arXiv subjects

Roberto Miatello

Publications and source records attributed to Roberto Miatello.

5 recordsLinked to original sources

arXiv2026

Cusp forms and parabolic cohomology classes for symmetric spaces of rank one

For any rank-one Riemannian symmetric space S of non-compact type and any discrete, cofinite, non-cocompact, torsion-free group $\Gamma$ of orientation-preserving Riemannian isometries on S, we develop a cohomological interpretation for the cusp forms of $\Gamma$. To that end, we identify certain $\Gamma$-submodules of smooth semi-analytic vectors in the spherical principal series representation with spectral parameter $\nu$ as well as certain subspaces of parabolic cohomology spaces of $\Gamma$ of degree dim S-1 with these $\Gamma$-submodules. We provide explicit isomorphisms between the spaces of cusp forms of spectral parameter $\nu$ and these specific cohomology subspaces. The isomorphisms from cusp forms to cohomology are given by an integral transform, and the explicit form of the inverse isomorphism takes advantage of a certain reproducing property of the integral transform. The result is uniform for all these symmetric spaces and does not rely on their classification.

math.NT↗

arXiv2009

Lattice points for products of upper half planes

Let $Γ$ be an irreducible lattice in $\PSL_2(\RR)^d$ ($d\in\NN$) and $z$ a point in the $d$-fold direct product of the upper half plane. We study the discrete set of componentwise distances ${\bf D}(\Gm,z)\subset \RR^d$ defined in (1). We prove asymptotic results on the number of $\gm\in\Gm$ such that $d(z,γz$ is contained in strips expanding in some directions and also in expanding hypercubes. The results on the counting in expanding strips are new. The results on expanding hypercubes % improve the error terms improve the existing error terms (by Gorodnick and Nevo) and generalize the Selberg error term for $d=1$. We give an asymptotic formula for the number of lattice points $γz$ such that the hyperbolic distance in each of the factors satisfies $d((γz)_j, z_j)\le T$. The error term, as $T \to \infty$ generalizes the error term given by Selberg for $d=1$, also we describe how the counting function depends on $z$. We also prove asymptotic results when the distance satisfies $A_j \le d((γz)_j, z_j) < B_j$, with fixed $A_j < B_j$ in some factors, while in the remaining factors $0 \le d((γz)_j, z_j) \le T$ is satisfied.

math.NT↗

arXiv2005

Spectral properties of 4-dimensional compact flat manifolds

We study the spectral properties of a large class of compact flat Riemannian manifolds of dimension 4, namely, those whose corresponding Bieberbach groups have the canonical lattice as translation lattice. By using the explicit expression of the heat trace of the Laplacian acting on $p$-forms, we determine all $p$-isospectral and $L$-isospectral pairs and we show that in this class of manifolds, isospectrality on functions and isospectrality on $p$-forms for all values of $p$ are equivalent to each other. The list shows for any $p$, $1 \le p \le 3$, many $p$-isospectral pairs that are not isospectral on functions and have different lengths of closed geodesics. We also determine all length isospectral pairs (i.e. with the same length multiplicities), showing that there are two weak length isospectral pairs that are not length isospectral, and many pairs, $p$-isospectral for all $p$ and not length isospectral.

math.DG↗

arXiv2004

The spectrum of twisted Dirac operators on compact flat manifolds

Let $M$ be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of $M$, and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group $\Z_2^k$, we give a very simple expression for the multiplicities of eigenvalues that allows to compute explicitly the $η$-series in terms of values of Riemann-Hurwitz zeta functions, and the $η$-invariant. We give the dimension of the space of harmonic spinors and characterize all $\Z_2^k$-manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs of $\Z_2^k$-manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat $n$-manifolds, pairwise non-homeomorphic to each other.

math.DG↗

arXiv2003

Spin structures and spectra of $Z_2^k$-manifolds

We give necessary and sufficient conditions for the existence of pin+, pin- and spin structures on Riemannian manifolds with holonomy group $Z_2^k$. For any n>3 (resp. n>5) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M_1, M_2, non homeomorphic to each other, that are Laplace isospectral on functions and on p-forms for any p and such that M_1 admits a pin+, or pin-, (resp. spin) structure whereas M_2 does not.

math.DG↗