On Schwarzschild’s Interior Solution and Perfect Fluid Star Model

We solve the boundary value problem for Einstein’s gravitational field equations in the presence of matter in the form of an incompressible perfect fluid of density ρ and pressure field p(r) located in a ball r≤r0. We find a 1-parameter family of time-independent and radially symmetric solutions ga,ρa,pa:−2m<a<a1 satisfying the boundary conditions g=gS and p=0 on r=r0, where gS is the exterior Schwarzschild solution (solving the gravitational field equations for a point mass M concentrated at r=0) and containing (for a=0) the interior Schwarzschild solution, i.e., the classical perfect fluid star model. We show that Schwarzschild’s requirement r0>9κM/(4c2) identifies the “physical” (i.e., such that pa(r)≥0 and pa(r) is bounded in 0≤r≤r0) solutions {pa:a∈U0} for some neighbourhood U0⊂(−2m,+∞) of a=0. For every star model {ga:a0<a<a1}, we compute the volume V(a) of the region r≤r0 in terms of abelian integrals of the first, second, and third kind in Legendre form.

Semistable models of elliptic curves over residue characteristic 2

Given an elliptic curve E in Legendre form $y^2 = x(x - 1)(x - \lambda )$ over the fraction field of a Henselian ring R of mixed characteristic $(0, 2)$ , we present an algorithm for determining a semistable model of E over R that depends only on the valuation of $\lambda $ . We provide several examples along with an easy corollary concerning $2$ -torsion.

Six unlikely intersection problems in search of effectivity

We investigate four properties related to an elliptic curve Et in Legendre form with parameter t: the curve Et has complex multiplication, E−t has complex multiplication, a point on Et with abscissa 2 is of finite order, and t is a root of unity. Combining all pairs of properties leads to six problems on unlikely intersections. Using a variety of techniques we solve these problems with varying degrees of effectivity (and for three of them we even present the list of all possible t).