SOME REMARKS ON THE TORSION OF ELLIPTIC CURVES

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.

We give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field K are modular and have L-functions which satisfy the Generalized Riemann Hypothesis, we show that the average analytic rank of isomorphism classes of elliptic curves over K is bounded above by $(9\deg (K)+1)/2$ , when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest.

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

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 $\varphi$ be an endomorphism of $\mathbb{P}^1_{\overline{\Q}}$ defined over a number field $K$. Given a discrete valuation $v$ of $K$, we consider here two notions of good reduction of $\varphi$ at $v$, called Standard Good Reduction (S.G.R., for short) and Critically Good Reduction (C.G.R.). If we consider the reduced map $\varphi_v$, in general its degree is smaller or equal to the degree of $\varphi$. We say that the map $\varphi$ has S.G.R. at $v$ if the degree of the reduced map $\varphi_v$ is equal to the degree of $\varphi$. This notion is frequently used in the study of arithmetical dynamical systems, allowing to reduce a global problem to a local problem. Another notion of good reduction has been recently introduced by Szpiro and Tucker to prove a finitess result about equivalence classes of endomorphisms of the projective line. We say that $\varphi$ has C.G.R. at $v$ if every pair of ramification points of $\varphi$ do not coincide modulo $v$ and the same holds for every pair of branch points. As an application of their result, Szpiro and Tucker showed that their theorem implies the well-known Shafarevich-Faltings theorem about the finiteness of the isomorphism classes of elliptic curves defined over a number field $K$ having good reduction outside a prescribed finite set of discrete valuations of $K$. Szpiro and Tucker already in their paper showed with same examples that these two notions are not equivalent. We prove here that if $\varphi$ has C.G.R. at $v$ and the reduced map $\varphi_v$ is separable, then $\varphi$ has S.G.R. at $v$.

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.

Elliptic curves, modular forms, and sums of Hurwitz class numbers

For each t∈Q∖{−1,0,1}, define an elliptic curve over Q by Et:y2=x(x+1)(x+t2).Using a formula for the root number W(Et) as a function of t and assuming some standard conjectures about ranks of elliptic curves, we determine (up to a set of density zero) the set of isomorphism classes of elliptic curves E/Q whose Mordell–Weil group contains Z×Z/2Z×Z/4Z, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.

We discuss the$\ell $-adic case of Mazur’s ‘Program B’ over$\mathbb {Q}$: the problem of classifying the possible images of$\ell $-adic Galois representations attached to elliptic curvesEover$\mathbb {Q}$, equivalently, classifying the rational points on the corresponding modular curves. The primes$\ell =2$and$\ell \ge 13$are addressed by prior work, so we focus on the remaining primes$\ell = 3, 5, 7, 11$. For each of these$\ell $, we compute the directed graph of arithmetically maximal$\ell $-power level modular curves$X_H$, compute explicit equations for all but three of them and classify the rational points on all of them except$X_{\mathrm {ns}}^{+}(N)$, for$N = 27, 25, 49, 121$and two-level$49$curves of genus$9$whose Jacobians have analytic rank$9$.Aside from the$\ell $-adic images that are known to arise for infinitely many${\overline {\mathbb {Q}}}$-isomorphism classes of elliptic curves$E/\mathbb {Q}$, we find only 22 exceptional images that arise for any prime$\ell $and any$E/\mathbb {Q}$without complex multiplication; these exceptional images are realised by 20 non-CM rationalj-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on$X_{\mathrm {ns}}^+(\ell )$with$\ell \ge 19$, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the$\ell $-adic images of Galois for any elliptic curve over$\mathbb {Q}$.In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on$\Gamma _1(N)$are of$\operatorname {GL}_2$-type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of$X_H$.

* Author writes in a clear and engaging style * Contains never before published elementary proofs * Author provides new results and detailed exposition * Self-contained, and suitable for use in a classroom setting or for self-study * A highly creative contribution to the theory of modular forms and dirichlet series The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books reiterate a single old result with a single old method. After a review of elementary Fourier analysis, the author presents completely new results with new methods, though old results will also be proved. No advanced knowledge of number theory is required up to this point. As applications, new formulas for the second factor of the class number of a cyclotomic field will be given. The second half of the book assumes familiarity with basic knowledge of modular forms. However, all definitions and facts are clearly stated, and precise references are given. The notion of nearly holomorphic modular forms is introduced and applied to the determination of the critical values of Hecke L-functions of an imaginary quadratic field. Other notable features of the book are: (1) some new results on classical Eisenstein series; (2) the discussion of isomorphism classes of elliptic curves with complex multiplication in connection with their zeta function and periods; (3) a new class of holomorphic differential operators that send modular forms to those of a different weight. The book will be of interest to graduate students and researchers who are interested in special values of L-functions, class number formulae, arithmetic properties of modular forms (especially their values), and the arithmetic properties of Dirichlet series. It treats in detail, from an elementary viewpoint, the simplest cases of a fundamental area of ongoing research, the only prerequisite being a basic course in algebraic number theory.

Let C be a curve of genus 2 that admits a nonhyperelliptic involution. We show that there are at most 2 isomorphism classes of elliptic curves that are quotients of degree 2 of the Jacobian of C. Our proof is constructive, and we present explicit formulae, classified according to the involutions of C, that give the minimal polynomial of the j-invariant of these curves in terms of the moduli of C. The coefficients of these minimal polynomials are given as rational functions of the moduli.

We give a method for expressing the modular j-invariant function J in a rational function of generators of the modular function field with respect to the modular group Γ0(N ). In the case the genus of the modular function field is positive, using this expression, we can determine isomorphism classes of elliptic curves corresponding to solutions of the defining equation, deduced from these generators, of the modular curve X0(N ). For every N from 6 to 50 and further for N = 52, we give computational results for the expression of J, the generators and the defining equation.