Hyperbolic manifolds and pseudo-arithmeticity

Abstract

We introduce and motivate a notion of pseudo-arithmeticity, which possibly applies to all lattices in PO ⁡ ( n , 1 ) \operatorname {PO}(n,1) with n > 3 n>3 . We further show that under an additional assumption (satisfied in all known cases), the covolumes of these lattices correspond to rational linear combinations of special values of L L -functions.

The Trace Field of Hyperbolic Gluings

“…Therefore M / ∈ A gl . However, it turns out that this manifold M is still pseudo-arithmetic (see [EM18] and Chapter 5).…”

“…The goal of this final chapter is to present the notion of pseudo-arithmeticity and motivate it using the trace field computations from the previous sections. The content of this chapter is a more detailed version of a joint work with Vincent Emery [EM18]. In the first section, we start by introducing pseudoarithmetic manifolds and deduce from the results established in the previous chapters that all gluings of arithmetic pieces are pseudo-arithmetic.…”

“…• The maps 1 and 2 exist when coefficients are extended to Q (this is Lemma 3.2 in [EM18]; see also [Em17, Proposition 6.1]).…”

“…In the final part of this thesis (based on a joint work [EM18] with Vincent Emery), we introduce and motivate the notion of pseudo-arithmeticity using the theory developed in the previous chapters. Pseudo-arithmetic manifolds can be seen as a generalized version of gluings of pieces of arithmetic manifolds (see the remark after their definition in Section 5.1.1).…”

“…After defining them in their most general form, we compute their trace field and derive the corollaries mentioned above. Finally Chapter 5 treats pseudoarithmeticity and its consequences, following [EM18].…”