##
**Counting points of schemes over finite rings and counting representations of arithmetic lattices.**
*(English)*
Zbl 1436.14044

The authors of the article continue their study of representation growth and rational singularities of moduli spaces of local systems
[A. Aizenbud and N. Avni, Invent. Math. 204, No. 1, 245–316 (2016; Zbl 1401.14057)], now from a more general point of view.
They relate algebraic geometry over finite rings with representation theory, and obtain two main results concerning estimates (bounds) for the number of points of schemes over finite rings (Theorem A) and estimates (bounds) for the number of irreducible representations of arithmetic lattices of algebraic group schemes (Theorem B).

Let \(X\) be a scheme of finite type over \({\mathbb Z}\) with the reduced, absolutely irreducible generic fiber \(X_{\mathbb Q} := X \times_{\operatorname{Spec}{\mathbb Z}} \operatorname{Spec}{\mathbb Q}\) of \(X\) which is a local complete intersection. Let \(\#X(R)\) (the authors use the notation \(|X(R)|\)) be the number of points of the scheme \(X\) over the finite ring \(R\).

Theorem A states, in particular, that the next conditions are equivalent: For any \(m\), \(\lim_{p \to \infty} \frac{\#X({\mathbb Z}/{p^m})}{p^{m \cdot \dim X_{\mathbb Q}}} = 1\); \(X_{\mathbb Q}\) has rational singularities.

Theorem B states that for any algebraic group scheme \(G\), whose generic fiber \(G_{\mathbb Q}\) is simple, connected, simply connected, and of \({\mathbb Q}\)-rank at least 2, and for every \(C > 40\), the number of irreducible representations of \(G({\mathbb Z})\) of dimension \(n\) is equal to \( o(n^C)\).

The main tools in proving these theorems and their generalizations are Poincare series by Borevich-Shafarevich, Igusa zeta functions and other \(p\)-adic integrals, Lang-Weil bounds, deformation schemes and the theorem of Frobenius. “For a topological group \(\Gamma\), let \(r_n(\Gamma)\) be the number of isomorphism classes of irreducible, \(n\)-dimensional, complex, continuous representations of \(\Gamma\)”, and let \(\zeta_{\Gamma}(s)\) be the representation zeta function of \(\Gamma\).

Let \(k\) be a global field and let \(T\) be a finite set of places of \(k\) containing all Archimedean places. By \({\mathcal O}_{k,T}\) authors denote the ring of \(T\)-integers of \(k\) and by \(\widehat{{\mathcal O}_{k,T}}\) the profinite completion of \({\mathcal O}_{k,T}\).

Let \(\alpha(\Gamma)\) be the abscissa of convergence of \(\zeta_{\Gamma}(s)\). The second section of the article deals with preliminaries, which include (along with the above) elements of singularities and theorems by J. Denef [Am. J. Math. 109, 991–1008 (1987; Zbl 0659.14017)], and by M. Mustaţă [Invent. Math. 145, No. 3, 397–424 (2001; Zbl 1091.14004)]. In the next sections authors of the article under review “study the number of points of schemes over finite rings” and prove (a generalization) of Theorem A.

The article closes with results on representation zeta functions of compact \(p\)-adic groups, of adelic groups and of arithmetic groups.

In the forth section the authors prove Theorem II on the abscissa of convergence \(\alpha(G({\mathcal O}_{k,T}))\), Theorem III on abscissa of convergence \(\alpha(G({\widehat{{\mathcal O}_{k,T}}}))\) and Theorem V on estimates of representation zeta function values at integer points \(2n - 2, \; n \ge 2\).

Reviewer’s remark: It is interesting to relay results of this article with results of the paper by B. Frankel [J. Algebra 510, 393–412 (2018; Zbl 1436.14041)].

Let \(X\) be a scheme of finite type over \({\mathbb Z}\) with the reduced, absolutely irreducible generic fiber \(X_{\mathbb Q} := X \times_{\operatorname{Spec}{\mathbb Z}} \operatorname{Spec}{\mathbb Q}\) of \(X\) which is a local complete intersection. Let \(\#X(R)\) (the authors use the notation \(|X(R)|\)) be the number of points of the scheme \(X\) over the finite ring \(R\).

Theorem A states, in particular, that the next conditions are equivalent: For any \(m\), \(\lim_{p \to \infty} \frac{\#X({\mathbb Z}/{p^m})}{p^{m \cdot \dim X_{\mathbb Q}}} = 1\); \(X_{\mathbb Q}\) has rational singularities.

Theorem B states that for any algebraic group scheme \(G\), whose generic fiber \(G_{\mathbb Q}\) is simple, connected, simply connected, and of \({\mathbb Q}\)-rank at least 2, and for every \(C > 40\), the number of irreducible representations of \(G({\mathbb Z})\) of dimension \(n\) is equal to \( o(n^C)\).

The main tools in proving these theorems and their generalizations are Poincare series by Borevich-Shafarevich, Igusa zeta functions and other \(p\)-adic integrals, Lang-Weil bounds, deformation schemes and the theorem of Frobenius. “For a topological group \(\Gamma\), let \(r_n(\Gamma)\) be the number of isomorphism classes of irreducible, \(n\)-dimensional, complex, continuous representations of \(\Gamma\)”, and let \(\zeta_{\Gamma}(s)\) be the representation zeta function of \(\Gamma\).

Let \(k\) be a global field and let \(T\) be a finite set of places of \(k\) containing all Archimedean places. By \({\mathcal O}_{k,T}\) authors denote the ring of \(T\)-integers of \(k\) and by \(\widehat{{\mathcal O}_{k,T}}\) the profinite completion of \({\mathcal O}_{k,T}\).

Let \(\alpha(\Gamma)\) be the abscissa of convergence of \(\zeta_{\Gamma}(s)\). The second section of the article deals with preliminaries, which include (along with the above) elements of singularities and theorems by J. Denef [Am. J. Math. 109, 991–1008 (1987; Zbl 0659.14017)], and by M. Mustaţă [Invent. Math. 145, No. 3, 397–424 (2001; Zbl 1091.14004)]. In the next sections authors of the article under review “study the number of points of schemes over finite rings” and prove (a generalization) of Theorem A.

The article closes with results on representation zeta functions of compact \(p\)-adic groups, of adelic groups and of arithmetic groups.

In the forth section the authors prove Theorem II on the abscissa of convergence \(\alpha(G({\mathcal O}_{k,T}))\), Theorem III on abscissa of convergence \(\alpha(G({\widehat{{\mathcal O}_{k,T}}}))\) and Theorem V on estimates of representation zeta function values at integer points \(2n - 2, \; n \ge 2\).

Reviewer’s remark: It is interesting to relay results of this article with results of the paper by B. Frankel [J. Algebra 510, 393–412 (2018; Zbl 1436.14041)].

Reviewer: Nikolaj M. Glazunov (Kyïv)

### MSC:

14G05 | Rational points |

14B05 | Singularities in algebraic geometry |

20F69 | Asymptotic properties of groups |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

20G05 | Representation theory for linear algebraic groups |

20G30 | Linear algebraic groups over global fields and their integers |

### Keywords:

representation growth; Igusa zeta function; points of schemes over finite ring; complete intersection; rational singularities; representation zeta function
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\textit{A. Aizenbud} and \textit{N. Avni}, Duke Math. J. 167, No. 14, 2721--2743 (2018; Zbl 1436.14044)

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