COUNIFORMIZATION of CURVES OVER NUMBER FIELDS by Fedor Bogomolov and Yuri Tschinkel

COUNIFORMIZATION OF CURVES OVER NUMBER FIELDS by Fedor Bogomolov and Yuri Tschinkel Abstract. — We study correspondences between projective curves over Q¯ . Contents 1. Introduction .............................................. 1 2. Minimal curves ............................................ 3 3. A graph on the set of elliptic curves ...................... 9 4. Collecting points .......................................... 14 References .................................................. 18 1. Introduction In this note we investigate correspondences between (geometrically irreducible) algebraic curves over number fields. Let C, C0 be two such curves. We say that C lies over C0 and write C ⇒ C0 if there exist an étalecover C˜ → C and a dominant map C˜ → C0. In particular, every curve lies over P1. Clearly, if C ⇒ C0 and C0 ⇒ C00 then C ⇒ C00. We say that a curve C0 is minimal for some class of curves C if every C ∈ C lies over C0. 2 FEDOR BOGOMOLOV and YURI TSCHINKEL Let n 2 (1.1) Cn : y = x + 1 and C be the set of such curves. For all n, m ∈ N we have Cmn ⇒ Cn. Belyi’s theorem [1] implies that for every curve C0 defined over a number 0 field there exists a curve C = Cn ∈ C such that C ⇒ C (see [3] for a simple proof of this corollary). A natural extremal statement is: ¯ Conjecture 1.1.— The curve C6 lies over every curve C over Q. Every hyperelliptic curve C of genus g(C) ≥ 2 lies over C6 (see Propo- sition 2.4 or [3]). The conjecture would imply that any hyperbolic hyperelliptic curve lies over any other curve. Our main result towards the above conjecture is Theorem 1.2.— For every m ≥ 6 and every n ∈ {2, 3, 5} the curve Cm lies over Cmn. The relevance of such geometric constructions to number theory comes from a theorem of Chevalley-Weil: if π : C˜ → C is an unramified map of proper algebraic curves over a number field K then there exists a finite extension K/K˜ such that π−1(C(K)) ⊂ C˜(K˜ ). Therefore, if C ⇒ C0 then Mordell’s conjecture (Faltings’ theorem) for C follows from Mordell’s conjecture for C0. Our constructions allow us, at least in the case of hyperelliptic curves, to control the degree and discriminant of the field K˜ in terms of the coefficients defining the curve. In particular, “effective” Mordell for C6 implies effective Mordell for every hyperelliptic curve (see also [11], [9],[6]). The proof of this theorem uses certain special properties of modular curves and related elliptic curves. In the construction of unramified cov- ers we need to exhibit maps from various intermediate curves onto P1 or elliptic curves with simultaneous restrictions on the local ramification indices and the branching points. This is very close, in spirit, to Belyi’s theorem which says that every projective algebraic curve defined over Q¯ has a map onto P1 ramified in 0, 1, ∞. In fact, there are many such maps. Our technique involves optimizing the choice of these maps by trading the freedom to impose ramification conditions for the degree of the map. COUNIFORMAZATION OF CURVES 3 An example of this is given in Section 4 where we prove the first part of Belyi’s theorem (reduction to Q-rational branching) under the restriction that the only prime dividing the local ramification indices is 2. Acknowledgments. The first author was partially supported by the NSF Grant DMS-0404715. The second author was partially supported by the NSF Grant DMS-0100277. 2. Minimal curves Notations 2.1.— For a surjective morphism of curves π : C0 → C of degree d we denote by Bran(π) ⊂ C the branching locus of π. For c ∈ Bran(π) put d2 d3 X dc := (2 , 3 ,...), idi ≤ d, i −1 where di is the number of points in π (c) with local ramification index i. Let RD(π) = {dc}c∈Bran(π) be the ramification datum. Example 2.2.— Let zn : P1 → P1 be the n-power map z 7→ zn. Then n n Bran(z ) = {0, ∞} and RD(z ) = {(n)0, (n)∞}. Notations 2.3.— Let E be an elliptic curve over Q¯ with a fixed 0 ∈ E, E[n] the set of n-torsion points and ∞ ¯ E[∞] := ∪n=1E[n] ⊂ E(Q) the set of all torsion points of E. Usually, we write σ : x → −x for the standard involution on E and 1 π = πσ : E → E/σ = P for the induced map. When we specify the elliptic curve by the branching locus we write E = E(Bran(π)). Proposition 2.4.— The curves C6 and C8 are minimal for the class of hyperbolic hyperelliptic curves. 4 FEDOR BOGOMOLOV and YURI TSCHINKEL Proof.