We prove a dynamical version of the Mordell-Lang conjecture for étale endomorphisms of quasiprojective varieties. We use p-adic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. As special cases of our result we obtain a new proof of the classical Mordell-Lang conjecture for cyclic subgroups of a semiabelian variety, and we also answer posi...

Elliptic curve cryptography has received great attention in recent years due to its high resistance against modern cryptanalysis. The aim of this article is present efficient generators generate substitution boxes (S-boxes) and pseudo random numbers which are essential for many well-known cryptosystems. These based on a special class ordered Mordell elliptic curves. Rigorous analyses performed ...

Given a genus 2 curve C with rational Weierstrass point defined over number field, we construct family of 5 curves that realize descent by maximal unramified abelian two-covers C, and describe explicit models the isogeny classes their Jacobians as restrictions scalars elliptic curves. All constructions this paper are accompanied formulas implemented in Magma and/or SageMath. We apply these algo...

We prove versions of the Mordell-Lang conjecture for semiabelian varieties defined over fields of positive characteristic.

with equality if and only if the triangle is equilateral and P is its center. This inequality was conjectured by Erdős [1] and proved by Mordell and Barrow [2]. Oppenheim [3] established a number of additional inequalities relating the six distances p, q, r , x , y, and z. Such an inequality will be referred to as an Erdős-Mordell-type inequality. A survey of some of these inequalities can be f...

1 The Tate{Lichtenbaum pairing In the paper F-R] it is shown how the Tate pairing on Abelian varieties in Licht-enbaum`s version can be used to relate the discrete logarithm in the group J m (F q) of m{torsion points of the Mordell-Weil group of the Jacobian J of a curve over a nite eld F q to the discrete logarithm in F q if q ? 1 is divisible by m. 1 More precisely the main result of F-R] can...

Given an L-function, one of the most important questions concerns its vanishing at the central point; for example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing there of an elliptic curve L-function equals the rank of the Mordell-Weil group. The Katz and Sarnak Density Conjecture states that this and other behavior is well-modeled by random matrix ensembles. This c...

Let E be an elliptic curve having Complex Multiplication by the full ring OK of integers of K = Q( √ −D), let H = K(j(E)) be the Hilbert class field of K. Then the Mordell-Weil group E(H) is an OK-module, and its Steinitz class St(E) is studied. When D is a prime number, it is proved that St(E) = 1 if D ≡ 3 (mod 4); and St(E) = [P]t if p ≡ 1 (mod 4), where [P] is the ideal class of K represente...

We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. Specifically, let φ be a rational function with no periodic critical points other than those that are totally invariant, and consider the diagonal action of φ on (P). If the coefficients of φ are algebraic, we show that the orbit of a point outside the union of the proper preperiodic subvarieties of (P) ha...

be a positive definite quadratic form with determinant D R and integer coefficients a ;Call it an even form if all a ; ; are even ., an odd form if at least one a ;; is odd . Then Í,, is called non-decomposable, if it cannot be expressed as a sum of two non-negative quadratic forms with integer coefficients . Mordell 1 ) proved that f, can always be decomposed into a sum of five squares of line...