Related papers: Computing Congruences Between an Elliptic Curve wi…

In this paper, we develop a new and efficient approach to the computation of envelope surfaces. We interpret one-parameter systems of surfaces as curves in the homogeneous spaces of suitable Lie groups. Using the formalism of Lie groups and…

We study a subclass of congruent elliptic curves $E^{(n)}: y^2=x^3-n^2x$, where $n$ is a positive integer congruent to $1\pmod 8$ with all prime factors congruent to $1\pmod 4$. We characterize such $E^{(n)}$ with Mordell-Weil rank zero and…

Let $p, q$ be twin prime numbers with $q-p=2$ . Consider the elliptic curves E=E_\sigma: y^2 = x (x+\sigma p)(x+\sigma q) . (\sigma =\pm 1). E=E_\sigma is also denoted as E_+ or E_- when \sigma = +1or $-1.Here the Mordell-Weil group and the…

We look for elliptic curves featuring rational points whose coordinates form two arithmetic progressions, one for each coordinate. A constructive method for creating such curves is shown, for lengths up to 5.

An isogeny between elliptic curves is an algebraic morphism which is a group homomorphism. Many applications in cryptography require evaluating large degree isogenies between elliptic curves efficiently. For ordinary curves of the same…

In Regge calculus space time is usually approximated by a triangulation with flat simplices. We present a formulation using simplices with constant sectional curvature adjusted to the presence of a cosmological constant. As we will show…

Let $X$ be an elliptic curve or a ramifying hyperelliptic curve over $\mathbb F_q$. We will discuss how to factorize the coefficients of the exponential and logarithm series for a Hayes module over such a curve. This allows us to obtain…

For an elliptic curve $E$ over $K$, the Birch and Swinnerton-Dyer conjecture predicts that the rank of Mordell-Weil group $E(K)$ is equal to the order of the zero of $L(E_{/ K},s)$ at $s=1$. In this paper, we shall give a proof for elliptic…

We propose a new method for accelerating the computation of a concurrency relation, that is all pairs of places in a Petri net that can be marked together. Our approach relies on a state space abstraction, that involves a mix between…

We present a deterministic and explicit algorithm to compute the endomorphism rings of supersingular elliptic curves. As an example we compute the endomorphism rings of all supersingular elliptic curves defined over characteristic…

We describe an algorithm to compute the local component at p of the Coleman-Gross p-adic height pairing on divisors on hyperelliptic curves. As the height pairing is given in terms of a Coleman integral, we also provide new techniques to…

We derive an integral formula for the linking number of two submanifolds of the n-sphere S^n, of the product S^n x R^m, and of other manifolds which appear as "nice" hypersurfaces in Euclidean space. The formulas are geometrically…

Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K)\otimes Q_p, known as a shadow line. When…

A computer which has access to a closed timelike curve, and can thereby send the results of calculations into its own past, can exploit this to solve difficult computational problems efficiently. I give a specific demonstration of this for…

For coprime positive integers $q$ and $e$, let $m(q,e)$ denote the least positive integer $t$ such that there exists a sum of $t$ powers of $q$ which is divisible by $e$. We prove an upper bound for $m(q.e)$ and investigate the case where…

We consider the problem of answering connectivity queries on a real algebraic curve. The curve is given as the real trace of an algebraic curve, assumed to be in generic position, and being defined by some rational parametrizations. The…

The writhe of a space curve fragment is considered for various boundary conditions. An expression for the writhe as a function of arclength for an arbitrary space curve is obtained. The formula is built on the base of closing the tangent…

We show that if $E/\mathbb{Q}$ is an elliptic curve with a rational $p$-torsion for $p=2$ or $3$, then there is a congruence relation between Ramanujan's tau function and $E$ modulo $p$. We make use of such congruences to compute the…

Orientations of supersingular elliptic curves encode the information of an endomorphism of the curve. Computing the full endomorphism ring is a known hard problem, so one might consider how hard it is to find one such orientation. We prove…

We prove the $p$-parity conjecture for elliptic curves over global fields of characteristic $p > 3$. We also present partial results on the $\ell$-parity conjecture for primes $\ell \neq p$.