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...

We produce explicit elliptic curves over Fp(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptoti...

All plane curves of degree less than 7 with coefficients in F2 are examined for curves with a large number of Fq rational points on their smooth model, for q = 2,m = 3, 4, ..., 11. Known lower bounds are improved, and new curves are found meeting or close to Serre’s, Lauter’s, and Ihara’s upper bounds for the maximal number of Fq rational points on a curve of genus g.

Steepest descent drives both theory and practice of nonsmooth optimization. We study slight relaxations of two influential notions of steepest descent curves — curves of maximal slope and solutions to evolution equations. In particular, we provide a simple proof showing that lower-semicontinuous functions that are locally Lipschitz continuous on their domains — functions playing a central role ...

Suppose that N is a geometrically finite hyperbolic 3-manifold such that the bending locus of its convex core is a collection of simple closed curves α. We prove that the map which associates to each structure the lengths of the curves in the bending locus is a global diffeomorphism from the set of all structures bent along α onto its image. If α is maximal, the traces of the curves αi ∈ α are ...

We show that the maximal number of singular moves required to pass between any two regularly homotopic plane or spherical curves with at most n crossings grows quadratically with respect to n. Furthermore, for any two regularly homotopic curves with at most n crossings, there exists such a sequence of singular moves, satisfying the quadratic bound, for which all curves along the way have at mos...

In this paper results about the Hilbert function and about the number of minimal generators stated in (Orecchia) for disjoint unions of rational smooth curves are generalized to disjoint unions of distinct smooth non special curves. Hence, the maximal rank and the minimal generation of such curves are studied. In particular, we consider elliptic curves and we describe a method to compute their ...