:

Theorem A.1

(Tamme-Kani) Let $m\in {\mathbb {N}}$. Let E be an elliptic curve over over a field k. Let C be a smooth projective geometrically connected curve of genus q over k. Then, up to composition with an element of ${{\,\mathrm{Aut}\,}}_k(E)$, the number of degree-m covers $h :C\rightarrow E$ is uniformly bounded by an effective function M(q, m) depending only on q and m.

Proof

It suffices to apply (Kani 1986, Theorem 4) (and the corollary that follows) and define $M(q,m)= 2^{6q^2-1}m^{4q^2-2}(\zeta (2)/2)m^2+(m/2)(\log (m)+1).$ $\square$

Given two covers $f_1 :X\rightarrow Y_1$ and $f_2 :X\rightarrow Y_2$ of compact Riemann surfaces, we say that $f_1$ and $f_2$ are equivalent if there is a biholomorphic map $p:Y_1 \rightarrow Y_2$ such that $f_2=p\circ f_1$.

1.2 A.2 Simple cyclic covers

Let k be an algebraically closed field of characteristic 0.

Proposition A.2

(Simple cyclic covers) Let $m\in {\mathbb {N}}$. Let X be an irreducible projective smooth variety over k. Let D be an effective divisor of X such that $D \sim L^{\otimes m}$ for some line bundle L of X. There exists a unique finite cyclic cover of degree m of irreducible projective k-varieties $f_{D, L, m} :X' \rightarrow X$ such that f is totally ramified and ramified only above D and every point of $f_{D,L,m}^{-1} (X \setminus D_{\text {sing}})$ is a regular point of $X'$.

Proof

See for example Pardini (1991), in particular (Pardini 1991, Proposition 3.1). $\square$

Moreover, it is clear that construction of simple cyclic covers is functorial:

Proposition A.3

Let X, Y be smooth irreducible projective varieties. Let $g :Y \rightarrow X$ be a morphism such that $g^*D$ is well defined where D is an effective divisor on X. Let $m \in {\mathbb {N}}$ and L a line bundle such that $L^{\otimes m}\sim {\mathcal {O}}_X(D)$. We have a cartesian square:

1.3 A.3 Geometry of elliptic surfaces

Theorem A.4

Let $f:X\rightarrow B$ be a minimal elliptic surface with a section (O) and let E be the associated elliptic curve over K. Then the following hold:

• (1) For each $P \in E(K)=X(B)$, the rational maps $\tau _P:X \rightarrow X$ induced by the translation by P on each regular fiber extends, by minimality of X, to an B-automorphism.

• (2) We have a homomorphism of groups $E(K) \rightarrow \text {Aut}_B(X)$, $P \mapsto \tau _P$. In particular, $\tau _{-P}$ is the inverse of $\tau _P$ for any $P\in E(K)$.

• (3) We have a canonical isomorphism of groups $\text {Isom}_K(E)\simeq {{\,\mathrm{Aut}\,}}_B(X)$.

• (4) The isomorphism group of E over K is given by $\text {Isom}_K(E)\simeq E(K) \rtimes {{\,\mathrm{Aut}\,}}_K(E)$ where $\#{{\,\mathrm{Aut}\,}}_{{\overline{K}}}(E)=2,4,6$ according to $j_E\ne 0,1728, j_E=1728$ or $j_E=0$.

Proof

See for example (Silverman 1994, Theorem III.9.1) for (1), (2), (3), (Parshin 1968, Theorem 0) for (3), and (Silverman 2009, Theorem III.10.1) for (4). $\square$

Appendix B

We explain the proof of Theorem 1.4 as well as the finiteness of the union $J_L$ defined in (4.2). By (Hindry and Silverman 1988, Corollary 8.5), the canonical height of an (S, O)-integral point is bounded by some linear function in g and $\#S$. Hence, the same proof of (Hindry and Silverman 1988, Theorem 8.1) with a counting lemma (e.g. Silverman (1982)) can be applied to conclude the first statement of Theorem 1.4. The constants $\alpha , \beta$ are functions of g and $\gamma$ is half of the Mordell-Weil rank of $X_K(K)$, which is bounded by $2\chi (f)=2(2g-2+t)$ by the Shioda-Tate formula (Shioda 1972, Theorem 2.5).

Let ${\mathcal {D}}\subset X$ be an integral curve which is finite over B. Let $C \rightarrow {\mathcal {D}}$ be the normalization morphism and let $h :C \rightarrow B$ be the induced finite morphism of degree d. Consider the elliptic surface $f' :X'= X \times _B C \rightarrow C$ which is also nonisotrivial (by Lemma 2.2). For each finite subset $S \subset B$ of cardinality $\#S\le s$, let $S' = f'^{-1}(S) \subset C$ then $\# S' \le s' := ds$. Let $T'$ be the type of $X'$ then $\# T' \le d t$. It is clear that ${\mathcal {D}}\times _B C$ splits into sections of $f'$. Let R be one of these sections and let $K'={\mathbb {C}}(C)$. It follows that

$$\begin{aligned} I_s \subset I'_{s'} := \cup _{S' \subset B, \#S' \le s'} \{(S', R)\text {-integral points of } X'_{K'} \} \subset X(K'). \end{aligned}$$

The desired properties of $I_s$ follows easily from those of $I'_{s'}$ and Theorem 1.4 is proved.

Suppose now that ${\mathcal {D}}\in \vert L\vert _{sm}$ so $C={\mathcal {D}}$. The genus of ${\mathcal {D}}$ and the degree d of h depend only on L. Thus, by (Hindry and Silverman 1988, Corollary 8.5), the canonical heights of integral points in $I'_{s'}$ are bounded by some fixed linear function in g and $\#S$ independent of the choice of $S \subset B$ and ${\mathcal {D}}\in \vert L \vert _{sm}$. We deduce that the integral points in $J_L$ also have bounded height and thus $J_L$ is finite.