Root numbers of a family of elliptic curves and two applications

On the number of isomorphism classes of CM elliptic curves defined over a number field

Let E / Q E/\mathbb {Q} be an elliptic curve and let Q ( 3 ∞ ) \mathbb {Q}(3^\infty ) be the compositum of all cubic extensions of Q \mathbb {Q} . In this article we show that the torsion subgroup of E ( Q ( 3 ∞ ) ) E(\mathbb {Q}(3^\infty )) is finite and we determine 20 possibilities for its structure, along with a complete description of the Q ¯ \overline {\mathbb {Q}} -isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many Q ¯ \overline {\mathbb {Q}} -isomorphism classes of elliptic curves, and a complete list of j j -invariants for each of the 4 that do not.

An Algorithm for the Rank of Elliptic Curves

By the theory of elliptic curves, we show that there are infinitely many integral right triangle-perpendicular quadrilateral, integral isosceles triangle-perpendicular quadrilateral, and Heron triangle-perpendicular quadrilateral pairs with a common area and a common perimeter. Moreover, for the elliptic curve associated to integral isosceles triangle and integral perpendicular quadrilateral pairs, we present several subfamilies of rank $\\geq 4$, and show the existence of infinitely many elliptic curves of rank $\\geq 5$, parameterized by the points of an elliptic curve of positive rank.

This paper presents explicit formulas for the number of isomorphism classes of elliptic curves with 2-torsion points over finite fields. These results also can be used in the elliptic curve cryptosystems and classification problems.Keywordselliptic curvecryptographyisomorphism classesfinite field

Let E ∕ ℚbe an elliptic curve defined over the rational field ℚ. We examine the rank of the Mordell–Weil group E(K) as Kranges over cubic extensions ofℚ.

Ranks of elliptic curves in cyclic sextic extensions of [formula omitted]

Let K be a number field, an algebraic closure of K and E/K an elliptic curve defined over K. In this paper, we prove that if E/K has a K-rational point P such that 2P ≠ O and 3P ≠ O, then for each σ ∈ Gal(/K), the Mordell–Weil group of E over the fixed subfield of under σ has infinite rank.

Edwards recently introduced a new normal form for elliptic curves. Every elliptic curve over a non-binary field is birationally equivalent to a curve in Edwards form over an extension of the field, and in many cases over the original field.This paper presents fast explicit formulas (and register allocations) for group operations on an Edwards curve. The algorithm for doubling uses only 3M + 4S, i.e., 3 field multiplications and 4 field squarings. If curve parameters are chosen to be small then the algorithm for mixed addition uses only 9M + 1S and the algorithm for non-mixed addition uses only 10M + 1S. Arbitrary Edwards curves can be handled at the cost of just one extra multiplication by a curve parameter.For comparison, the fastest algorithms known for the popular “a 4 = −3 Jacobian” form use 3M + 5S for doubling; use 7M + 4S for mixed addition; use 11M + 5S for non-mixed addition; and use 10M + 4S for non-mixed addition when one input has been added before.The explicit formulas for non-mixed addition on an Edwards curve can be used for doublings at no extra cost, simplifying protection against side-channel attacks. Even better, many elliptic curves (approximately 1/4 of all isomorphism classes of elliptic curves over a non-binary finite field) are birationally equivalent — over the original field — to Edwards curves where this addition algorithm works for all pairs of curve points, including inverses, the neutral element, etc.This paper contains an extensive comparison of different forms of elliptic curves and different coordinate systems for the basic group operations (doubling, mixed addition, non-mixed addition, and unified addition) as well as higher-level operations such as multi-scalar multiplication.

Let k be a global field, $\bar{k}$ a separable closure of k, and $G_k$ the absolute Galois group $\Gal(\bar{k}/k)$ of $\bar{k}$ over k. For every g in $G_k$, let $\bar{k}^g$ be the fixed subfield of $\bar{k}$ under g. Let E/k be an elliptic curve over k. We show that for each g in $G_k$, the Mordell-Weil group $E(\bar{k}^g)$ has infinite rank in the following two cases. Firstly when k is a global function field of odd characteristic and E is parametrized by a Drinfeld modular curve, and secondly when k is a totally real number field and E/k is parametrized by a Shimura curve. In both cases our approach uses the non-triviality of a sequence of Heegner points on E defined over ring class fields.

In this note, we construct an infinite family of elliptic curves $E$ defined over $\mathbf{Q}$ whose Mordell-Weil group $E(\mathbf{Q})$ has rank exactly two under the parity conjecture.

In this chapter, we count the isomorphism classes of elliptic curves over finite fields K. For the case K = F 2 m, we list a representative, in Weierstrass form, of each isomorphism class. We determine #E(F 2 m) for each supersingular curve E defined over F 2 m.

We propose a public-key encryption scheme and key agreement protocols based on a group action on a set. We construct an implementation of these schemes for the action of the class group $\mathcal{CL}(\mathcal{O}_K)$ of an imaginary quadratic field $K$ on the set $\mathcal{ELL}$p,n$(\mathcal{O}_K)$ of isomorphism classes of elliptic curves over $\mathbb{F}_p$ with $n$ points and the endomorphism ring $\mathcal{O}_K$.This introduces a novel way of using elliptic curves for constructing asymmetric cryptography.

For a prime p and a given square box, B, we consider all elliptic curves Er,s : Y 2 = X 3 + rX + s defined over a field Fp of p elements with coefficients (r, s) ∈ B. We obtain a nontrivial upper bound for the number of such curves which are isomorphic to ag iven one overFp, in terms of the size of B. We also give an optimal lower bound on the number of distinct isomorphic classes represented.

We discuss recent applications of analytic number theory to the study of ranks of elliptic curves. Introduction This article is meant to be a sampling of techniques and interesting results on the (analytic) ranks of elliptic curves. The main result discussed is an upper bound on the average rank of the family of all elliptic curves. The bound obtained is less than 2, which implies (by work of Kolyvagin) that a positive proportion of elliptic curves have finite Tate-Shafarevich group and algebraic rank equal to analytic rank. The synergy here between algebraic and analytic methods is extremely pleasant. We also discuss the problem of showing that a large number of elliptic curve L -functions do not vanish at the central point. Many of the techniques used in bounding the average rank are useful in this direction. Our exposition is meant to be somewhat colloquial. The interested reader should consult [Y1] and [Y2] for all technical details. In this volume E. Kowalski has given a broad overview of what is known on ranks of elliptic curves in families. We shall refer to his article for general background knowledge on elliptic curves. We have attempted to minimize overlap with his article without loss of coherence of this paper. We shall assume the Generalized Riemann Hypothesis throughout this article. Acknowledgements I would like to thank Henryk Iwaniec for supporting my last-minute decision to attend the workshops. I also thank the organizers of the Newton Institute program for inviting me to attend.