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

Aizenbud, Avraham; Avni, Nir

Duke Math. J. 167, No. 14, 2721-2743 (2018).

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)].

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

Citations:

Zbl 1401.14057; Zbl 0659.14017; Zbl 1091.14004; Zbl 1436.14041

References:

[1]	A. Aizenbud and N. Avni, Representation growth and rational singularities of the moduli space of local systems, Invent. Math. 204 (2016), 245–316. · Zbl 1401.14057
[2]	N. Avni, Arithmetic groups have rational representation growth, Ann. of Math. (2) 174 (2011), 1009–1056. · Zbl 1244.20043
[3]	N. Avni, B. Klopsch, U. Onn, and C. Voll, Representation zeta functions of compact \(p\)-adic analytic groups and arithmetic groups, Duke Math. J. 162 (2013), 111–197. · Zbl 1281.22005
[4]	N. Avni, B. Klopsch, U. Onn, and C. Voll, Arithmetic groups, base change, and representation growth, Geom. Funct. Anal. 26 (2016), 67–135. · Zbl 1348.20054
[5]	A. I. Borevich and I. R. Shafarevich, Number Theory, Pure Appl. Math. 20, Academic Press, New York, 1966.
[6]	R. Cluckers and F. Loeser, Constructible exponential functions, motivic Fourier transform and transfer principle, Ann. of Math. (2) 171 (2010), 1011–1065. · Zbl 1246.14025
[7]	A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. · Zbl 0916.14005
[8]	J. Denef, On the degree of Igusa’s local zeta function, Amer. J. Math. 109 (1987), 991–1008. · Zbl 0659.14017
[9]	J. Denef, Report on Igusa’s local zeta function, Astérisque 201–203 (1992), 359–386, Séminaire Bourbaki 1990/1991, no. 741. · Zbl 0749.11054
[10]	M. P. F. Du Sautoy and F. Grunewald, Analytic properties of zeta functions and subgroup growth, Ann. of Math. (2) 152 (2000), 793–833. · Zbl 1006.11051
[11]	R. Elkik, Rationalite des singularites canoniques, Invent. Math. 64 (1981), 1–6. · Zbl 0498.14002
[12]	H. Flenner, Rationale quasihomogene Singularitäten, Arch. Math. (Basel) 36 (1981), 35–44. · Zbl 0454.14001
[13]	A. Grothendieck, Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, III, Inst. Hautes Études Sci. Publ. Math. 28 (1966).
[14]	R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
[15]	H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, Ann. of Math. (2) 79 (1964), 109–203; II, 205–326. · Zbl 0122.38603
[16]	A. Jaikin-Zapirain, Zeta function of representations of compact \(p\)-adic analytic groups, J. Amer. Math. Soc. 19 (2006), 91–118. · Zbl 1092.20023
[17]	G. Kempf, F. F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal Embeddings, I, Lecture Notes in Math. 339, Springer, Berlin, 1973. · Zbl 0271.14017
[18]	S. Lang and A. Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827. · Zbl 0058.27202
[19]	M. Larsen and A. Lubotzky, Representation growth of linear groups, J. Eur. Math. Soc. (JEMS) 10 (2008), 351–390. · Zbl 1142.22006
[20]	M. W. Liebeck and A. Shalev, Character degrees and random walks in finite groups of Lie type, Proc. London Math. Soc. (3) 90 (2005), 61–86. · Zbl 1077.20020
[21]	M. W. Liebeck and A. Shalev, Fuchsian groups, finite simple groups and representation varieties, Invent. Math. 159 (2005), 317–367. · Zbl 1134.20059
[22]	A. Lubotzky and B. Martin, Polynomial representation growth and the congruence subgroup problem, Israel J. Math. 144 (2004), 293–316. · Zbl 1134.20056
[23]	M. Mustaţă, Jet schemes of locally complete intersection canonical singularities, Invent. Math. 145 (2001), 397–424. · Zbl 1091.14004
[24]	M. Mustaţă, Zeta functions in algebraic geometry, lecture notes, 2011, http://www.math.lsa.umich.edu/ mmustata/zeta_book.pdf.
[25]	V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure Appl. Math. 139, Academic Press, Boston, 1994. · Zbl 0841.20046
[26]	M. S. Raghunathan, The congruence subgroup problem, Proc. Indian Acad. Sci. Math. Sci. 114 (2004), 299–308. · Zbl 1086.20024