Unramified correspondences.pdfVIP

a r X i v : m a t h / 0 2 0 2 2 2 3 v 1 [ m a t h .A G ] 2 1 F e b 2 0 0 2 UNRAMIFIED CORRESPONDENCES by Fedor Bogomolov and Yuri Tschinkel Abstract. — We study correspondences between algebraic curves defined over the separable closure of Q or Fp. Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Main construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. The case of characteristic 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4. Geometric constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Introduction A class C(Q) of complete algebraic curves over Q will be called domi- nating if for every algebraic curve C ′ over Q there exist a curve C? ∈ C(Q) and a birational surjective map C? → C ′. A curve C will be called uni- versal if the class UC(Q) of its unramified covers is dominating. Theorem 1.1 (Belyi). — Every algebraic curve C defined over a num- ber field admits a surjective map onto P1 which is unramified outside (0, 1,∞). In 1978 Manin pointed out that Belyi’s theorem implies the following 2 FEDOR BOGOMOLOV and YURI TSCHINKEL Proposition 1.2. — The class MU(Q) consisting of modular curves and their unramified covers is dominating. There are many other classes of curves with the same property, for example: 1. hyperelliptic curves and their unramified coverings; 2. the class CU(Q) := ∪n∈NCn(Q), with Cn(Q) consisting of curves with function field Q(z, n √ z(1? z)) and their unramified coverings. 3. the class CN (Q) := ∪n∈NCNn(Q) where CNn(Q) consists of all un- ramified covers of any curve Cn with the property that Cn → P1 is ramified in (0, 1,∞) only and all local ramification indices of Cn over 0 are divisible by 3, over 1 divisible by 2 and over ∞ divisible by n. In particu