One-parameter Families Of Elliptic Curves Research Articles

Applications of moments of Dirichlet coefficients in elliptic curve families

The moments of the coefficients of elliptic curve [Formula: see text]-functions are related to numerous important arithmetic problems. Rosen and Silverman proved a conjecture of Nagao relating the first moment of one-parameter families satisfying Tate’s conjecture to the rank of the corresponding elliptic surface over [Formula: see text]; one can also construct families of moderate rank by finding families with large first moments. Michel proved that if [Formula: see text] is not constant, then the second moment of the family is of size [Formula: see text]; these two moments show that for suitably small support the behavior of zeros near the central point agree with that of eigenvalues from random matrix ensembles, with the higher moments impacting the rate of convergence. In his thesis, Miller noticed a negative bias in the second moment of every one-parameter family of elliptic curves over [Formula: see text] whose second moment had a (by him) calculable closed-form expression, specifically the first lower-order term which does not average to zero is on average negative. This Bias Conjecture has now been confirmed for many families; however, these are highly non-generic families as they are specially chosen so that the resulting Legendre sums can be determined. For cohomological reasons, each subsequent term in the second moment expansion is smaller than the previous by a factor on the order of [Formula: see text], and thus numerically, it is hard to see a term of size [Formula: see text] with a small negative average as it can be masked by a term of size [Formula: see text] which averages to zero. Inspired by the recent successes by Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, Alexey Pozdnyakov and others in investigations of murmurations of elliptic curve coefficients with machine learning techniques, we pose a similar problem for trying to understand the Bias Conjecture. As a start to this program, we numerically investigate the bias conjecture and provide a visual representation of the bias for the second moment. We find a one-parameter family of elliptic curves whose bias is positive for half the primes. However, the numerics do not offer conclusive evidence that negative bias for the other half is enough to overwhelm the positive bias. Without an explicit expansion for the second moment, we are not able to extract potential negative bias of the order [Formula: see text] term.

Elliptic loops

Given a local ring (R,m) and an elliptic curve E(R/m), we define its elliptic loop as the points of P2(R) projecting to E under the canonical modulo-m reduction, endowed with an operation that extends the curve's addition. While its subset of points satisfying the curve's Weierstrass equation is a group, this larger object is proved to be a power-associative abelian algebraic loop, which is seldom completely associative. When an elliptic loop has no points of order 3, its affine part is obtained as a stratification of a one-parameter family of elliptic curves defined over R, which we call layers. Stronger associativity properties are established when me vanishes for small values of e∈Z. When the underlying ring is R=Z/peZ, the infinity part of an elliptic loop is generated by two elements, the group structure of layers may be established and the points with the same projection and same order possess a geometric description.

Biases in Moments of the Dirichlet Coefficients in One-Parameter Families of Elliptic Curves

Elliptic curves arise in many important areas of modern number theory. One way to study them is to take local data, the number of solutions modulo a prime p, and create an L-function. The behavior of this global object is related to two of the seven Clay Millenial Problems: the Birch and Swinnerton-Dyer Conjecture and the Generalized Riemann Hypothesis. We study one-parameter families over Q(T). We look at the r-th moment of the series expansion of the L-function (the p-th coefficient is related to the number of solutions to the elliptic curve modulo p). Rosen and Silverman showed biases in the first moment equal the rank of the Mordell-Weil group of rational solutions.
 Michel proved the main term of the second moment is universal, with the lower order terms smaller by at least a factor of the square-root of the prime. Based on several special families where computations can be done in closed form, Miller in his thesis conjectured that the largest lower-order term in the second moment that does not average to 0 is on average negative. He further showed that such a negative bias has implications in the distribution of zeros of the elliptic curve L-function near the central point. To date, evidence for this conjecture is limited to special families. In this paper, we explore the first and second moments of some one-parameter families of elliptic curves, looking to see if the biases persist and exploring the consequence these have on fundamental properties of elliptic curves. We observe that in all of the one-parameter families where we can compute in closed form that the first term that does not average to zero in the second-moment expansion of the Dirichlet coefficients has a negative average. In addition to studying some additional families where the calculations can be done in closed form, we also systematically investigate families of various ranks. These are the first general tests of the conjecture; while we cannot in general obtain closed form solutions, we discuss computations which support or contradict the conjecture. We then generalize to higher moments, and see evidence that the bias continues in the even moments.

Families of cubic elliptic curves containing sequences of consecutive powers

Let E be an elliptic curve over ℚ defined by y2=ax3+bx2+cx+d. We say a sequence of rational points (xi,yi)∈E(ℚ), i=0,1,…,ℓ, forms a sequence of consecutive n-th powers on E of length ℓ whenever the sequence of x-coordinates, xi, i=0,1,…,ℓ, consists of consecutive powers of degree n in the form xi=(g+i)n, for some rational g. Applying the known Mestre’s theorem, for an arbitrary natural number n≥2, we produce a one-parameter family of elliptic curves over ℚ which contains an 8-term sequence of consecutive n-th powers. Furthermore, we show that for n=2 and 3 the associated families of elliptic curves are of generic rank ≥6 and 7, respectively. We also provide an explicit set of linearly independent points for those families. Finally, according to our limited trial conducted for 4≤n≤50, we discovered that the generic rank of the corresponding families is ≥7. We guess that this holds for all n≥4; however, we are not able to prove it at this time.

