For each $t\in\mathbb{Q}\setminus\{-1,0,1\}$, define an elliptic curve over $\mathbb{Q}$ by \begin{align*}
E_t:y^2=x(x+1)(x+t^2). \end{align*} Using a formula for the root number $W(E_t)$ 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 $E/\mathbb{Q}$ whose Mordell-Weil group contains $\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.
In this paper, we show that an action on the set of elliptic curves with j= 1728 preserves a certain kind of symmetry on the local root number of Hecke characters attached to such elliptic curves. As a consequence, we give results on the distribution of the root numbers and their average of the aforementioned Hecke characters.
To determine the global root number of an elliptic curve defined over a number field, one needs to understand all the local root numbers. These have been classified except at places above 2, and in this paper we attempt to complete the classification. At places above 2, we express the local root numbers in terms of norm residue symbols (resp. root numbers of explicit 1-dimensional characters) in case when wild inertia acts through a cyclic (resp. quaternionic) quotient.
In a previous article, the author proves that the value of the root number varies in a non-isotrivial family of elliptic curves indexed by one parameter $t$ running through $\mathbb{Q}$. However, a well-known example of Washington has root number $-1$ for every fiber when $t$ runs through $\mathbb{Z}$. Such examples are rare since, as proven in this paper, the root number of the integer fibers varies for a large class of families of elliptic curves. Our results depends on the squarefree conjecture and Chowla's conjecture, and are unconditional in many cases.
By considering a one-parameter family of elliptic curves defined over $\mathbb{Q}$, 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, we know (at least conjecturally) that the average root number of an elliptic curve defined over $\mathbb{Q}(T)$ is zero as soon as there is a place of multiplicative reduction over $\mathbb{Q}(T)$ other than $-$deg. Recently, Helfgott's work was extended by Desjardins, where she relaxes some of Helfgott's hypotheses and is able to provide unconditional results on the…
Given an elliptic curve over a field of characteristic different from 2,3, its discriminant defines a $μ_{12}$-torsor over the field. In this paper, we give an explicit description of this $μ_{12}$-torsor in terms of the 3-torsion points and of the 4-torsion points on the given elliptic curve. %In addition, we show that such a description involves the Weil pairing in a certain way. As an application, we generalize a result of Coates on the 12-th root of the discriminant of an elliptic curve.
The purpose of the paper is to complete several global and local results concerning parity of ranks of elliptic curves. Primarily, we show that the Shafarevich-Tate conjecture implies the parity conjecture for all elliptic curves over number fields, give a formula for local and global root numbers of elliptic curves and complete the proof of a conjecture of Kramer and Tunnell in characteristic 0. The method is to settle the outstanding local formulae by deforming from local fields to totally real number fields and then using global parity results.
We give an explicit description of the behaviour of the root number in the family given by twists of an elliptic curve $E$ by the rational values of a polynomial $f(T)$. In particular, we give a criterion (on $f$ depending on $E$) for the family to have a constant root number over $\mathbb{Q}$. This completes a work of Rohrlich: we detail the behaviour of the root number when $E$ has bad reduction over $\mathbb{Q}^{ab}$ and we treat the cases $j(E)=0,1728$ which were not considered previously.
We consider the problem of finding $1$-parameter families of elliptic curves whose root number does not average to zero as the parameter varies in $\mathbb{Z}$. We classify all such families when the degree of the coefficients (in the parameter $t$) is less than or equal to $2$ and we compute the rank over $\mathbb{Q}(t)$ of all these families. Also, we compute explicitly the average of the root numbers for some of these families highlighting some special cases. Finally, we prove some results on the possible values average root numbers can take, showing for example that all rational number in …
Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$ of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the $x$-coordinate of a point of maximal order in the group $E(\mathbb{F}_p)$. We prove unconditionally that $r(E,p)> 0.72\log\log p$ for infinitely many $p$, and $r(E,p) > 0.36 \log p$ under the assumption of the Generalized Riemann Hypothesis. This can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime.
