Let k be a global field, k a separable closure of k, and Gk the absolute Galois group Gal(k/k) of k over k. For every σ ∈ Gk, let k σ be the fixed subfield of k under σ. Let E/k be an elliptic curve over k. We show that for each σ ∈ Gk, the Mordell-Weil group E(k σ ) has infinite rank in the following two cases. Firstly when k is a global function field of odd characteristic and E is parametriz...

Although this formula can be obtained by a limiting process from (0.1), it was found before [11] by the paper of Kiepert [13]. If we set y(u) = 1 2℘ ′(u) and x(u) = ℘(u), then we have an equation y(u) = x(u)+ · · · , that is a defining equation of the elliptic curve to which the functions ℘(u) and σ(u) are attached. Here the complex number u and the coordinate (x(u), y(u)) correspond by the equ...

Let Lε = −div ` A ` x ε ́ ∇ ́ , ε > 0 be a family of second order elliptic operators with real, symmetric coefficients on a bounded Lipschitz domain Ω in Rn, subject to the Dirichlet boundary condition. Assuming that A(x) is periodic and belongs to VMO, we show that there exists δ > 0 independent of ε such that Riesz transforms ∇(Lε)−1/2 are uniformly bounded on Lp(Ω), where 1 < p < 3+δ if n ≥ 3,...

The modular curve X1(N) parametrizes elliptic curves with a point of order N . For N ≤ 50 we obtain plane models for X1(N) that have been optimized for fast computation, and provide explicit birational maps to transform a point on our model of X1(N) to an elliptic curve. Over a finite field Fq, these allow us to quickly construct elliptic curves containing a point of order N , and can accelerat...

For every prime p we give infinitely many examples of torsors under abelian varieties over Q that are locally trivial but not divisible by p in the Weil-Châtelet group. We also give an example of a locally trivial torsor under an elliptic curve over Q which is not divisible by 4 in the Weil-Châtelet group. This gives a negative answer to a question of Cassels.

We consider F-theory compactifications on a mirror pair of elliptic Calabi–Yau threefolds. This yields two different six-dimensional theories, each of them being nonperturbatively equivalent to some compactification of heterotic strings on a K3 surface S with certain bundle data E → S. We find evidence for a transformation of S together with the bundle that takes one heterotic model to the othe...

A new technique for proving uniqueness of martingale problems is introduced. The method is illustrated in the context of elliptic diffusions in Rd.

In this paper we describe a family of locally conformal Kähler metrics on class 1 Hopf surfaces Hα,β containing some recent metrics constructed in [GO98]. We study some canonical foliations associated to these metrics, in particular a 2-dimensional foliation Eα,β that is shown to be independent of the metric. We elementary prove that Eα,β has compact leaves if and only if α = β for some integer...

An open problem in affine geometry is whether an affine complete locally uniformly convex hypersurface in Euclidean (n + 1)-space is Euclidean complete for n ≥ 2. In this paper we give the affirmative answer. As an application, it follows that an affine complete, affine maximal surface in R3 must be an elliptic paraboloid.

This paper presents new explicit formulae for the point doubling, tripling and addition for ordinary Weierstraß elliptic curves with a point of order 3 and their equivalent Hessian curves over finite fields of characteristic three. The cost of basic point operations is lower than that of all previously proposed ones. The new doubling, mixed addition and tripling formulae in projective coordinat...