— Fix a hyperbolic hyperelliptic curve C. Notice that for any such C there exists an étalecover C1 → C of degree 2 and a degree two surjection C1 → E onto an elliptic curve. For example, we can take E to be any elliptic curve ramified in 4 of the ramification points of the initial hyperelliptic map C → P1. Fix such an E. We use the following simple fact about elliptic curves: Let π : E → P1 be an elliptic curve. Then π(E[3]) is (projectively equivalent to) the union of one point from Bran(π) and {1, ζ, ζ2, ∞} ⊂ P1 (where ζ is a fixed third root of 1). Similarly, π(E[4]) is (projectively equivalent to) −1 −1 1 Bran(π) = {λ, λ , −λ, −λ } ∪ {1, −1, i, −i, 0, ∞} ⊂ P . Now consider the natural (multiplication by m) isogeny ϕm : E → E where m = 3 or 4. The map ϕm is 2-ramified in E[m], for m = 3, 4. Consider the diagram τ2 τ3 τ4 τ5 C o C1 o C2 C2 o C3 o C4 o C5 ι1 ι2 σ2 ι3 ι4 EEo / 1 o E o E o C . ϕm π P πm m ϕm m ιm 2m Here 3 – Bran(π3) = {1, ζ, ζ , ∞} ⊂ Bran(σ2); – Bran(π4) = {1, −1, i, −i} ⊂ Bran(σ2); – ιm : C2m → Em = Cm is the standard map, it is ramified in two points (whose difference is) in Em[m]; – C2 is an irreducible component of the fiber product C1 ×E E; – σ2 = π ◦ ι2; 1 – C3 := C2 ×P Em; – C4 is an irreducible component of C3 ×Em Em; – C5 := C4 ×Em C2m; Observe that for q ∈ Bran(πm) the local ramification indices in the preim- −1 age σ2 (q) are all even. Therefore, τ3 is unramified and ι3 has even local ramification indices over (the preimage of) q ∈ {π(E[m]) \ Bran(πm)} (such a point exists). Note that q ∈ Bran(π). The map ι4 is ramified COUNIFORMAZATION OF CURVES 5 −1 over the preimages (πm ◦ ϕm) (q), with even local ramification indices, which implies that τ5 is unramified. Finally, C5 has a dominant map onto C2m and is unramified over C4 (and consequently, C1). This shows that every hyperelliptic curve lies over C2m, for m = 3, 4. Theorem 2.5.— For all m ≥ 6 and n ∈ {2, 3} one has Cm ⇒ Cmn. Proof.— We first assume that m = 2n is even and ≥ 8, since C6 ⇒ C8. First we show that C := Cm lies over C2m. Consider the diagram: τ1 τ2 τ3 τ4 C2n o C1 o C2 o C3 C3 o C4 0 ι0 ι1 ι2 ι3 ι3 1 1 1 o o E o E / o C4n. P zn P π ϕ2 π0 P θ Here – π is a double cover whose branch locus consists of 3 points in the preimage of 1 under zn and the preimage of 0; 1 1 – C1 is the fiber product C2n ×P P , note that τ1 is unramified and that ι1 is evenly ramified over all points in Bran(π); 1 – C2 = C1 ×P E, note that τ2 is unramified since ι1 has ramification 1 of order two over 0 and even ramification over all ζn ∈ P ; – τ3 is unramified; – since n ≥ 4, the map ι2 has ramification points of order 2n and ι3 is branched with ramification index 2n over all points in E[2]; 0 0 0 0 0 – π is the map such that Bran(π ) = π (E[2]), then ι3 := π ◦ ι3 is 4n-ramified over all points in Bran(π0); – θ is the map branched in three of the above points, in particular, τ4 is unramified. Now we assume that m is odd, m ≥ 5 and consider the diagram: 6 FEDOR BOGOMOLOV and YURI TSCHINKEL τ1 τ2 τ3 Cm o C1 C1 o C2 o C3 0 ι0 ι1 ι1 ι2 ι3 1 1 1 1 o m / o o E. P z P ψ1 P ψ2 P π Here −1 0 1 – ψ1 : z 7→ (z + z )/2, then ι1 = ψ1 ◦ ι1 : C1 → P is 2-ramified i −i over -1, 2m-ramified over 1 and m-ramified over ξi := (ζm + ζm )/2; pm – ψ2 = (z − ξ1)/(z − ξ2), it has 2-ramification over all m preimages of −1 and 2m-ramification over the preimages of 1; – π is a double cover ramified over (arbitrary) 4 points in the preimage of −1 under ψ2, then ι3 : C3 → E is m-ramified over all other points and we can continue as above. Now we show that Cm lies over C3m (m even, this suffices for our pur- poses). Consider: τ1 τ2 τ3 τ4 τ5 τ6 C2n o C1 o C2 o C3 C3 o C4 o C5 C5 o C6 0 ι0 ι1 ι2 ι3 ι3 ι4 ι5 1 1 1 1 o n o E o E 0 / o E0 o E0 / o C6n P z P π ϕ6 π P π0 ϕ3 θ0 P Here – π is a double cover whose branch locus consists of 3 points in the preimage of 1 under zn and the preimage of 0; 1 1 – C1 = C2n ×P P , note that τ1 is unramified and that ι1 is evenly ramified over all points in Bran(π); 1 – C2 is the fiber product C1 ×P E, note that τ2 is unramified since ι1 has ramification of order two over 0 and even ramification over all 1 ζn ∈ P ; – τ3 is unramified; – since n ≥ 4, the map ι2 has ramification points of order 2n and ι3 is branched with ramification index 2n over all points in E[6]; 0 1 0 0 0 – π : E → P is the map such that Bran(π ) = π (E[2]), then ι3 = 0 0 π ◦ ι3 is 4n-ramified over all points of Bran(π ); COUNIFORMAZATION OF CURVES 7 0 0 1 – Bran(π3) = π (E[3]) \ π (0) and the fiber product C4 = C3 ×P C3 is unramified over C3, since the all the preimages of Bran(π3) in C3 have even ramifications (for ι3); −1 – note that there is a point q0 ∈ E0 such that every point in ι4 (q0) ∈ C4 has ramification of order 2n (for example, take a point q of order −1 0 exactly 6 in E and take any q0 ∈ π0 (π (q)) ∈ E0).

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