Frey proposed the idea of ‘disguising’ an elliptic curve. This is a method to obtain a ‘black box’ representation of a group. We adapt this notion to finite fields and tori and study the question of whether such systems are secure. Our main result is an algebraic attack which shows that it is not secure to disguise the torus T2. We also show that some methods for disguising an elliptic curve ar...

A covariant functor on the elliptic curves with complex multiplication is constructed. The functor takes values in the noncommutative tori with real multiplication. A conjecture on the rank of an elliptic curve is formulated.

Let E be an elliptic curve over Q. Using Iwasawa theory, we give what seems to be the first general upper bound for the order of vanishing of the p-adic L-function at s = 0, and the Zp-corank of the Tate-Shafarevich group for all sufficiently large good ordinary primes p.

grows exponentially. Thus, monodromy groups of elliptic fibrations over P constitute a small, but still very significant fraction of all subgroups of finite index in SL(2,Z). Our goal is to introduce some structure on the set of monodromy groups of elliptic fibrations which would help to answer some natural questions. For example, we show how to describe the set of groups corresponding to ratio...

We prove that there are at most finitely many complex λ = 0, 1 such that two points on the Legendre elliptic curve Y2 = X(X − 1)(X − λ) with coordinates X = 2, 3 both have finite order. This is a very special case of some conjectures on unlikely intersections in semiabelian schemes.

Let E/Q be an elliptic curve. Silverman and Stange define primes p and q to be an elliptic amicable pair if #E(Fp) = q and #E(Fq) = p. More generally, they define the notion of aliquot cycles for elliptic curves. Here we study the same notion in the case that the elliptic curve is defined over a number field K. We focus on proving the existence of an elliptic curve E/K with aliquot cycle (p1, ....

We discuss microscopic origin of integrability in Seiberg-Witten theory, following the results of [1], as well as present their certain extension and consider several explicit examples. In particular, we discuss in more detail the theory with the only switched on higher perturbationin the ultraviolet, where extra explicit formulas are obtained using bosonization and elliptic uniformization of t...

A celebrated theorem of Bogomolov asserts that the l-adic Lie algebra attached to the Galois action on the Tate module of an abelian variety over a number field contains all homotheties. This is not the case in characteristic p: a “counterexample” is provided by an ordinary elliptic curve defined over a finite field. In this note we discuss (and explicitly construct) more interesting examples o...

In this paper we reformulate the question of whether the ranks of the quadratic twists of an elliptic curve over Q are bounded, into the question of the whether certain infinite series converge. Our results were inspired by ideas in a paper of Gouvêa and Mazur [2]. Fix a, b, c ∈ Z such that f(x) = x + ax + bx+ c has 3 distinct complex roots, and let E be the elliptic curve y = f(x). For D ∈ Z−{...

This paper proposes new explicit formulae for the point doubling, tripling and addition on ordinary Weierstrass elliptic curves with a point of order 3 over finite fields of characteristic three. The cost of basic point operations is lower than that of all previously proposed ones. The new doubling, mixed addition and tripling formulae in projective coordinates require 3M + 2C, 8M + 1C + 1D and...