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A recent paper of Shekhar compares the ranks of elliptic curves E1 and E2 for which there is an isomorphism E1[p] ∼ E2[p]asGal(.../Q)- modules, where p is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where p is a prime of good supersingular reduction. [ABSTRACT FROM AUTHOR]

Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges areℓ-isogenies. Algorithms allowing to travel on these graphs were developed by Kohel in his thesis (1996) and later on, by Fouquet and Morain (2001). However, up to now, no method was known, to predict, before taking a step on the volcano, the direction of this step. Hence, in Kohel's and Fouquet- Morain's algorithms, many steps are taken before choosing the right direction. In particular, ascending or horizontal isogenies are usually found using a trialand- error approach. In this paper, we propose an alternative method that efficiently finds all points P of order ℓ such that the subgroup generated by P is the kernel of a horizontal or an ascending isogeny. In many cases, our method is faster than previous methods. This is an extended version of a paper published in the proceedings of ANTS 2010. In addition, we treat the case of 2-isogeny volcanoes and we derive from the group structure of the curve and the pairing a new invariant of the endomorphism class of an elliptic curve. Our benchmarks show that the resulting algorithm for endomorphism ring computation is faster than Kohel's method for computing the ℓ-adic valuation of the conductor of the endomorphism ring for small ℓ. [ABSTRACT FROM AUTHOR]

This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] ≃ Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[2]) with kernel containing E(K)[2], then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of E. If E has a cyclic 4-isogeny with kernel containing E(K)[2] defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r2, the number of complex places of K. We also obtain results about the minimum number of twists of E with rank 0 and, subject to standard conjectures, the number of twists with rank 1, provided E does not have a cyclic 4-isogeny defined over K. [ABSTRACT FROM AUTHOR]

In this paper we construct infinite families of elliptic curves with given torsion group structures over cubic number fields. This result provides explicit examples of the theoretical result recently developed by the first two authors and A. Schweizer; they determined all the group structures which occur infinitely often as the torsion of elliptic curves over cubic number fields. In fact, this paper presents an efficient way of constructing such families of elliptic curves with prescribed torsion group structures over cubic number fields. [ABSTRACT FROM AUTHOR]

Isogenies are the morphisms between elliptic curves and are, accordingly, a topic of interest in the subject. As such, they have been well studied, and have been used in several cryptographic applications. V'elu's formulas show how to explicitly evaluate an isogeny, given a specification of the kernel as a list of points. However, V'elu's formulas only work for elliptic curves specified by a Weierstrass equation. This paper presents formulas similar to V'elu's that can be used to evaluate isogenies on Edwards curves and Huff curves, which are normal forms of elliptic curves that provide an alternative to the traditional Weierstrass form. Our formulas are not simply compositions of V'elu's formulas with mappings to and from Weierstrass form. Our alternate derivation yields efficient formulas for isogenies with lower algebraic complexity than such compositions. In fact, these formulas have lower algebraic complexity than V'elu's formulas on Weierstrass curves. [ABSTRACT FROM AUTHOR]

In this paper we construct infinite families of elliptic curves with given torsion group structures over cyclic cubic number fields. [ABSTRACT FROM AUTHOR]

Given natural numbers n and k, with n > k, the Prouhet-Tarry- Escott (PTE) problem asks for distinct subsets of ℤ, say X = {x1,...xn} and Y = {y1,...yn}, such that xi1 + ... + xin = yi1 + ... + yin for i = 1, ... ,k. Many partial solutions to this problem were found in the late 19th century and early 20th century. When n = k - 1, we call a solution X =n-1 Y ideal. This is considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves, and computational techniques. In 2007, Alpers and Tijdeman gave examples of solutions to the PTE problem over the Gaussian integers. This paper extends the framework of the problem to this setting. We prove generalizations of results from the literature, and use this information along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers. [ABSTRACT FROM AUTHOR]

In this paper we give a sufficient condition on the class numbers of distinct quadratic imaginary fields so that on a given CM elliptic curve over Q with fixed modular parameterization, the Heegner points associated to (the maximal orders of) these quadratic imaginary fields are linearly independent. This extends results of Rosen and Silverman from the non-CM to the CM case. [ABSTRACT FROM AUTHOR]

In this paper, by using the theory of elliptic curves, we prove that for every k, there exists infinitely many primitive sets of k n-tuples of positive integers with the same sum and the same product. [ABSTRACT FROM AUTHOR]

Computing a lower bound for the canonical height is a crucial step in determining a Mordell-Weil basis for elliptic curves. This paper presents an algorithm for computing such a lower bound for elliptic curves over number fields without searching for points. The algorithm is illustrated by some examples. [ABSTRACT FROM AUTHOR]

Stroeker and Tzanakis gave convincing numerical and heuristic evidence for the fact that in their $mathcal {E}llog$ method a certain parameter $lambda $ plays a decisive role in the size of the final bound for the integral points on elliptic curves. Furthermore, they provided an algorithm to determine the Mordell-Weil basis of the curve which corresponds to the optimal choice of $lambda $. In this paper we show that working with more Mordell-Weil bases simultaneously, the final bound for the integral points can be further decreased. [ABSTRACT FROM AUTHOR]