Helly-type theorems for homothets of planar convex curves

Helly's theorem implies that if is a finite collection of (positive) homothets of a planar convex body , any three having non-empty intersection, then has non-empty intersection. We show that for collections of homothets (including translates) of the boundary , if any four curves in have non-empty intersection, then has non-empty intersection. We prove the following dual version: If any four points of a finite set in the plane can be covered by a translate [homothet] of , then can be covered by a translate [homothet] of . These results are best possible in general.

     

Iwasawa μ-invariants of elliptic curves and their symmetric powers

In an earlier paper we considered the effects that finite submodules can have on μ-invariants of Selmer groups. In this paper we examine some of the consequences of that theory to elliptic curves and their symmetric powers. One main result is the construction of an isogeny class of elliptic curves, all of which have positive μ-invariants. A second result is a connection between the behaviors of μ-invariants associated with the symmetric powers of an elliptic curve over , and the behavior of the μ-invariant of that elliptic curve over different extensions of .     

Higher Weierstrass points on

We study the arithmetic properties of higher Weierstrass points on modular curves for primes . In particular, for , we obtain a relationship between the reductions modulo of the collection of -Weierstrass points on and the supersingular locus in characteristic .

     

Taylor expansion of an Eisenstein series

In this paper, we give an explicit formula for the first two terms of the Taylor expansion of a classical Eisenstein series of weight for . Both the first term and the second term have interesting arithmetic interpretations. We apply the result to compute the central derivative of some Hecke -functions.

     

Sieving for rational points on hyperelliptic curves

We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even when our curve has points over all localizations . We illustrate the practicality of the method with some examples of hyperelliptic curves of genus .

     

A Note on Shiffman's Theorems

Shiffman proved his famous first theorem, that if A R³ is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by strictly convex horizontal Jordan curves. In this article we use Perron's method to construct minimal annuli which have a planar end and are bounded by two convex Jordan curves in horizontal planes, but the horizontal level sets of the surfaces are not all convex Jordan curves or straight lines. These surfaces show that unlike his second and third theorems, Shiffman's first theorem is not generalizable without further qualification.     

Sums of squares of regular functions on real algebraic varieties

Let be an affine algebraic variety over (or any other real closed field ). We ask when it is true that every positive semidefinite (psd) polynomial function on is a sum of squares (sos). We show that for the answer is always negative if has a real point. Also, if is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if is a smooth surface with only real divisors at infinity. The ``compact' case is harder. We completely settle the case of smooth curves of genus : If such a curve has a complex point at infinity, then every psd function is sos, provided the field is archimedean. If is not archimedean, there are counter-examples of genus .

     

Kamienny's criterion and the method of Coleman and Chabauty

This paper gives a new proof of Kamienny's Criterion using the method of Coleman and Chabauty.

     

Global Geometry of 3-Body Motions with Vanishing Angular Momentum Ⅰ

Following Jacobi＇s geometrization of Lagrange＇s least action principle, trajectories of classical mechanics can be characterized as geodesics on the configuration space M with respect to a suitable metric which is the conformal modification of the kinematic metric by the factor （U ＋ h）, where U and h are the potential function and the total energy, respectively. In the special case of 3-body motions with zero angular momentum, the global geometry of such trajectories can be reduced to that of their moduli curves, which record the change of size and shape, in the moduli space of oriented m-triangles, whose kinematic metric is, in fact, a Riemannian cone over the shape space M^＊≌S^2 （1/2）.
In this paper, it is shown that the moduli curve of such a motion is uniquely determined by its shape curve （which only records the change of shape） in the case of h≠0, while in the special case of h = 0 it is uniquely determined up to scaling. Thus, the study of the global geometry of such motions can be further reduced to that of the shape curves, which are time-parametrized curves on the 2-sphere characterized by a third order ODE. Moreover, these curves have two remarkable properties, namely the uniqueness of parametrization and the monotonieity, that constitute a solid foundation for a systematic study of their global geometry and naturally lead to the formulation of some pertinent problems.