The Kodaira dimension of complex hyperbolic manifolds with cusps

Bakker, Benjamin; Tsimerman, Jacob

doi:10.1112/s0010437x1700762x

Cited by 8 publications

(15 citation statements)

References 34 publications

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“…Combining (6.2) with (5.3) and (6.7), we finally obtain the lower bound (1.1) on vol(E GG k,• Ω X ), which proves Theorem 1.1. The expression (1.1) being valid for any k, we can use the results of [3] to determine an order k after which the algebra E GG k,• Ω X has maximal growth. For example, it is not hard to obtain an asymptotic expansion of (1.1), with leading coefficient…”

Section: Uniform Lower Bound In K On Vol(e Gg Km ω X )mentioning

confidence: 99%

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Jet differentials on toroidal compactifications of ball quotients

Cadorel¹

2021

Annales De l'Institut Fourier

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Section: Uniform Lower Bound In K On Vol(e Gg Km ω X )mentioning

confidence: 99%

“…Using the results of [3], it is not hard to derive explicit orders k for E GG k,• Ω X to have maximal growth:…”

Section: Introductionmentioning

confidence: 99%

Jet differentials on toroidal compactifications of ball quotients

Cadorel¹

2021

Annales De l'Institut Fourier

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“…We also expect Theorem C to be true uniformly in X(1), and that the same idea for the proof should work with some modifications. The methods investigated here apply more generally to rank one lattices (see [BT15] for an application to complex ball quotients), and we expect the torsion conjecture in the case of abelian varieties parametrized by a rank one Shimura variety to be proven similarly.…”

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confidence: 94%

The geometric torsion conjecture for abelian varieties with real multiplication

Bakker

Tsimerman

2018

J. Differential Geom.

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The geometric torsion conjecture asserts that the torsion part of the Mordell-Weil group of a family of abelian varieties over a complex quasiprojective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture for abelian varieties with real multiplication, uniformly in the field of multiplication. Fixing the field, we furthermore show 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 is a hybrid technique employing both the hyperbolic and algebraic geometry of the toroidal compactifications of the Hilbert modular varieties X(1) parametrizing such abelian varieties. We show that only finitely many torsion covers X1(n) contain d-gonal curves outside of the boundary for any fixed d. We further show the same is true for entire curves C → X1(n). Statement of ResultsFor any elliptic curve E/Q, the group of rational points E(Q) is finitely generated by Mordell's theorem. The free part behaves wildly; it is expected that there are elliptic curves E/Q with arbitrarily large rank rk E(Q), and the record to date is an elliptic curve E with rk E(Q) ≥ 28 found by Elkies. On the other hand, by a celebrated theorem of Mazur [MG78] the torsion part E(Q) tor is uniformly bounded:Theorem (Mazur). For any elliptic curve E/Q, |E(Q) tor | ≤ 16.Mazur's theorem was subsequently generalized to arbitrary number fields K/Q by Merel [Mer96] (building on partial results of [Kam92]) who showed a stronger uniformity: there is an integer N = N (d) such that for any degree d number field K and any elliptic curve E/K, every K-rational torsion point has order dividing N , i.e. E(K) tor ⊂ E(K) [N ].Similarly, it is expected that the torsion part of the Mordell-Weil group of an abelian variety A/K is uniformly bounded, though there are few results in this direction. The same question can be asked for K = k(C) the function field of a curve C over any field k, and though k = F p is most closely analogous to the number field case, k = Q is also interesting:Conjecture (Geometric torsion conjecture). Let k be an algebraically closed field of characteristic 0. There is an integer N = N (g, n) such that for any Date: April 9, 2015.

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“…The noncompact manifold X admits toroidal compactification X by [1] in case Γ is arithmetic and by [12] for non-arithmetic Γ. It is shown that X possesses many hyperbolic properties (see for example, [5], [6], [7], [16]).…”

Section: Introductionmentioning

confidence: 99%

“…From our point of view, the proof boils down to the construction of holomorphic symmetric differentials vanishing at infinity, which can also be thought of as a generalization of cusps form in modular curves. As inspired by the works of Mumford [14], Nadel [15] and Bakker-Tsimerman [5], an important element for the existence of the desired symmetric differentials is to make sure the size of the lattice "Γ is sufficiently small" or the infinity "D is not too big". Hence we introduce the notion of canonical radius of a cusp, which measures the size of the infinity corresponding to a cusp of Γ.…”

Section: Introductionmentioning

confidence: 99%

On effective existence of symmetric differentials of complex hyperbolic space forms

Wong

2018

Math. Z.

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For a noncompact complex hyperbolic space form of finite volume X = B n /Γ, we consider the problem of producing symmetric differentials vanishing at infinity on the Mumford compactification X of X similar to the case of producing cusp forms on hyperbolic Riemann surfaces. We introduce a natural geometric measurement which measures the size of the infinity X − X called canonical radius of a cusp of Γ. The main result in the article is that there is a constant r * = r * (n) depending only on the dimension, so that if the canonical radii of all cusps of Γ are larger than r * , then there exist symmetric differentials of X vanishing at infinity. As a corollary, we show that the cotangent bundle T X is ample modulo the infinity if moreover the injectivity radius in the interior of X is larger than some constant d * = d * (n) which depends only on the dimension.

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The Kodaira dimension of complex hyperbolic manifolds with cusps

Cited by 8 publications

References 34 publications

Jet differentials on toroidal compactifications of ball quotients

Jet differentials on toroidal compactifications of ball quotients

The geometric torsion conjecture for abelian varieties with real multiplication

On effective existence of symmetric differentials of complex hyperbolic space forms

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