We prove local results on the $p$-adic density of elliptic curves over $\mathbb{Q}_p$ with different reduction types, together with global results on densities of elliptic curves over $\mathbb{Q}$ with specified reduction types at one or more (including infinitely many) primes. These global results include: the density of integral Weierstrass equations which are minimal models of semistable elliptic curves over $\mathbb{Q}$ (that is, elliptic curves with square-free conductor) is $1/ζ(2)\approx60.79\%$, the same as the density of square-free integers; the density of semistable elliptic curves over $\mathbb{Q}$ is $ζ(10)/ζ(2)\approx60.85\%$; the density of integral Weierstrass equations which have square-free discriminant is $\prod_p\left(1-\frac{2}{p^2}+\frac{1}{p^3}\right) \approx 42.89\%$, which is the same (except for a different factor at the prime $2$) as the density of monic integral cubic polynomials with square-free discriminant (and agrees with a previous result of Baier and Browning for short Weierstrass equations); and the density of elliptic curves over $\mathbb{Q}$ with square-free minimal discriminant is $ζ(10)\prod_p\left(1-\frac{2}{p^2}+\frac{1}{p^3}\right)\approx42.93\%$.
The local results derive from a detailed analysis of Tate's Algorithm, while the global ones are obtained through the use of the Ekedahl Sieve, as developed by Poonen, Stoll, and Bhargava.
We give new bounds on the zeros of polynomials in $z$ and the $j$--function, and $z$ and Weierstrass elliptic functions with rectangular associated lattice, controlling the zeros of these functions by their tame behaviour on the boundaries of their respective fundamental domains.
For primes $p>3$ we produce a new derivation of the universal $p$-adic sigma function and $p$-adic Weierstrass zeta functions of Mazur and Tate for ordinary elliptic curves by a method that highlights congruences among coefficients in Laurent expansions of elliptic functions, and works simultaneously for generalized elliptic curves defined by Weierstrass equations.
We prove that there are only finitely many complex numbers $a$ and $b$ with $4a^3+27b^2\not=0$ such that the three points $(1,*),(2,*),$ and $(3,*)$ are simultaneously torsion on the elliptic curve defined in Weierstrass form by $y^2=x^3+ax+b$. This gives an affirmative answer to a question raised by Masser and Zannier. We thus confirm a special case in two dimensions of the relative Manin-Mumford Conjecture formulated by Pink and Masser-Zannier.
Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves $E/\mathbb{Q}$. We show that mock modular forms which arise from Weierstrass $ζ$-functions encode the central $L$-values and $L$-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently studied by Hövel, we obtain canonical weight 1/2 harmonic Maass forms whose Fourier coefficients encode the vanishing of these values…
We give explicit definitions of the Weierstrass elliptic functions $\wp$ and $ζ$ in terms of pfaffian functions, with complexity independent of the lattice involved. We also give such a definition for a modification of the Weierstrass function $σ$. As immediate applications, we give an explicit uniform zero estimate for $\wp$ and answer a question of Corvaja, Masser and Zannier on additive extensions of elliptic curves.
We study the arithmetic properties of Weierstrass points on the modular curves $X_0^+(p)$ for primes $p$. In particular, we obtain a relationship between the Weierstrass points on $X_0^+(p)$ and the $j$-invariants of supersingular elliptic curves in characteristic $p$.
We prove new sharp asymptotic for counting the semistable elliptic curves with two marked Weierstrass points at $\infty$ and $0$ and also the cases where $0$ is a 2-torsion or a 3-torsion marked Weierstrass point over $\mathbb{F}_q(t)$ by the bounded height of discriminant $Δ(X)$. We consider the motivic probabilities over any basefield $K$ with $\text{char}(K) \neq 2,3$ of picking a nonsingular semistable elliptic surface over $\mathbb{P}^{1}$ with two marked Weierstrass sections at $\infty$ and $0$ such that marked Weierstrass section at $0$ is 2-torsion or 3-torsion. In the end, we formulat…
In the theory of elliptic functions and elliptic curves, the Weierstrass $zeta$ function (which is essentially an antiderivative of the Weierstrass $\wp$ function) plays a prominent role. Although it is not an elliptic function, Eisenstein constructed a simple (non-holomorphic) completion of this form which is doubly periodic. This theorem has begun to play an important role in the theory of harmonic Maass forms, and was crucial to work of Guerzhoy as well as Alfes, Griffin, Ono, and the author. In particular, this simple completion of $ζ$ provides a powerful method to construct harmonic Maass…
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…
We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These new results were inspired by new addition formulae found in the case of an equianharmonic curve, which we can now observe as a specialisation of the results here. The new formulae, and the techniques used to find them, also follow the recent work for the generalisation of Weierstrass' functions to curves of higher genus.
