Related papers: Integer sequences from elliptic curves

Let $E_{m,n}$ be an elliptic curve over $\mathbb{Q}$ of the form $y^2=x^3-m^2x+n^2$, where $m$ and $n$ are positive integers. Brown and Myers showed that the curve $E_{1,n}$ has rank at least two for all $n$. In the present paper, we…

An infinite $\pm 1$-sequence is called {\it Apwenian} if its Hankel determinant of order $n$ divided by $2^{n-1}$ is an odd number for every positive integer $n$. In 1998, Allouche, Peyri\`ere, Wen and Wen discovered and proved that the…

Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note, we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in…

Major controversy surrounds the use of Elliptic Curves in finite fields as Random Number Generators. There is little information however concerning the "randomness" of different procedures on Elliptic Curves defined over fields of…

Many discrete integrable systems exhibit the Laurent phenomenon. In this paper, we investigate three integrable systems: the Somos-4 recurrence, the Somos-5 recurrence and a system related to so-called $A_1$ $Q$-system, whose general…

We present recursive formulas which compute the recently defined "higher symplectic capacities" for all convex toric domains. In the special case of four-dimensional ellipsoids, we apply homological perturbation theory to the associated…

We present a method to calculate the rank of $E(\oQ(s,t))$ for the elliptic curve mentioned in the title. This method uses a generalization of a method from Van Geemen and Werner to calculate $h^4(Y)$ for nodal hypersurfaces $Y$.

We give two explicit sets of generators of the group of invertible regular functions over QQ on the modular curve Y1(N). The first set of generators is very surprising. It is essentially the set of defining equations of Y1(k) for k <= N/2…

In this paper we relate some classical normal forms for complex elliptic curves in terms of 4-point sets in the Riemann sphere. Our main result is an alternative proof that every elliptic curve is isomorphic as a Riemann surface to one in…

In this paper we deal with the dimension of multisequences and related properties. For a given multisequence W and an m tuple of positive integers R, we define the R extension of W. Further we count the number of multisequences W whose R…

For the integer $ D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\mathbb Q$, where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity…

For a given elliptic curve, its associated $L$-function evaluated at $1$ is closely related to its real period. In this article, we generalize this principle to a rational curve. We count the rational points over all finite fields and use…

We study rational points on the elliptic surface given by the equation: $$y^2 = x^3 + AxQ(u,v)^2 + BQ(u,v)^3,$$ where $A,B\in \mathbb{Z}$ satisfy that $4A^3-27B^2\neq 0$ and $Q(u,v)$ is a positive-definite quadratic form. We prove…

Let E be a modular elliptic curve defined over a rational function field k of odd characteristic. We construct a sequence of Heegner points on E, defined over a $Z_p^{\infty}$-tower of finite extensions of k, and show that these Heegner…

We introduce a class of iterated integrals that generalize multiple polylogarithms to elliptic curves. These elliptic multiple polylogarithms are closely related to similar functions defined in pure math- ematics and string theory. We then…

This paper provides the connection between the Hankel transform and aerating transforms of a given integer sequence. Results obtained are used to establish a completely different Hankel transform evaluation of the series reversion of a…

We show that there are infinitely many elliptic curves $E/\mathbb{Q}$, up to isomorphism over $\overline{\mathbb{Q}}$, for which the finitely generated group $E(\mathbb{Q})$ has rank exactly $2$. Our elliptic curves are given by explicit…

We construct invariants under deformation of real symplectic 4-manifolds. These invariants are obtained by counting three different kinds of real rational J-holomorphic curves which realize a given homology class and pass through a given…

We prove useful necessary and sufficient conditions for an elliptic curve over a number field to admit a surjective adelic Galois representation. Using these conditions, we compute an example of a number field K and an elliptic curve E/K…

We propose several procedures for creating new families of integer sequences based on the method of Cantor diagonalization. Then we modify and generalize this method. The paper includes explicit formulas for most proposed families of…