Torsion homology growth in arithmetic groups

  • Nicolas Bergeron

    Université Pierre et Marie Curie, Paris, France
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Various recent works show that certain arithmetic groups -- that generalize the modular group -- can have `a lot' of torsion in their homology. Among these groups are the finite index (congruence) subgroups of SL3(Z)\mathrm{SL}_3 (\mathbb{Z}) or SL2(Z[i])\mathrm{SL}_2 (\mathbb{Z} [i]). In the latter case homology reduces to abelianization. In particular, for

Γ0(N)={(abcd)SL2(Z[i])    Nc}(NZ[i]),\textstyle \Gamma_0 (N) = \left\{ \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \mathrm{SL}_2 (\mathbb{Z} [i]) \; \big| \; N | c \right\} \quad (N \in \mathbb{Z}[i]),

one may ask about the structure of the finitely generated Z\mathbb{Z}-module Γ0(N)ab=Γ0(N)/[Γ0(N):Γ0(N)].\Gamma_0 (N) ^{\rm ab} = \Gamma_0 (N) / [\Gamma_0 (N) : \Gamma_0 (N)]. It has a finite torsion part Γ0(N)torsab\Gamma_0 (N)^{\rm ab}_{\rm tors} and Akshay Venkatesh and I have conjectured that, as NN tends to \infty among primes, we have:

logΓ0(N)torsabN2λ18π,λ=L(2,χQ(i))=119+125149+\frac{\log |\Gamma_0 (N)^{\rm ab}_{\rm tors}|}{|N|^2} \to \frac{\lambda }{18\pi}, \quad \lambda=L(2, \chi_{\mathbb{Q} (i)}) = 1- \frac{1}{9} + \frac{1}{25} - \frac{1}{49} + \ldots

More generally one may ask: \emph{How does the amount of torsion in the homology of an arithmetic group grow with the level NN?} We propose a conjectural partial answer. This contribution presents ideas for how to attack this conjecture and discusses recent progress towards it. This topic interacts with more classical questions of geometry (analytic torsion, Gromov--Thurston norm, (higher) cost, rank and deficiency gradient \dots) and number theory (BSD conjecture, ABC conjecture~\dots). A big motivation is provided by (one of) Peter Scholze's recent breakthrough(s): \emph{very roughly} a mod pp torsion class in Γ0(N)torsab\Gamma_0 (N)^{\rm ab}_{\rm tors} parametrizes a field extension K/Q(i)K/\mathbb{Q} (i) whose Galois group is a subgroup of GL2(Fp)\mathrm{GL}_2 (\overline{\mathbb{F}}_p). Moreover, it is anticipated that there is a corresponding 'torsion Langlands program'.