Duke Mathematical Journal

Torsion homology growth and cycle complexity of arithmetic manifolds

Nicolas Bergeron, Mehmet Haluk Şengün, and Akshay Venkatesh

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $M$ be an arithmetic hyperbolic $3$-manifold, such as a Bianchi manifold. We conjecture that there is a basis for the second homology of $M$, where each basis element is represented by a surface of “low” genus, and we give evidence for this. We explain the relationship between this conjecture and the study of torsion homology growth.

Article information

Duke Math. J. Volume 165, Number 9 (2016), 1629-1693.

Received: 30 January 2014
Revised: 24 August 2015
First available in Project Euclid: 22 February 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Secondary: 57M50: Geometric structures on low-dimensional manifolds

hyperbolic geometry modular forms cycle complexity


Bergeron, Nicolas; Şengün, Mehmet Haluk; Venkatesh, Akshay. Torsion homology growth and cycle complexity of arithmetic manifolds. Duke Math. J. 165 (2016), no. 9, 1629--1693. doi:10.1215/00127094-3450429. https://projecteuclid.org/euclid.dmj/1456150040.

Export citation


  • [1] M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, and I. Samet, On the growth of $L^{2}$-invariants for sequences of lattices in Lie groups, preprint, arXiv:1210.2961v2 [math.RT].
  • [2] U. K. Anandavardhanan and D. Prasad, On the $\operatorname{SL}(2)$ period integral, Amer. J. Math. 128 (2006), 1429–1453.
  • [3] U. K. Anandavardhanan and D. Prasad, A local-global question in automorphic forms, Compos. Math. 149 (2013), 959–995.
  • [4] Y. Benoist and H. Oh, Effective equidistribution of $S$-integral points on symmetric varieties, Ann. Inst. Fourier (Grenoble) 62 (2010), 1889–1942.
  • [5] N. Bergeron, P. Linnell, W. Lück, and R. Sauer, On the growth of Betti numbers in $p$-adic analytic towers, Groups Geom. Dyn. 8 (2014), 311–329.
  • [6] N. Bergeron and A. Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12 (2013), 391–447.
  • [7] B. J. Birch, “Elliptic curves over $Q$: A progress report” in 1969 Number Theory Institute (Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, 1971, 396–400.
  • [8] B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, II, J. Reine Angew. Math. 218 (1965), 79–108.
  • [9] W. Bley, Numerical evidence for the equivariant Birch and Swinnerton-Dyer conjecture, Exp. Math. 20 (2011), 426–456.
  • [10] V. Blomer and G. Harcos, Twisted $L$-functions over number fields and Hilbert’s eleventh problem, Geom. Funct. Anal. 20 (2010), 1–52.
  • [11] J. Bolte, G. Steil, and F. Steiner, Arithmetical chaos and violation of universality in energy level statistics, Phys. Rev. Lett. 69, no. 15 (1992), 2188–2191.
  • [12] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2nd ed., Math. Surveys Monogr. 67, Amer. Math. Soc., Providence, 2000.
  • [13] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin, 1990.
  • [14] J. F. Brock and N. M. Dunfield, Injectivity radii of hyperbolic integer homology $3$-spheres, Geom. Topol. 19 (2015), 497–523.
  • [15] F. Brumley, Effective multiplicity one on $\mathrm{GL}_{N}$ and narrow zero-free regions for Rankin-Selberg $L$-functions, Amer. J. Math. 128 (2006), 1455–1474.
  • [16] D. Bump, Automorphic Forms and Representations, Cambridge Stud. Adv. Math. 55, Cambridge Univ. Press, Cambridge, 1997.
  • [17] C. J. Bushnell, Hereditary orders, Gauss sums and supercuspidal representations of $\mathrm{GL}_{N}$, J. Reine Angew. Math. 375/376 (1987), 184–210.
