Exponential growth of homological torsion for towers of congruence subgroups of Bianchi groups

Pfaff, Jonathan

doi:10.1007/s10455-013-9400-2

Cited by 15 publications

(7 citation statements)

References 18 publications

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“…In fact, at least with nontrivial coefficient systems, there are precise bounds on the growth of the torsion in H i (X K , Z), showing exponential growth in the case that n = 2 and F is imaginary-quadratic (while H i (X K , C) stays small), cf. [12], [50]. In other words, Conjecture I.2 predicts the existence of many more Galois representations than Conjecture I.…”

Section: Contentsmentioning

confidence: 96%

On Torsion in the Cohomology of Locally Symmetric Varieties

2015

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The main result of this paper is the existence of Galois representations associated with the mod p (or mod p m ) cohomology of the locally symmetric spaces for GL n over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for GL n as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of Shimura varieties.Thus, we prove new results on the p-adic geometry of Shimura varieties (of Hodge type). Namely, the Shimura varieties become perfectoid when passing to the inverse limit over all levels at p, and a new period map towards the flag variety exists on them, called the Hodge-Tate period map. It is roughly analogous to the embedding of the hermitian symmetric domain (which is roughly the inverse limit over all levels of the complex points of the Shimura variety) into its compact dual. The Hodge-Tate period map has several favorable properties, the most important being that it commutes with the Hecke operators away from p (for the trivial action of these Hecke operators on the flag variety), and that automorphic vector bundles come via pullback from the flag variety.

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Section: Contentsmentioning

confidence: 96%

On Torsion in the Cohomology of Locally Symmetric Varieties

2015

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“…In [Ra2] he applied this to study the growth of torsion in the cohomology for certain sequences of congruence subgroups of Bianchi groups. His result generalized the exponential growth of torsion, obtained in [Pf2] for local systems induced from the even symmetric powers of the standard representation of SL 2 (C), to all strongly acyclic local systems and furthermore they implied that the limit of the normalized torsion size exists. The main purpose of the present paper is to extend (1.3) to hyperbolic manifolds of finite volume and arbitrary dimension.…”

Section: Introductionmentioning

confidence: 71%

“…The groups Γ(a) are torsion-free and satisfy (3.1) for N(a) sufficiently large. This was shown for example in the proof of Lemma 4.1 in [Pf2]. Since [Γ(D) : Γ(a)] tends to ∞ if N(a) tends to ∞ and since each γ 0 ∈ Γ(D), γ 0 = 1, is contained in only finitely many Γ(a), Corollary 1.4 follows from Proposition 11.1 and Theorem 1.2.…”

Section: Principal Congruence Subgroups and Hecke Subgroups Of Bianchmentioning

confidence: 78%

The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume

Müller

Pfaff

2014

Journal of Functional Analysis

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Abstract. In this paper we study the regularized analytic torsion of finite volume hyperbolic manifolds. We consider sequences of coverings X i of a fixed hyperbolic orbifold X 0 . Our main result is that for certain sequences of coverings and strongly acyclic flat bundles, the analytic torsion divided by the index of the covering, converges to the L 2 -torsion. Our results apply to certain sequences of arithmetic groups, in particular to sequences of principal congruence subgroups of SO 0 (d, 1)(Z) and to sequences of principal congruence subgroups or Hecke subgroups of Bianchi groups.

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Section: Introductionmentioning

confidence: 96%

On torsion in the cohomology of locally symmetric varieties

Scholze

2015

Ann. Math.

134

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Abstract. The main result of this paper is the existence of Galois representations associated with the mod p (or mod p m ) cohomology of the locally symmetric spaces for GL n over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for GL n as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of Shimura varieties.Thus, we prove new results on the p-adic geometry of Shimura varieties (of Hodge type). Namely, the Shimura varieties become perfectoid when passing to the inverse limit over all levels at p, and a new period map towards the flag variety exists on them, called the Hodge-Tate period map. It is roughly analogous to the embedding of the hermitian symmetric domain (which is roughly the inverse limit over all levels of the complex points of the Shimura variety) into its compact dual. The Hodge-Tate period map has several favorable properties, the most important being that it commutes with the Hecke operators away from p (for the trivial action of these Hecke operators on the flag variety), and that automorphic vector bundles come via pullback from the flag variety.

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Exponential growth of homological torsion for towers of congruence subgroups of Bianchi groups

Cited by 15 publications

References 18 publications

On Torsion in the Cohomology of Locally Symmetric Varieties

On Torsion in the Cohomology of Locally Symmetric Varieties

The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume

On torsion in the cohomology of locally symmetric varieties

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