Elliptic Divisibility Sequences Research Articles

Some effectivity results for primitive divisors of elliptic divisibility sequences

Some effectivity results for primitive divisors of elliptic divisibility sequences

The elliptic sieve and Brauer groups

AbstractA theorem of Serre states that almost all plane conics over have no rational point. We prove an analogue of this for families of conics parametrised by elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve for elliptic curves. We also give more general results for specialisations of Brauer groups, which yields applications to norm form equations.

On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences

We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the form y 2 =x(x 2 +b), where b is any positive integer. We achieve this by using the modularity of elliptic curves over real quadratic number fields.

Orders of vanishing and U(1) charges in F-theory

Many interesting questions about F-theory models, including several concerning the F-theory swampland, involve massless matter charged under U(1) gauge symmetries. It is therefore important to better understand the geometric properties of F-theory models realizing various U(1) charges. We propose that, for F-theory models described by elliptic fibrations in Weierstrass form, the U(1) charge of light matter is encoded in the orders of vanishing of the section components corresponding to the U(1) gauge symmetry. We give specific equations relating the U(1) charges to the orders of vanishing that seem to hold for both U(1)-charged singlets and for matter additionally charged under a simply-laced nonabelian gauge algebra. Our formulas correctly describe properties of F-theory models in the prior literature, and we give an argument that they should describe the orders of vanishing for arbitrarily high U(1) charges. They also resemble formulas for the p-adic valuations of elliptic divisibility sequences developed by Stange [1]. These proposals could serve as a U(1) analogue of the Katz-Vafa method, allowing one to determine U(1) charges without resolution. Additionally, they predict geometric information about F-theory models with general U(1) charges, which may be useful for exploring the F-theory landscape and swampland.

Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces

We consider elliptic surfaces \({\mathcal {E}}\) over a field k equipped with zero section O and another section P of infinite order. If k has characteristic zero, we show there are only finitely many points where O is tangent to a multiple of P. Equivalently, there is a finite list of integers such that if n is not divisible by any of them, then nP is not tangent to O. Such tangencies can be interpreted as unlikely intersections. If k has characteristic zero or \(p>3\) and \({\mathcal {E}}\) is very general, then we show there are no tangencies between O and nP. We apply these results to square-freeness of elliptic divisibility sequences and to geography of surfaces. In particular, we construct mildly singular surfaces of arbitrary fixed geometric genus with K ample and \(K^2\) unbounded.

Primitive divisors of sequences associated to elliptic curves with complex multiplication

Let P and Q be two points on an elliptic curve defined over a number field K. For alpha in {text {End}}(E), define B_alpha to be the mathcal {O}_K-integral ideal generated by the denominator of x(alpha (P)+Q). Let mathcal {O} be a subring of {text {End}}(E), that is a Dedekind domain. We will study the sequence {B_alpha }_{alpha in mathcal {O}}. We will show that, for all but finitely many alpha in mathcal {O}, the ideal B_alpha has a primitive divisor when P is a non-torsion point and there exist two endomorphisms gne 0 and f so that f(P)= g(Q). This is a generalization of previous results on elliptic divisibility sequences.

Elliptic divisibility sequences over the Edwards model of elliptic curves

In this work, we use division polynomials on Edwards elliptic curves to construct elliptic nets of rank one called elliptic divisibility sequence on this curve. Introduced by Morgan Ward, elliptic divisibility sequence on elliptic curve are used for solving the elliptic curve discrete logarithm problem (ECDLP). Moved on rank 2, Elliptic Nets algorithm via elliptic nets associated to Weierstrass elliptic curves are useful for the computation of bilinear maps (called pairings) defined on the groups of points of elliptic curves and very useful to the construction of cryptographic protocols.

Primitive divisors of elliptic divisibility sequences for elliptic curves with $j=1728$

Take a rational elliptic curve defined by the equation $y^2=x^3+ax$ in minimal form and consider the sequence $B_n$ of the denominators of the abscissas of the iterate of a non-torsion point; we show that $B_{5m}$ has a primitive divisor for every $m$. Then, we show how to generalize this method to the terms in the form $B_{mp}$ with $p$ a prime congruent to $1$ modulo $4$.

Sequences associated to elliptic curves with non-cyclic torsion subgroup

Let $E$ be an elliptic curve defined over $K$ given by a Weierstrass equation and let $P=(x,y)\in E(K)$ be a point. Then for each $n$ $\geq 1$ we can write the $x$- and $y$-coordinates of the point $[n]P$ as\[ [n]P=\left( \frac{G_{n}(P)}{F_{n}^{2}(P)},\frac{H_{n}(P)}{F_{n}^{3}(P)}\right)\]where $F_{n}$, $G_{n}$, and $H_{n}\in K[x,y]$ are division polynomials of $E$. In this work we give explicit formulas for sequences\[(F_{n}(P))_{n\geq 0},\,(G_{n}(P))_{n\geq 0},\,\text{and}\,(H_{n}(P))_{n\geq 0}\]associated to an elliptic curve $E$ defined over $\mathbb{Q}$ with non-cyclic torsion subgroup. As applications we give similar formulas for elliptic divisibility sequences associated to elliptic curves with non-cyclic torsion subgroup and determine square terms in these sequences.