  • [18] F. Calegari and M. Emerton, Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms, Ann. of Math. (2) 170 (2009), 1437–1446.
  • [19] F. Calegari and A. Venkatesh, A torsion Jacquet–Langlands correspondence, preprint, arXiv:1212.3847v1 [math.NT].
  • [20] B. Casselman, “The asymptotic behavior of matrix coefficients” in Introduction to Admissible Representations of $p$-Adic Groups, preprint, https://www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf (accessed 2 February 2016).
  • [21] J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), 259–322.
  • [22] L. Clozel, “Motifs et formes automorphes: applications du principe de fonctorialité” in Automorphic Forms, Shimura Varieties, and $L$-Functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math. 10, Academic Press, Boston, 1990, 77–159.
  • [23] J. E. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, Compos. Math. 51 (1984), 275–324.
  • [24] J. E. Cremona and E. Whitley, Periods of cusp forms and elliptic curves over imaginary quadratic fields, Math. Comp. 62 (1994), 407–429.
  • [25] B. Edixhoven, Néron models and tame ramification, Compos. Math. 81 (1992), 291–306.
  • [26] T. Finis, E. Lapid, and W. Müller, On the degrees of matrix coefficients of intertwining operators, Pacific J. Math. 260 (2012), 433–456.
  • [27] M. Flach, “The equivariant Tamagawa number conjecture: A survey” with an appendix by C. Greither in Stark’s Conjectures: Recent Work and New Directions, Contemp. Math. 358, Amer. Math. Soc., Providence, 2004, 79–125.
  • [28] Y. Z. Flicker, Twisted tensors and Euler products, Bull. Soc. Math. France 116 (1988), 295–313.
  • [29] Y. Z. Flicker, On distinguished representations, J. Reine Angew. Math. 418 (1991), 139–172.
  • [30] S. Friedberg and J. Hoffstein, Nonvanishing theorems for automorphic $L$-functions on $\mathrm{GL}(2)$, Ann. of Math. (2) 142 (1995), 385–423.
  • [31] D. Gabai, Foliations and the topology of $3$-manifolds, J. Differential Geom. 18 (1983), 445–503.
  • [32] D. Goldfeld, “Modular forms, elliptic curves and the $ABC$-conjecture” in A Panorama of Number Theory or the View from Baker’s Garden (Zürich, 1999), Cambridge Univ. Press, Cambridge, 2002, 128–147.
  • [33] B. H. Gross, “On the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication” in Number Theory Related to Fermat’s Last Theorem (Cambridge, Mass., 1981), Progr. Math. 26, Birkhäuser, Boston, 1982, 219–236.
  • [34] F. Grunewald and J. Mennicke, SL(2,$\mathcal{O}$) and elliptic curves, preprint, 1978.
  • [35] G. Harder, R. P. Langlands, and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal-Flächen, J. Reine Angew. Math. 366 (1986), 53–120.
  • [36] M. Hindry and J. H. Silverman, Diophantine Geometry, Grad. Texts in Math. 201, Springer, New York, 2000.
  • [37] J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero with an appendix by D. Goldfeld, J. Hoffstein, and D. Lieman, Ann. of Math. (2) 140 (1994), 161–181.
  • [38] H. Jacquet, I. I. Piatetski-Shapiro, and J. Shalika, Conducteur des représentations du groupe linéaire, Math. Ann. 256 (1981), 199–214.
  • [39] H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms, II, Amer. J. Math. 103 (1981), 777–815.
  • [40] B. W. Jordan and R. Livné, Integral Hodge theory and congruences between modular forms, Duke Math. J. 80 (1995), 419–484.
  • [41] S.-i. Kato and K. Takano, Subrepresentation theorem for $p$-adic symmetric spaces, Int. Math. Res. Not. IMRN 2008, no. 11, art. ID rnn028.
