@article{31982,
abstract = {{AbstractWe show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $$\Sigma $$
Σ
with Betti number $$b_1$$
b
1
, the order of vanishing of the Ruelle zeta function at zero equals $$4-b_1$$
4
-
b
1
, while in the hyperbolic case it is equal to $$4-2b_1$$
4
-
2
b
1
. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle $$S\Sigma $$
S
Σ
with harmonic 1-forms on $$\Sigma $$
Σ
.}},
author = {{Cekić, Mihajlo and Delarue, Benjamin and Dyatlov, Semyon and Paternain, Gabriel P.}},
issn = {{0020-9910}},
journal = {{Inventiones mathematicae}},
keywords = {{General Mathematics}},
number = {{1}},
pages = {{303--394}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds}}},
doi = {{10.1007/s00222-022-01108-x}},
volume = {{229}},
year = {{2022}},
}
@article{34890,
abstract = {{We prove that the 4-rank of class groups of quadratic number fields behaves as predicted in an extension due to Gerth of the Cohen–Lenstra heuristics. }},
author = {{Fouvry, Étienne and Klüners, Jürgen}},
issn = {{0020-9910}},
journal = {{Inventiones mathematicae}},
keywords = {{General Mathematics}},
number = {{3}},
pages = {{455--513}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{On the 4-rank of class groups of quadratic number fields}}},
doi = {{10.1007/s00222-006-0021-2}},
volume = {{167}},
year = {{2006}},
}