As a consequence of Wan's theorem about Dwork's conjecture, the unit root $L$-functions of the $n$-dimensional Kloosterman family are $p$-adic meromorphic. By studying the symmetric power $L$-functions associated to the Kloosterman family, we prove that for each $0\le j\le n$, the unit root $L$-function coming from slope $j$ has a unique unit root which equals 1.
In this article we perform an extensive study of the spaces of automorphic forms for GL(2) of weight two and level N, for N an ideal in the ring of integers of the quartic CM field generated by the twelfth roots of unity. This study is conducted through the computation of the group cohomology of the congruence subgroup of upper triangular matrices modulo N, and the corresponding Hecke action which this space admits. Combining this Hecke data with the Faltings-Serre method for proving equivalence of Galois representations, we are able to provide the first known examples of modular elliptic curv…
Rizzo showed that the family of elliptic curves $\mathcal{W}(t) :y^2=x^3+tx^2-(t+3)x+1$, a well-known example of Washington, has root number $W(\mathcal{W}(t))=-1$ for all $t\in\mathbb{Z}$. In this paper we generalize this example and identify the families of small degree on which this phenomenon happens. Motivated by results from David, Bettin and Delaunay (arXiv:1612.03095) and Desjardins (arXiv:1810.12787), we study in detail the two families $\mathcal{F}_s(t):y^2=x^3+3tx^2+3sx+st$ and $\mathcal{L}_{w,s,v}(t): wy^2=x^3+3(t^2+v)x^2+3sx+s(t^2+v)$ and describe necessary and sufficient conditio…
The Thue-Siegel method is used to obtain an upper bound for the number of primitive integral solutions to a family of quartic Thue's inequalities. This will provide an upper bound for the number of integer points on a family of elliptic curves with j-invariant equal to 1728.
The parity of the analytic rank of an elliptic curve is given by the root number in the functional equation L(E,s). Fixing an elliptic curve over any number field and considering the family of its quadratic twists, it is natural to ask what the average analytic rank in this family is. A lower bound on this number is given by the average root number. In this paper, we investigate the root number in such families and derive an asymptotic formula for the proportion of curves in the family that have even rank. Our results are then used to support a conjecture about the average analytic rank in thi…
The aim of this paper is to introduce a new family of elliptic curves in the form of $y^2=x(x-a^2)(x-b^2)$ that have positive ranks. We first generate a list of pythagorean triples $(a,b,c)$ and then construct this family of elliptic curves. It turn out that this new family have positive ranks and search for the upper bound for their ranks.
We conjecture that, for a fixed prime $p$, rational elliptic curves with higher rank tend to have more points mod $p$. We show that there is an analogous bias for modular forms with respect to root numbers, and conjecture that the order of the rank bias for elliptic curves is greater than that of the root number bias for modular forms.
Let E be a one-parameter family of elliptic curves over Q. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any family E with at least one point of multiplicative reduction over Q(t) has average root number 0, provided that two classical arithmetical conjectures hold for two polynomials constructed explicitly in terms of E. The behaviour of the root number in any family E without multiplicative reduction over Q(t) is shown to be rather regular and non-random; we give expressions for the average root…
The aim of this paper is to introduce a new family of elliptic curves with positive ranks. These elliptic curves have been constructed with certain rational numbers, namely a, b, and c as sides of Heron triangles having rational areas $k$. It turned out that the torsion groups of this family are of the form $\frac{\Bbb{Z}}{2\Bbb{Z}}\times \frac{\Bbb{Z}}{2\Bbb{Z}}$ and also the rank is positive.
