We generalize a construction of families of moderate rank elliptic curves over $\mathbb{Q}$ to number fields $K/\mathbb{Q}$. The construction, originally due to Steven J. Miller, Álvaro Lozano-Robledo and Scott Arms, invokes a theorem of Rosen and Silverman to show that computing the rank of these curves can be done by controlling the average of the traces of Frobenius, the construction for number fields proceeds in essentially the same way. One novelty of this method is that we can construct families of moderate rank without having to explicitly determine points and calculating determinants of height matrices.
We present a method for constructing optimized equations for the modular curve X_1(N) using a local search algorithm on a suitably defined graph of birationally equivalent plane curves. We then apply these equations over a finite field F_q to efficiently generate elliptic curves with nontrivial N-torsion by searching for affine points on X_1(N)(F_q), and we give a fast method for generating curves with (or without) a point of order 4N using X_1(2N).
Elliptic curves over finite fields with predefined conditions in the order are practically constructed using the theory of complex multiplication. The stage with longest calculations in this method reconstructs some polynomial with integer coefficients. We will prove theoretical results and give a detailed account of the method itself and how one can use a divisor of the mentioned polynomial with coefficients in some extension of the field of rational numbers.
A positive integer that is the area of some rational right triangle is called a congruent number. In an algebraic point of view, being a congruent number means satisfying a system of equations. As early as the 1800s, it is understood that if $n$ is a congruent number, then the equation $nm^2 = uv(u^2 - v^2)$ has a solution in $\mathbb{Z}$. Using the relation between congruent numbers and elliptic curves $E_n: y^2 = x^3 - n^2 x$ which was established in the 1900s, we will prove that the converse of this two century-old result holds as well. In addition to this, we present another proof of the c…
Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.
I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number field equipped with a rational point, (resp. with two rational points) of infinite order over the given number field, and elliptic curves over the rationals with two rational points over `simplest cubic fields.' I also provide hyperelliptic curves of genus exceeding any given number over any given number fields with points (over the given number field) which sp…
Let O be a maximal order in the quaternion algebra B_p over Q ramified at p and infinity. The paper is about the computational problem: Construct a supersingular elliptic curve E over F_p such that End(E) = O. We present an algorithm that solves this problem by taking gcds of the reductions modulo p of Hilbert class polynomials. New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice O^T of O, the order O is effectively characterized by the three successive minima and two other short vectors…
For a given elliptic curve $E_0$ defined over a number field $k$, we construct two families of elliptic curves whose mod 3 representations are isomorphic to that of $E_0$. The isomorphisms in the first family are symplectic, and those in the second family are anti-symplectic. Our construction is based on the notion of Hessian and Cayleyan curves in classical geometry.
The second part of the Birch and Swinnerton-Dyer (BSD) conjecture gives a conjectural formula for the order of the Shafarevich-Tate group of an elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing non-trivial elements of the Shafarevich-Tate group of an elliptic curve by means of the Mordell-Weil group of an ambient curve. In this paper, we generalize Cremona and Mazur's work and give precise conditions under which such a construction can be made for the Shafarevich-Tate gro…
We study the existence of rational points on modular curves of $\cal{D}$-elliptic sheaves over local fields and the structure of special fibres of these curves. We discuss some applications which include finding presentations for arithmetic groups arising from quaternion algebras, finding the equations of modular curves of $\cal{D}$-elliptic sheaves, and constructing curves violating the Hasse principle.
We compute a lower bound of the canonical height on quadratic twists of certain elliptic curves. Also we show a simple method for constructing families of quadratic twists with an explicit rational point. % from cubic polynomials. Using the above lower bound, we show that the explicit rational point is primitive as an element of the Mordell--Weil group.
In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/k\mathbb{Z}$ for $k = 2, 4, 6, 8$, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.
Given two elliptic curves over a finite field having the same cardinality and endomorphism ring, it is known that the curves admit an isogeny between them, but finding such an isogeny is believed to be computationally difficult. The fastest known classical algorithm takes exponential time, and prior to our work no faster quantum algorithm was known. Recently, public-key cryptosystems based on the presumed hardness of this problem have been proposed as candidates for post-quantum cryptography. In this paper, we give a subexponential-time quantum algorithm for constructing isogenies, assuming th…
We introduce a procedure for constructing a generalized elliptic curve from a genus-one twisted curve, and we use this procedure to define an explicit, modular isomorphism between two compactifications of moduli stacks of elliptic curves with level structure.
