For every odd prime number p, we give examples of non-constant smooth families of genus 2 curves over fields of characteristic p which have pro-Galois (pro-étale) covers of infinite degree with geometrically connected fibers. The Jacobians of the curves are isomorphic to products of elliptic curves.

Lagrangian curves in R entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangrian curves with constant symplectic curvatures, to construct a class of Lagrangian tori in R and determ...

In this paper, we give a ‘direct’ construction of the endomorphism ring of supersingular elliptic curves over a prime field Fp from ‘ideal classes’ of Q( √−p). We use the result to prove that the result of Kaneko on ‘minimal’ CM liftings of such supersingular elliptic curves is a best possible result. We also prove that the result of Elkies on ‘minimal’ CM liftings of all supersingular elliptic...

The Griiths formalism is applied to nd constant torsion curves which are extremal for arclength with respect to variations preserving torsion, xing the endpoints and the binormals at the endpoints. The critical curves are elastic rods of constant torsion, which are shown to not realize certain boundary conditions. In the calculus of variations under nonholonomic constraints, one tries to nd the...

A renormalization group (RG) improvement of the Einstein-Hilbert action is performed which promotes Newton's constant and the cosmological constant to scalar functions on spacetime. They arise from solutions of an exact RG equation by means of a " cutoff identification " which associates RG scales to the points of spacetime. The resulting modified Einstein equations for spherically symmetric, s...

Introduction: Raising the knowledge of skeletal and soft tissue changes with aging has been highly essential due to an increasing demand for aesthetic facial surgery following aging. The aim of this study is to evaluate the three dimensional computed tomographic images and process of changes in mandible with aging. Materials and Methods: In this descriptive study, the facial CT scans were obta...

in this note we review a simple criterion, due to ekedahl, for superspecial curves defined over finite fields.using this we generalize and give some simple proofs for some well-known superspecial curves.