Unramified covers of Galois covers of low genus curves

Abstract

Abstract. Let X → Y be a Galois covering of curves, where the genus of X is ≥ 2 and the genus of Y is ≤ 2. We prove that under certain hypotheses, X has an unramified cover that dominates a hyperelliptic curve; our results apply, for instance, to all tamely superelliptic curves. Combining this with a theorem of Bogomolov and Tschinkel shows that X has an unramified cover that dominates y 2 = x 6 − 1, if char k is not 2 or 3.

Un peu d'effectivité pour les variétés modulaires de Hilbert-Blumenthal

“…(For example, all solvable covers of P 1 over a number field are şirin, via e.g. the family given in Section 6 of [2] and the main theorem of Poonen's [13]. More generally, all curves C/K admitting a diagram C ϕ ← − C f − → P 1 over a finite L/K with ϕ étale and f Belyi with all ramification indices above 0, 1, ∞ respectively divisible by a, b, c with 1 a + 1 b + 1 c < 1 are also şirin, by pulling back a suitable hypergeometric family of abelian varieties.…”

“…Let us now prove Theorem 3.2. 14 13 Alternatively, combine Theorem 1.1, Lemma 3.4, and Propositions 4.1 and 6.2 to upper bound the index of End K (A) in one of an explicit finite set of rings. 14 Because we chose to state Theorem 3.2 in a previous section (namely immediately after Lemma 3.1 in Section 3.1), it is important to note that no preceding results depend on Theorem 3.2, and so we are e.g.…”

On Contraction of Algebraic Points

“…(This map is coming from a search using Belyi's formula (See Definition 21 and the proof of Proposition 23).) Ram(h 6 )=(0, 4, 6, 13, −14, 256, ∞) with corresponding ramification indices 3 9 5 4 , 2 6 3 4 5 4 13, 2 7 3 9 13, 2 12 5 4 , 2 7 3 2 5 2 13, 13, 5 and Bran(h 6 ) ∪h 6 (D 5 ) = (0, 1, ∞).…”

Rational points on solvable curves over $\mathbb{Q}$ via non-abelian Chabauty

“…Fortunately, such a fund of examples is supplied by a theorem of Bogomolov and Tschinkel [BT02] (see also [BQ17]) which shows that every hyperelliptic curve has an étale cover which geometrically dominates a curve over Q with CM Jacobian. Poonen [Poo05] generalized this theorem to a more general class of curves, including all superelliptic curves. 1 By now it is well-understood that one can control the rational points of a variety Y by controlling the rational points of the twists of an étale cover of Y .…”

“…In §6 and §7, we prove the bounds needed for the dimension hypothesis, which we prove in §8. Finally, in §9, we combine our results with a theorem of Poonen [Poo05] to deduce finiteness of Y (Q) for several classes of curves, including hyperelliptic and superelliptic curves.…”