### Abstract

We relate the singularities of a scheme X to the asymptotics of the number of points of X over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if Γ is an arithmetic lattice whose Q-rank is greater than 1, then let r_{n}.(Γ)be the number of irreducible n-dimensional representations of Γ up to isomorphism. We prove that there is a constant C (in fact, any C > 40 suffices) such that r_{n}(Γ) = O(n^{C}) for every such Γ. This answers a question of Larsen and Lubotzky.

Original language | English (US) |
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Pages (from-to) | 2721-2743 |

Number of pages | 23 |

Journal | Duke Mathematical Journal |

Volume | 167 |

Issue number | 14 |

DOIs | |

State | Published - Oct 1 2018 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Duke Mathematical Journal*,

*167*(14), 2721-2743. https://doi.org/10.1215/00127094-2018-0021

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*Duke Mathematical Journal*, vol. 167, no. 14, pp. 2721-2743. https://doi.org/10.1215/00127094-2018-0021

**Counting points of schemes over finite rings and counting representations of arithmetic lattices.** / Aizenbud, Avraham; Avni, Nir.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Counting points of schemes over finite rings and counting representations of arithmetic lattices

AU - Aizenbud, Avraham

AU - Avni, Nir

PY - 2018/10/1

Y1 - 2018/10/1

N2 - We relate the singularities of a scheme X to the asymptotics of the number of points of X over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if Γ is an arithmetic lattice whose Q-rank is greater than 1, then let rn.(Γ)be the number of irreducible n-dimensional representations of Γ up to isomorphism. We prove that there is a constant C (in fact, any C > 40 suffices) such that rn(Γ) = O(nC) for every such Γ. This answers a question of Larsen and Lubotzky.

AB - We relate the singularities of a scheme X to the asymptotics of the number of points of X over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if Γ is an arithmetic lattice whose Q-rank is greater than 1, then let rn.(Γ)be the number of irreducible n-dimensional representations of Γ up to isomorphism. We prove that there is a constant C (in fact, any C > 40 suffices) such that rn(Γ) = O(nC) for every such Γ. This answers a question of Larsen and Lubotzky.

UR - http://www.scopus.com/inward/record.url?scp=85055254228&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055254228&partnerID=8YFLogxK

U2 - 10.1215/00127094-2018-0021

DO - 10.1215/00127094-2018-0021

M3 - Article

VL - 167

SP - 2721

EP - 2743

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 14

ER -