Asymptotic expansions for a class of generalized holomorphic Eisenstein series, Ramanujan's formula for $ζ(2k+1)$, Weierstrass' elliptic and allied functions
Starting from the classical division polynomials we construct homogeneous polynomials $α_n$, $β_n$, $γ_n$ such that for $P = (x:y:z)$ on an elliptic curve in Weierstrass form over an arbitrary ring we have $nP = \bigl(α_n(P):β_n(P):γ_n(P)\bigr)$. To show that $α_n,β_n,γ_n$ indeed have this property we use the a priori existence of such polynomials, which we deduce from the Theorem of the Cube. We then use this result to show that the equations defining the modular curve $Y_1(n)_{\mathbb C}$ computed for example by Baaziz, in fact are equations of $Y_1(n)$ over $\mathbb Z[1/n]$.
In this note we give an algorithm to explicitly construct the modular parametrization of an elliptic curve over the rationals given the Weierstrass function $\wp (z)$.
This paper, motivated by problems in Diophantine analysis which can be formulated as problems of finding rational points on the intersection of two quadrics, presents an explicit construction of a rationally defined isomorphism (biregular mapping) between a rationally defined smooth intersection of two quadrics in projective three-space and an elliptic curve in Weierstrass form which maps a distinguished rational point to the point at infinity. The usual approach of transforming a smooth plane cubic to a curve in Weierstrass form by mapping an inflection point to the point at infinity in a par…
Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves
Temple University（天普大学）；University of Connecticut（康涅狄格大学）；Korea Institute for Advanced Study（韩国高等研究院）；University of California, Berkeley（加州大学伯克利分校）；University of Westminster（威斯敏斯特大学）；Princeton University（普林斯顿大学）
Consider the Weierstrass family of elliptic curves $E_λ:y^2=x^3+λ$ parametrized by nonzero $λ\in\overline{\mathbb{Q}_2}$, and let $P_λ(x)=(x,\sqrt{x^3+λ})\in E_λ$. In this article, given $α,β\in\overline{\mathbb{Q}_2}$ such that $\fracαβ\in\mathbb{Q}$, we provide an explicit description for the set of parameters $λ$ such that $P_λ(α)$ and $P_λ(β)$ are simultaneously torsion for $E_λ$. In particular we prove that the aforementioned set is empty unless $\fracαβ\in\{-2,-\frac{1}{2}\}$. Furthermore, we show that this set is empty even when $\fracαβ\notin\mathbb{Q}$ provided that $α$ and $β$ have d…
Let $\wp $ be a Weierstrass $\wp $-function with algebraic $g_2$ and $g_3$, which has a non-zero real period $ω$. We show that at least one of $πr$ and $\wp(ωr)$ is transcendental for any non-zero real number $r$.
The Langlands Programme predicts that a weight 2 newform f over a number field K with integer Hecke eigenvalues generally should have an associated elliptic curve E_f over K. In our previous paper, we associated, building on works of Darmon and Greenberg, a p-adic lattice to f, under certain hypothesis, and implicitly conjectured that this lattice is commensurable with the p-adic Tate lattice of E_f . In this paper, we present this conjecture in detail and discuss how it can be used to compute, directly from f, an explicit Weierstrass equation for the conjectural E_f . We develop algorithms to…
In this paper, we find a power series expansion of the invariant differential $ω_E$ of an elliptic curve $E$ defined over $\mathbb{Q}$, where $E$ is described by certain families of Weierstrass equations. In addition, we introduce several congruence relations satisfied by the trace of the Frobenius endomorphism of $E$.
