Cubic structures, equivariant Euler characteristics and lattices of modular forms

We use the theory of cubic structures to give a fixed point Riemann-Roch formula for the equivariant Euler characteristics of coherent sheaves on projective flat schemes over Z with a tame action of a finite abelian group. This formula supports a conjecture concerning the extent to which such equivariant Euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus. Our results are applied to study the module structure of modular forms having Fourier coefficients in a ring of algebraic integers, as well as the action of diamond Hecke operators on the Mordell-Weil groups and Tate-Shafarevich groups of Jacobians of modular curves.Comment: 40pp, Final version, to appear in the Annals of Mathematic

Matching of analytical and numerical solutions for neutron stars of arbitrary rotation

We demonstrate the results of an attempt to match the two-soliton analytical solution with the numerically produced solutions of the Einstein field equations, that describe the spacetime exterior of rotating neutron stars, for arbitrary rotation. The matching procedure is performed by equating the first four multipole moments of the analytical solution to the multipole moments of the numerical one. We then argue that in order to check the effectiveness of the matching of the analytical with the numerical solution we should compare the metric components, the radius of the innermost stable circular orbit ($R_{ISCO}$), the rotation frequency $\Omega\equiv\frac{d\phi}{dt}$ and the epicyclic frequencies $\Omega_{\rho},\;\Omega_z$. Finally we present some results of the comparison.Comment: Contribution at the 13th Conference on Recent Developments in Gravity (NEB XIII), corrected typo in $M_4$ of eq. 5 of the published versio

Rescuing Quartic and Natural Inflation in the Palatini Formalism

When considered in the Palatini formalism, the Starobinsky model does not provide us with a mechanism for inflation due to the absence of a propagating scalar degree of freedom. By (non)--minimally coupling scalar fields to the Starobinsky model in the Palatini formalism we can in principle describe the inflationary epoch. In this article, we focus on the minimally coupled quartic and natural inflation models. Both theories are excluded in their simplest realization since they predict values for the inflationary observables that are outside the limits set by the Planck data. However, with the addition of the $R^2$ term and the use of the Palatini formalism, we show that these models can be rendered viable.Comment: JCAP accepted version, 16 pages, 7 figure

Faithful transformation of quasi-isotropic to Weyl-Papapetrou coordinates: A prerequisite to compare metrics

We demonstrate how one should transform correctly quasi-isotropic coordinates to Weyl-Papapetrou coordinates in order to compare the metric around a rotating star that has been constructed numerically in the former coordinates with an axially symmetric stationary metric that is given through an analytical form in the latter coordinates. Since a stationary metric associated with an isolated object that is built numerically partly refers to a non-vacuum solution (interior of the star) the transformation of its coordinates to Weyl-Papapetrou coordinates, which are usually used to describe vacuum axisymmetric and stationary solutions of Einstein equations, is not straightforward in the non-vacuum region. If this point is \textit{not} taken into consideration, one may end up to erroneous conclusions about how well a specific analytical metric matches the metric around the star, due to fallacious coordinate transformations.Comment: 18 pages, 2 figure