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The geometric torsion conjecture for abelian varieties with real multiplication. (English) Zbl 1423.11109
In the paper under review some instances of the Geometric Torsion Conjecture are proven. The geometric torsion conjecture asserts that the torsion part of the Mordell-Weil group of a family of abelian varieties over a complex quasi-projective curve is uniformly bounded in terms of the genus of the curve. The conjecture for abelian varieties with real multiplication is proven uniformly in the field of multiplication in this paper. Fixing the field, it is furthermore shown that the torsion is bounded in terms of the gonality of the base curve, which is the closer analog of the arithmetic conjecture.
The proof follows the general strategy developed in other paper by the same authors [B. Bakker and J. Tsimerman, Ann. Math. (2) 184, No. 3, 709–744 (2016; Zbl 1386.14093)] to prove the geometric analog of another arithmetic uniformity conjecture, the Frey-Mazur conjecture. First, they detail the local structure of cusps of cofinite-volume quotients of $$\mathbb{H}^n$$ and their toroidal compactifications and they prove a volume bound on the boundary multiplicity of curves in the toroidal compactification. These bounds are similar to those proven by [J.-M. Hwang and W.-K. To, Am. J. Math. 124, No. 6, 1221–1246 (2002; Zbl 1024.32013)] for interior points. Later they show that the torsion covers of Hilbert modular varieties hyperbolically expand, from which it follows that the multiplicity bound previously found improves in the torsion tower. Then they deduce some geometric results as $$X(1)$$ being of general type provided $$n>6$$ that was already proved in [S. Tsuyumine, Invent. Math. 80, 269–281 (1985; Zbl 0576.14036)] with the theory of modular forms. Here the proof only involves the metric geometry. After they assemble the previous results to prove the following:
Theorem. For each $$\mathfrak{n}\subset\mathcal{O}_F$$ let $$X_1(\mathfrak{n})^*$$ be the Baily-Borel compactification of the $$\mathfrak{n}$$-torsion level cover of the Hilbert modular variety $$X(1)$$. Then for any $$g$$, $$X_1(\mathfrak{n})^*$$ contains no genus $$g$$ curves for all but finitely many $$\mathfrak{n}$$, uniformly for all $$F$$ of a fixed degree. For a fixed $$F$$, the same is true of $$d$$-gonal curves.
The proof is conceptually similar to that of [J.-M. Hwang and W.-K. To, Math. Ann. 335, No. 2, 363–377 (2006; Zbl 1090.14013)], where a version of this theorem is shown for full-level covers $$X(\mathfrak{n})$$. However, the boundary of $$X_1(\mathfrak{n})$$ does not totally ramify over $$X(1)$$ and therefore a new technique is required. The core idea is to prove a bound relating the volume of a curve in a toroidal compactification $$\bar{X}_1(\mathfrak{n})$$ to its multiplicity along the boundary using again the metric geometry.
From here two versions of the Geometric Torsion Conjecture for abelian varieties with real multiplication is proven: one uniformly in the field of multiplication and other in terms of the gonality of the base curve.

##### MSC:
 11G10 Abelian varieties of dimension $$> 1$$ 11G15 Complex multiplication and moduli of abelian varieties 14G35 Modular and Shimura varieties 14K10 Algebraic moduli of abelian varieties, classification 14K15 Arithmetic ground fields for abelian varieties 14J25 Special surfaces
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