Using explicit constructions of the Weierstrass mock modular form, we offer a closed formula for generating the values of shifted convolution $L$-values for certain elliptic curves that can be computed to arbitrary precision. These identities provide a surprising relation between weight $2$ newforms and shifted convolution $L$-values.
In this note we combine the advantages of the methods of Siegel-Baker-Coates and of Lang-Zagier for the computation of S-integral points on elliptic curves in Weierstrass normal form over the rationals. In this way we are able to overcome the absence of an explicit lower bound for linear forms in q-adic elliptic logarithms. We present an efficient algorithm for determining all S-integral points on such curves.
Based on an equation for the rank of an elliptic surface over $\mathbb{Q}$ which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank $0$ when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain $L$-functions, or from understanding of the local behaviour of the surfaces. Finally, we make a conjecture about ranks of elliptic surfaces over finite fields, and highlight some experimental …
We study elliptic surfaces over $\mathbb{Q}(T)$ with coefficients of a Weierstrass model being polynomials in $\mathbb{Q}[T]$ with degree at most 2. We derive an explicit expression for their rank over $\mathbb{Q}(T)$ depending on the factorization and other simple properties of certain polynomials. Finally, we give sharp estimates for the ranks of the considered families and we present several applications, among which there are lists of rational points, generic families with maximal rank and generalizations of former results.
Consider the elliptic curve $E$ given by the Weierstrass equation $y^2 = x^3 - 11x - 14$, which has complex multiplication by the order of conductor $2$ inside $\mathbb{Z}[i]$. It was recently observed in a paper of Daniels and Lozano-Robledo that, for each $n \geq 2$, $\mathbb{Q}(μ_{2^{n+1}}) \subseteq \mathbb{Q}(E[2^n])$. In this note, we prove that this (a priori surprising) ``tower of vertical entanglements'' is actually more a feature than a bug: it holds for any elliptic curve $E$ over $\mathbb{Q}$ with complex multiplication by any order of even discriminant.
In this paper we give an algorithm of how to determine a Weierstrass equation with minimal discriminant for superelliptic curves generalizing work of Tate for elliptic curves and Liu for genus 2 curves.
Using the division polynomials for elliptic curves in Weierstrass form, it shown that the group of rational points on the curve $H: ky(yy - 1) = lx(xx - 1)$ is torsion-free.
Let $E$ be an elliptic curve defined over the rationals and in minimal Weierstrass form, and let $P=(x_1/z_1^2,y_1/z_1^3)$ be a rational point of infinite order on $E$, where $x_1,y_1,z_1$ are coprime integers. We show that the integer sequence $(z_n)$ defined by $nP=(x_n/z_n^2,y_n/z_n^3)$ for all $n\ge 1$ does not eventually coincide with $(u_{n^2})$ for any choice of linear recurrence sequence $(u_n)$ with integer values.
There is a lifting from a non-CM elliptic curve $E/\mathbb{Q}$ to a paramodular form $f$ of degree $2$ and weight $3$ given by the symmetric cube map. We find the level of $f$ in an explicit way in terms of the coefficients of the Weierstrass equation of $E$. In order to compute the paramodular level, we use the available description of the local representations of $\mathrm{GL}(2,\mathbb{Q}_p)$ attached to $E$ for $p \ge 5$ and determine the local representation of $\mathrm{GL}(2,\mathbb{Q}_3)$ attached to $E$.
We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that for a fixed power greater than 1, there are only finitely many rational points. Where descent via an isogeny is possible we show, with no restrictions on the power, that there are only finitely many rational points, these points are bounded in number in an explicit fashion, and that they are effectively computable.