Lower-Order Biases in the Second Moment of Dirichlet Coefficients in Families of L-Functions

Let be a nontrivial one-parameter family of elliptic curves over , with . Consider the k th moments of the Dirichlet coefficients . Rosen and Silverman proved Nagao’s conjecture relating the first moment to the family’s rank over , and Michel proved if j(T) is not constant then the second moment equals . Cohomological arguments show the lower order terms are of sizes and 1. In every case, we can analyze in closed form, the largest lower order term in the second moment expansion that does not average to zero is on average negative, though numerics suggest this may fail for families of moderate rank. We prove this Bias Conjecture for several large classes of families, including families with rank, complex multiplication, and constant j(T)-invariant. We also study the analogous Bias Conjecture for families of Dirichlet characters, holomorphic forms on GL, and their symmetric powers and Rankin-Selberg convolutions. We identify all lower order terms in large classes of families, shedding light on the arithmetic objects controlling these terms. The negative bias in these lower order terms has implications toward the excess rank conjecture and the behavior of zeros near the central point.

Computing the average root number of an elliptic surface

By considering a one-parameter family of elliptic curves defined over Q, we might ask ourselves if there is any bias in the distribution (or parity) of the root numbers at each specialization. From the work of Helfgott [8], we know (at least conjecturally) that the average root number of an elliptic curve defined over Q(T) is zero as soon as there is a place of multiplicative reduction over Q(T) other than −deg.In this paper, we are concerned with elliptic curves defined over Q(T) with no place of multiplicative reduction over Q(T), except possibly at −deg. In [1], the authors classify all such one-parameter families of elliptic curves whose coefficients, in the parameter T, have degree less than or equal to 2; they also use the work of Helfgott to compute the average root number of two particular subfamilies. We complement the work in [1] by computing the average root number of one of these “potentially parity-biased” families and show that it is “parity-biased” infinitely-often.

The $$\mathbb {Q}$$ Q -curve Construction for Endomorphism-Accelerated Elliptic Curves

We give a detailed account of the use of $$\mathbb {Q}$$Q-curve reductions to construct elliptic curves over $$\mathbb {F}_{p^2}$$Fp2 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 $$p$$p is fixed for efficient implementation. Unlike GLS, we also offer the possibility of constructing twist-secure curves. We construct several one-parameter families of elliptic curves over $$\mathbb {F}_{p^2}$$Fp2 equipped with efficient endomorphisms for every $$p > 3$$p>3, and exhibit examples of twist-secure curves over $$\mathbb {F}_{p^2}$$Fp2 for the efficient Mersenne prime $$p = 2^{127}-1$$p=2127-1.

Singularities of elliptic curves in $$K3$$ K 3 surfaces and the Beauville–Voisin zero-cycle

Under some hypotheses on the singular type of the one-parameter family of elliptic curves in a primitively polarized $$K3$$ surface $$S$$ determined by its polarization (which is expected to be true for a very general polarized $$K3$$ surface), we give a more geometric proof of the fact that the second Chern class of $$S$$ is equal to $$24 \cdot o_S$$ in the Chow group of $$0$$ -cycles where $$o_S$$ is the Beauville–Voisin canonical $$0$$ -cycle.

On the ranks of the Jacobians of curves defined by Jacobi polynomials

We show that the degree-3 Jacobi Polynomials define two one-parameter families of elliptic curves $\E_{r}/\q(r)$ and $\F_{s}/\q(s)$ with positive rank outside of an explicit finite set of $(r,s) \in \q \times \q$. In addition we initiate a program to study the ranks of the Jacobians of these curves in higher degree.

Elliptic curves and Triangles with three rational medians

Abstract In his paper Triangles with three rational medians , Buchholz proves that each such triangle corresponds to a point on a one-parameter family of elliptic curves whose rank is at least 2. We prove that in fact the exact rank of the family in Buchholz paper is 3. We also exhibit a subfamily whose rank is at least 4 and we prove the existence of infinitely many curves of rank 5 over Q parametrized by an elliptic curve of positive rank. Finally, we show particular examples of curves within those families having rank 9 and 10 over Q .

One-parameter families of elliptic curves over ℚ with maximal Galois representations

Let E be an elliptic curve over ℚ and let ℚ(E[n]) be its nth division field. In 1972, Serre showed that if E is without complex multiplication, then the Galois group of ℚ(E[n])/ℚ is as large as possible, that is, GL2(ℤ/nℤ), for all integers n coprime to a constant integer m(E, ℚ) depending (at most) on E/ℚ. Serre also showed that the best one can hope for is to have |GL2(ℤ/nℤ) : Gal(ℚ(E[n])/ℚ)| ⩽ 2 for all positive integers n. We study the frequency of this optimal situation in a one-parameter family of elliptic curves over ℚ, and show that in essence, for almost all one-parameter families, almost all elliptic curves have this optimal behavior.