Primitive divisors of elliptic divisibility sequences over function fields with constant j-invariant

We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the j-invariant of the elliptic curve is constant.In more detail, given an elliptic curve E with a point P of infinite order over a global field, the sequence D1, D2,… of denominators of multiples P, 2P,… of P is a strong divisibility sequence in the sense that gcd⁡(Dm,Dn)=Dgcd⁡(m,n). This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences.A number N is called a Zsigmondy bound of the sequence if each term Dn with n>N presents a new prime factor. The optimal uniform Zsigmondy bound for the genus-zero sequences over Q is 30 by Bilu-Hanrot-Voutier [2], but finding such a bound remains an open problem in genus one, both over Q and over function fields.We prove that the optimal Zsigmondy bound for ordinary elliptic divisibility sequences over function fields is 2 if the j-invariant is constant. In the supersingular case, we give a complete classification of which terms can and cannot have a new prime factor.

Scalar Multiplication via Elliptic Net using Generalized Equivalent Sequences

Chord and tangent is a classical method to calculate the elliptic curve scalar multiplication. Alternatively, the scalar multiplication can be calculated by dividing polynomials over certain finite fields and the first elliptic net scalar multiplication was implemented on a short Weierstrass curve. The net was originated from non-linear recurrence sequences, namely as elliptic divisibility sequence. It is well known that the linear recurrence sequences have been applied in the cryptosystem as a cipher in the encryption and decryption process. From the perspective of cryptographic application, the elliptic divisibility sequence is used generally for integer factorization, solving elliptic curve discrete logarithm problem and computation of pairing or scalar multiplication. But there is a lack of contribution of these non-linear recurrence sequences in scalar multiplication. Therefore, this paper aims to discuss a generalization of the equivalent sequence of elliptic divisibility for computing scalar multiplication. The experimental results of scalar multiplication via the net and its coding in computer programming are presented. The future direction of scalar multiplication via the elliptic net is also discussed.

The density of the terms in an elliptic divisibility sequence having a fixed G.C.D. with their indices

Let D=(Dn)n≥1 be an elliptic divisibility sequence associated to the pair (E,P). For a fixed integer k, we define AE,k={n≥1:gcd⁡(n,Dn)=k}. We give an explicit structural description of AE,k. Also, we explain when AE,k has positive asymptotic density using bounds related to the distribution of trace of Frobenius of E. Furthermore, we obtain an explicit density for AE,k using the Möbius function.

PERFECT POWERS IN PRODUCTS OF TERMS OF ELLIPTIC DIVISIBILITY SEQUENCES

Diophantine problems involving recurrence sequences have a long history. We consider the equation $B_{m}B_{m+d}\cdots B_{m+(k-1)d}=y^{\ell }$ in positive integers $m,d,k,y$ with $\gcd (m,d)=1$ and $k\geq 2$, where $\ell \geq 2$ is a fixed integer and $B=(B_{n})_{n=1}^{\infty }$ is an elliptic divisibility sequence, an important class of nonlinear recurrences. We prove that the equation admits only finitely many solutions. In fact, we present an algorithm to find all possible solutions, provided that the set of $\ell$th powers in $B$ is given. We illustrate our method by an example.

Somos-4 and Somos-5 are arithmetic divisibility sequences

We provide an elementary proof to a conjecture by Robinson that multiples of (powers of) primes in the Somos-4 sequence are equally spaced. We also show, almost as a corollary, for the generalized Somos-4 sequence defined by and initial values , that the polynomial is a divisor of for all and establish a similar result for the generalized Somos-5 sequence. The proofs involve elliptic divisibility sequences, for which we also show that primes are equally spaced, and a nice property of subsets for which .

On symmetries of elliptic nets and valuations of net polynomials

We give a generalization of a theorem of Ayad on valuations of division polynomials in the context of net polynomials. We also present a generalization of a theorem of Ward on symmetry of elliptic divisibility sequences to the case of elliptic nets.

On common factors within a series of consecutive terms of an elliptic divisibility sequence

We prove that for any elliptic divisibility sequence and any sufficiently large integer k, one can find k consecutive terms of the sequence such that none of these terms is coprime to all the others. In other words, elliptic divisibility sequences are Pillai sequences, named for a problem posed originally by Pillai for the sequence of integers. In fact we give an upper bound for the smallest value k0 past which this property is valid. We also provide a more general theorem where the coprimality condition is severely relaxed. In case of some particular sequences we give the values of k0, as well.

The elliptic curve Diffie-Hellman problem and an equivalent hard problem for elliptic divisibility sequences

The elliptic curve Diffie-Hellman problem and an equivalent hard problem for elliptic divisibility sequences

Prime power terms in elliptic divisibility sequences

We study a problem on specializations of multiples of rational points on elliptic curves analogous to the Mersenne problem. We solve this problem when descent via isogeny is possible by giving explicit bounds on the indices of prime power terms in elliptic divisibility sequences associated to points in the image of a nontrivial isogeny. We also discuss the uniformity of these bounds assuming the Hall–Lang conjecture.

Elliptic divisibility sequences, squares and cubes

Elliptic divisibility sequences (EDSs) are generalizations of a class of integer divisibility sequences called Lucas sequences. There has been much interest in cases where the terms of Lucas sequences are squares or cubes. In this work, using the Tate normal form having one parameter of elliptic curves with torsion points, the general terms and periods of all elliptic divisibility sequences with a zero term are given in terms of this parameter by means of Mazur's theorem, and it is shown that which term of h_{n} of an EDS can be a square or a cube by using the general terms of these sequences.

Perfect powers in elliptic divisibility sequences

It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of Mordell curves and families of congruent number curves are given with corresponding elliptic divisibility sequences having no perfect power terms. The proofs combine primitive divisor results with modular methods for Diophantine equations.