We explicitly construct the Dirichlet series $$L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},$$ where $P_{\mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form with Tamagawa product $m.$ Although there are no $E/\mathbb{Q}$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $P_{\mathrm{Tam}}(1)=0.5053\dots.$ As a corollary, we find that $L_{\mathrm{Tam}}(-1)=1.8193\dots$ is the average Tamagawa product for elliptic curves over $\mathbb{Q}.$ We give an application of these results to c…
Let $E/\mathbb Q(t)$ be an elliptic curve and let $t_0 \in \mathbb Q$ be a rational number for which the specialization $E_{t_0}$ is an elliptic curve. Given a subgroup $M$ of $E(\mathbb Q(t))$ with mild conditions and $t_0 \in \mathbb Q$ coming from a relatively large subset $S_M \subset \mathbb Q$, we provide an algorithm that can show that the specialization map $σ_{t_0} : E(\mathbb Q(t)) \to E_{t_0}(\mathbb Q)$ is injective when restricted to $M$. The set $S_M$ is effectively computable in certain cases, and we carry out this computation for some explicit examples where $E$ is given by a W…
In this paper, we examine certain maximal rank sublattices of the Mordell-Weil lattices of two families of elliptic curves over fields of characteristic $p > 0$. We compute explicit lower bounds on the densest sphere packings of these sublattices by finding lower bounds on the minimal norms of the sublattices and explicitly computing the volumes of their fundamental domains.
The study of alternative models for elliptic curves has found recent interest from cryptographic applications, once it was recognized that such models provide more efficiently computable algorithms for the group law than the standard Weierstrass model. Examples of such models arise via symmetries induced by a rational torsion structure. We analyze the module structure of the space of sections of the addition morphisms, determine explicit dimension formulas for the spaces of sections and their eigenspaces under the action of torsion groups, and apply this to specific models of elliptic curves w…
Let q be a prime power and E a non-isotrivial elliptic curve over Fq(T) given by a Weierstrass model. We survey the construction, with an explicit point of view, of the modular parametrization of E by the associated Drinfeld modular curve. We then prove a formula that allows us to evaluate this modular parametrization at cusps and we produce an explicit method to compute these values. Finally we illustrate our results with several examples in characteristic 2 and 3.
We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree $\ell$ ($\ell$ different from the characteristic) in time quasi-linear with respect to $\ell$. This is based in particular on fast algorithms for power series expansion of the Weierstrass $\wp$-function and related functions.
Let $p$ be a prime and let $K$ be a finite extension of $\mathbb{Q}_p$. Let $E/K$ be an elliptic curve with additive reduction. In this paper, we study the topological group structure of the set of points of good reduction of $E(K)$. In particular, if $K/\mathbb{Q}_p$ is unramified, we show how one can read off the topological group structure from the Weierstrass coefficients defining $E$.
We give several new constructions for moderate rank elliptic curves over $\mathbb{Q}(T)$. In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over $\mathbb{Q}$ using polynomials of degree two in $T$. While our method generates linearly independent points, we are able to show the rank is exactly 6 \emph{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.
Elliptic curve multiplications can be improved by replacing the standard ladder algorithm's base 2 representation of the scalar multiplicand, with mixed-base representations with power-of-2 bases, processing the n bits of the current digit in one optimized step. For this purpose, we also present a new methodology to compute, for Weierstrass form elliptic curves in the affine plane, operations of the type mP+nQ where m and n are small integers. This provides implementations with the lower cost than previous algorithms, using only one inversion. In particular, the proposed techniques enable more…
We determine the $\ell$-adic étale cohomology and the eigenvalues of the geometric Frobenius for the moduli stack $\mathcal{L}_{1,12n} := \mathrm{Hom}_{n}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$ of stable elliptic fibrations over $\mathbb{P}^{1}$ with $12n$ nodal singular fibers and a marked Weierstrass section over $\overline{\mathbb{F}}_q$ with $\mathrm{char}(\overline{\mathbb{F}}_q) \neq 2,3$.
Following Katz-Sarnak, Iwaniec-Luo-Sarnak, and Rubinstein, we use the 1- and 2-level densities to study the distribution of low lying zeros for one-parameter rational families of elliptic curves over Q(t). Modulo standard conjectures, for small support the densities agree with Katz and Sarnak's predictions. Further, the densities confirm that the curves' L-functions behave in a manner consistent with having r zeros at the critical point, as predicted by the Birch and Swinnerton-Dyer conjecture. By studying the 2-level densities of some constant sign families, we find the first examples of fami…