The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$ in $\mathbb{P}^r$. By means of a study of the finite dimensional vector spaces of global sections, we reduce the problem of constructing and finding efficiently computable polynomial maps defining the addition morphism or isogenies to linear algebra. We demonstrate the effectiveness of the method by improving the best known complexity for doubling and tripli…
We first normalize the derivative Weierstrass $\wp'$-function appearing in Weierstrass equations which give rise to analytic parametrizations of elliptic curves by the Dedekind $η$-function. And, by making use of this normalization of $\wp'$ we associate certain elliptic curve to a given imaginary quadratic field $K$ and then generate an infinite family of ray class fields over $K$ by adjoining to $K$ torsion points of such elliptic curve. We further construct some ray class invariants of imaginary quadratic fields by utilizing singular values of the normalization of $\wp'$, as the $y$-coordin…
A low storage algorithm for constructing isogenies between ordinary elliptic curves was proposed by Galbraith, Hess and Smart (GHS). We give an improvement of this algorithm by modifying the pseudorandom walk so that lower-degree isogenies are used more frequently. This is motivated by the fact that high degree isogenies are slower to compute than low degree ones. We analyse the running time of the parallel collision search algorithm when the partitioning is uneven. We also give experimental results. We conclude that our algorithm is around 14 times faster than the GHS algorithm when construct…
The "defect" of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre bound for the curve. We present a construction for producing genus-4 double covers of genus-2 curves over finite fields such that the defect of the double cover is not much more than the defect of the genus-2 curve. We give an algorithm that uses this construction to produce genus-4 curves with small defect. Heuristically, for all sufficiently large primes and for almost all prime powers q, the algorithm is expected to produce a genus-4 curve over F_q with defect …
We study the failure of a local-global principle for the existence of $l$-isogenies for elliptic curves over number fields $K$. Sutherland has shown that over $\mathbb{Q}$ there is just one failure, which occurs for $l=7$ and a unique $j$-invariant, and has given a classification of such failures when $K$ does not contain the quadratic subfield of the $l$'th cyclotomic field. In this paper we provide a classification of failures for number fields which do contain this quadratic field, and we find a new `exceptional' source of such failures arising from the exceptional subgroups of $\mbox{PGL}_…
We give a new algorithm for constructing Picard curves over a finite field with a given endomorphism ring. This has important applications in cryptography since curves of genus 3 allow for smaller key sizes than elliptic curves. For a sextic CM-field $K$ containing the cube roots of unity, we define and compute certain class polynomials modulo small primes and then use the Chinese Remainder Theorem to construct the class polynomials over the rationals. We also give some examples.
In this note we study numbers which occur as conductors of elliptic curves over Q. We show, by constructing families of elliptic curves with quadratic discriminant and invoking a theorem of Iwaniec, that this set contains infinitely many almost primes. We show, assuming a strong version of the Cohen-Lenstra heuristics, that the set of prime conductors has an explicitly bounded density in the set of primes. Studying the Cremona and Stein-Watkins databases of elliptic curves we conjecture that the set of conductors should be of density zero in the set of natural numbers and that the set of prime…
This paper aims at studying the Iwasawa $λ$-invariant of the $p$-primary Selmer group. We study the growth behaviour of $p$-primary Selmer groups in $p$-power degree extensions over non-cyclotomic $\mathbb{Z}_p$-extensions of a number field. We prove a generalization of Kida's formula in such a case. Unlike the cyclotomic $\mathbb{Z}_p$-extension, where all primes are finitely decomposed; in the $\mathbb{Z}_p$-extensions we consider, primes may be infinitely decomposed. In the second part of the paper, we study the relationship for Iwasawa invariants with respect to congruences, obtaining refi…
We give a detailed account of the use of $\mathbb{Q}$-curve reductions to construct elliptic curves over $\mathbb{F}\_{p^2}$ with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant--Lambert--Vanstone (GLV) and Galbraith--Lin--Scott (GLS) endomorphisms. Like GLS (which is a degenerate case of our construction), we offer the advantage over GLV of selecting from a much wider range of curves, and thus finding secure group orders when \(p\) is fixed for efficient implementation. Unlike GLS, we also offer the possibilit…
We construct new families of elliptic curves over \(\FF_{p^2}\) with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant-Lambert-Vanstone (GLV) and Galbraith-Lin-Scott (GLS) endomorphisms. Our construction is based on reducing \(\QQ\)-curves-curves over quadratic number fields without complex multiplication, but with isogenies to their Galois conjugates-modulo inert primes. As a first application of the general theory we construct, for every \(p > 3\), two one-parameter families of elliptic curves over \(\FF_{p^2}\…
A permutation rational function $f\in \mathbb{F}_q(x)$ is a rational function that induces a bijection on $\mathbb{F}_q$, that is, for all $y\in\mathbb{F}_q$ there exists exactly one $x\in\mathbb{F}_q$ such that $f(x)=y$. Permutation rational functions are intimately related to exceptional rational functions, and more generally exceptional covers of the projective line, of which they form the first important example. In this paper, we show how to efficiently generate many permutation rational functions over large finite fields using isogenies of elliptic curves, and discuss some cryptographic …
We study the arithmetic of abelian varieties over $K=k(t)$ where $k$ is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield completely explicit points on elliptic curves with unbounded rank over $\Fpbar(t)$ and a new construction of elliptic curves with moderately high rank over $\C(t)$.
We describe a framework for constructing an efficient non-interactive key exchange (NIKE) protocol for n parties for any n >= 2. Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believed to be difficult. We do not obtain a working protocol because of a missing step that is currently an open mathematical problem. What we need to complete our protocol is an efficient algorithm that takes as input an abelian variety presented as a product of isogenous elliptic curves, and outputs an isomorphism invariant of the abelian variety. Our framework …
