Related papers: On simultaneous arithmetic progressions on ellipti…
In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic…
In this paper we study curves in Lorentz-Minkowski space $\mathbb{L}^2$ that are critical points of the moment of inertia with respect to the origin. This extends a problem posed by Euler in the Lorentzian setting. We obtain explicit…
We classify the possible torsion structures of rational elliptic curves over sextic number fields.
An elliptic curve may be immersed in ${\mathbf{P}}^{N-1}$ as a degree $N$ curve using level $N$ structure. In the case where $N$ is odd, there are well known classical results dating back to Bianchi and Klein. In this paper we study the…
We give an infinite family of congruent number elliptic curves, each with rank at least two, which are related to integral solutions of $m^2=n^2+nl+l^2$.
We consider a particular case of an analog for elliptic curves to the Mersenne problem : finding explicitely all prime power terms in an elliptic divisibility sequence when descent via isogeny is possible. We explain how this question can…
We observe that there are elliptic curves over number fields all of whose quadratic twists must have positive rank, assuming the Birch-Swinnerton-Dyer conjecture. We give a classification of such curves in terms of their local behaviour,…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
Asymptotic formula is derived for the behavior of the fundamental solution of the second-order elliptic self-adjoint operator with a piecewise-smooth coefficient in front of the senior derivatives near the discontinuity surface of the…
We consider homogenization problems for linear elliptic equations in divergence form. The coecients are assumed to be a local perturbation of some periodic background. We prove $W^{1,p}$ and Lipschitz convergence of the two-scale expansion,…
The parity conjecture has a long and distinguished history. It gives a way of predicting the existence of points of infinite order on elliptic curves without having to construct them, and is responsible for a wide range of unexplained…
We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…
We develop an arithmetic analogue of linear partial differential equations in two independent ``space-time'' variables. The spatial derivative is a Fermat quotient operator, while the time derivative is the usual derivation. This allows us…
The conjectures of Deligne, Be\u\i linson, and Bloch-Kato assert that there should be relations between the arithmetic of algebro-geometric objects and the special values of their $L$-functions. We make a numerical study for symmetric power…
This article deals with a survey of recent developments and results on Choquard equations where we focus on the existence and multiplicity of solutions of the partial differential equations which involve the nonlinearity of convolution…
We produce explicit elliptic curves over \Bbb F_p(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…
We study the interaction between the group law on an elliptic curve and the additive structure of $x$-coordinates of rational points on an elliptic curve. Let $E/\mathbb{Q}$ be an elliptic curve of Mordell-Weil rank $r \geq 1$, $d \geq 1$…
A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y^2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we…
In this paper, we deal with an elliptic problem with the Dirichlet boundary condition. We operate in Sobolev spaces and the main analytic tool we use is the Lax-Milgram lemma. First, we present the variational approach of the problem which…
These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and…