Toric forms of elliptic curves and their arithmetic

We scan a large class of one-parameter families of elliptic curves for efficient arithmetic. The construction of the class is inspired by toric geometry, which provides a natural framework for the study of various forms of elliptic curves. The class both encompasses many prominent known forms and includes thousands of new forms. A powerful algorithm is described that automatically computes the most compact group operation formulas for any parameterized family of elliptic curves. The generality of this algorithm is further illustrated by computing uniform addition formulas and formulas for generalized Montgomery arithmetic.

Towards an ‘average’ version of the Birch and Swinnerton-Dyer conjecture

Text The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell–Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L ( s , E ) . Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a family. Mestre (1986) [Mes] showed the first zero occurs by O ( 1 / log log N E ) , where N E is the conductor of E, though we expect the correct scale to study the zeros near the central point is the significantly smaller 1 / log N E . We significantly improve on Mestre's result by averaging over a one-parameter family of elliptic curves E over Q ( T ) . We assume GRH, Tate's conjecture if E is not a rational surface, and either the ABC or the Square-Free Sieve Conjecture if the discriminant has an irreducible polynomial factor of degree at least 4. We find non-trivial upper and lower bounds for the average number of normalized zeros in intervals on the order of 1 / log N E (which is the expected scale). Our results may be interpreted as providing further evidence in support of the Birch and Swinnerton-Dyer conjecture, as well as the Katz–Sarnak density conjecture from random matrix theory (as the number of zeros predicted by random matrix theory lies between our upper and lower bounds). These methods may be applied to additional families of L-functions. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3EVYPNi_LG0.

Lower order terms in the 1-level density for families of holomorphic cuspidal newforms

The Katz-Sarnak density conjecture states that, in the limit as the conductors tend to infinity, the behavior of normalized zeros near the central point of families of L-functions agree with the N -> oo scaling limits of eigenvalues near 1 of subgroups of U(N). Evidence for this has been found for many families by studying the n-level densities; for suitably restricted test functions the main terms agree with random matrix theory. In particular, all one-parameter families of elliptic curves with rank r over Q(T) and the same distribution of signs of functional equations have the same limiting behavior. We break this universality and find family dependent lower order correction terms in many cases; these lower order terms have applications ranging from excess rank to modeling the behavior of zeros near the central point, and depend on the arithmetic of the family. We derive an alternate form of the explicit formula for GL(2) L-functions which simplifies comparisons, replacing sums over powers of Satake parameters by sums of the moments of the Fourier coefficients lambda_f(p). Our formula highlights the differences that we expect to exist from families whose Fourier coefficients obey different laws (for example, we expect Sato-Tate to hold only for non-CM families of elliptic curves). Further, by the work of Rosen and Silverman we expect lower order biases to the Fourier coefficients in families of elliptic curves with rank over Q(T); these biases can be seen in our expansions. We analyze several families of elliptic curves and see different lower order corrections, depending on whether or not the family has complex multiplication, a forced torsion point, or non-zero rank over Q(T).

Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields

With the goal of producing elliptic curves and higher-dimensional abelian varieties of large rank over function fields, we provide a geometric construction of towers of surfaces dominated by products of curves; in the case where the surface is defined over a finite field our construction yields families of smooth, projective curves whose Jacobians satisfy the conjecture of Birch and Swinnerton-Dyer. As an immediate application of our work we employ known results on analytic ranks of abelian varieties defined in towers of function field extensions, producing a one-parameter family of elliptic curves over F q ( t 1 / d ) whose members obtain arbitrarily large rank as d → ∞ .

Constructing one-parameter families of elliptic curves with moderate rank

We give several new constructions for moderate rank elliptic curves over Q ( T ) . In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over Q using polynomials of degree two in T. While our method generates linearly independent points, we are able to show the rank is exactly 6 without having to verify the points are independent. The method generalizes; however, the higher rank surfaces are not rational, and we need to check that the constructed points are linearly independent.

Uniform results for Serre's theorem for elliptic curves

A celebrated theorem of Serre from 1972 asserts that if E is an elliptic curve defined over ℚ and without complex multiplication, then its associated mod l representation is surjective for all sufficiently large primes l. In this paper we address the question of what “sufficiently large” means in Serre's theorem. More precisely, we obtain a uniform version of Serre's theorem for nonconstant elliptic curves defined over function fields, and a uniform version of Serre's theorem for one-parameter families of elliptic curves defined over ℚ.

On Shioda–Inose structures of one-parameter families of K3 surfaces

We develop an algorithm to determine a one-parameter family of elliptic curves associated to a one-parameter family of K3 surfaces with generic Picard number 19 by a Shioda–Inose structure. The family of elliptic curves is determined up to an isomorphism and an isogeny. An application to a generalized congruence number problem is also discussed.

Triangles with Three Rational Medians

We present a characterization of all rational sided triangles with three rational medians. It turns out that they each correspond to a point on a one-parameter family of elliptic curves. It is possible to show that the rank of this family is at least two and in fact some reasonably high rank curves appear among them.