  • [42] N. Lagier, Terme constant de fonctions sur un espace symétrique réductif $p$-adique, J. Funct. Anal. 254 (2008), 1088–1145.
  • [43] J. Lansky and A. Raghuram, On the correspondence of representations between $\mathrm{GL}(n)$ and division algebras, Proc. Amer. Math. Soc. 131 (2003), 1641–1648.
  • [44] The LMFDB Collaboration, The $L$-functions and modular forms database, preprint, http://www.lmfdb.org (accessed 16 September 2013).
  • [45] W. Lück, Survey on approximating $L^{2}$-invariants by their classical counterparts: Betti numbers, torsion invariants and homological growth, preprint, arXiv:1501.0746v1 [math.GT].
  • [46] W. Luo and P. Sarnak, Number variance for arithmetic hyperbolic surfaces, Comm. Math. Phys. 161 (1994), 419–432.
  • [47] S. Marshall, Bounds for the multiplicities of cohomological automorphic forms on $\mathrm{GL}_{2}$, Ann. of Math. (2) 175 (2012), 1629–1651.
  • [48] D. Masser and G. Wüstholz, Isogeny estimates for abelian varieties, and finiteness theorems, Ann. of Math. (2) 137 (1993), 459–472.
  • [49] J. J. Millson, On the first Betti number of a constant negatively curved manifold, Ann. of Math. (2) 104 (1976), 235–247.
  • [50] C. J. Moreno, The strong multiplicity one theorem for $\mathrm{GL}_{n}$, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 180–182.
  • [51] W. Müller, Analytic torsion and ${R}$-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), 233–305.
  • [52] D. Prasad, Invariant forms for representations of $\mathrm{GL}_{2}$ over a local field, Amer. J. Math. 114 (1992), 1317–1363.
  • [53] A. D. Rahm and M. H. Şengün, On level one cuspidal Bianchi modular forms, LMS J. Comput. Math. 16 (2013), 187–199.
  • [54] J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, 2nd ed., Grad. Texts in Math. 149, Springer, New York, 2006.
  • [55] Z. Rudnick, A central limit theorem for the spectrum of the modular domain, Ann. Henri Poincaré 6 (2005), 863–883.
  • [56] Y. Sakellaridis, On the unramified spectrum of spherical varieties over $p$-adic fields, Compos. Math. 144 (2008), 978–1016.
  • [57] Y. Sakellaridis and A. Venkatesh, Periods and harmonic analysis on spherical varieties, preprint, arXiv:1203.0039v3 [math.RT].
  • [58] P. Sarnak, “Arithmetic quantum chaos” in The Schur Lectures (1992) (Tel Aviv), Israel Math. Conf. Proc. 8, Bar-Ilan Univ., Ramat Gan, 1995, 183–236.
  • [59] P. Sarnak, Letter to Z. Rudnick on multiplicities of eigenvalues for the modular surface, preprint, http://publications.ias.edu/sarnak/paper/500 (accessed 2 February 2016).
  • [60] M. H. Şengün, On the integral cohomology of Bianchi groups, Exp. Math. 20 (2011), 487–505.
  • [61] M. H. Şengün, On the torsion homology of non-arithmetic hyperbolic tetrahedral groups, Int. J. Number Theory 8 (2012), 311–320.
  • [62] M. H. Şengün and P. Tsaknias, Dimension formulae for spaces of lifted Bianchi modular forms, preprint, arXiv:1310.5385v2 [math.NT].
  • [63] J. H. Silverman, A lower bound for the canonical height on elliptic curves over abelian extensions, J. Number Theory 104 (2004), 353–372.
  • [64] R. Taylor, “Representations of Galois groups associated to modular forms” in Proceedings of the International Congress of Mathematicians, Vol. 1, 2(Zürich, 1994), Birkhäuser, Basel, 1995, 435–442.
  • [65] S. Ullom, Integral normal bases in Galois extensions of local fields, Nagoya Math. J. 39 (1970), 141–148.
  • [66] A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2) 172 (2010), 989–1094.