Let n be a positive integer and t a non-zero integer. We consider the elliptic curve over Q given by E : y 2 = x 3 + tx 2 -- n 2 (t + 3n 2)x + n 6. It is a special case of an elliptic surface studied recently by Bettin, David and Delaunay [2] and it generalizes Washington's family. The point (0, n 3) belongs to E(Q) and we obtain some results about its nondivisibility in E(Q). Our work extends to this two-parameter family of elliptic curves a previous study of Duquesne (mainly stated for n = 1 and t > 0).
In this paper, we construct a family of elliptic curves with rank $\geq 5$. To do this, we use the Heron formula for a triple $(A^2, B^2, C^2)$ which are not necessarily the three sides of a triangle. It turns out that as parameters of a family of elliptic curves, these three positive integers $A$, $B$, and $C$, along with the extra parameter $D$ satisfy the quartic Diophantine equation $A^4+D^4=2(B^4+D^4)$.
Given any positive integer n, it is well known that there always exist triangles with rational sides a, b and c such that the area of the triangle is n. Assuming finiteness of the Shafarevich-Tate group, we first construct a family of infinitely many Heronian elliptic curves of rank exactly 1 from Heron triangles of a certain type. We also explicitly produce a separate family of infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and 3.
We investigate $\mathbb{Q}$-ranks of the elliptic curve $E_t$: $y^2+txy=x^3+tx^2-x+1$ where $t$ is a rational parameter. We prove that for infinitely many values of $t$ the rank of $E_t(\mathbb{Q})$ is at least 4.
We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring of integers. In particular, for the partial $3$-torsion fields for a certain one-parameter family of non-CM elliptic curves, we describe a power basis. As a result, we show that the one-parameter family of quartic $S_4$ fields given by $T^4 - 6T^2 - αT - 3$ for $α\in \mathbb{Z}$ such that $α\pm 8$ are squarefree, are monogenic.
The parity conjecture has a long and distinguished history. It gives a way of predicting the existence of points of infinite order on elliptic curves without having to construct them, and is responsible for a wide range of unexplained arithmetic phenomena. It is one of the main consequences of the Birch and Swinnerton-Dyer conjecture and lets one calculate the parity of the rank of an elliptic curve using root numbers. In this handbook, we explain how to use local root numbers of elliptic curves to realise some of these phenomena, with an emphasis on explicit calculations. The text is aimed at…
For any number field K with a complex place, we present an infinite family of elliptic curves defined over K such that $dim \mathbb{F}_2 Sel_2(E^F/K) \ge dim \mathbb{F}_2 E^F(K)[2] + r_2$ for every quadratic twist E^F of every curve E in this family, where r_2 is the number of complex places of K. This provides a counterexample to a conjecture appearing in work of Mazur and Rubin.
Let $C_{m} : y^{2} = x^{3} - m^{2}x +p^{2}q^{2}$ be a family of elliptic curves over $\mathbb{Q}$, where $m$ is a positive integer and $p, q$ are distinct odd primes. We study the torsion part and the rank of $C_m(\mathbb{Q})$. More specifically, we prove that the torsion subgroup of $C_{m}(\mathbb{Q})$ is trivial and the $\mathbb{Q}$-rank of this family is at least $2$, whenever $m \not \equiv 0 \pmod 4$ and $m \equiv 2 \pmod {64}$ with neither $p$ nor $q$ divides $m$.
In this paper, we study a certain Artin--Schreier family of elliptic curves over the function field $\mathbb{F}_q(t)$. We prove an asymptotic estimate on the size of the special value of their $L$-function in terms of the degree of their conductor; loosely speaking, we show that the special values are "asymptotically as large as possible". We also provide an explicit expression for the $L$-function of the elliptic curves in the family. The proof of the main result uses this expression and a detailed study of the distribution of some character sums related to Kloosterman sums. Via the BSD conje…
This work is a sequel of a previous work of one of the authors (Y.Ô), which treated certain congruence relation between an elliptic Gauss sum and a coefficient of power series expansion at the origin of the lemniscate sine function. We extend the previous result (in \cite{O}) which concerned only for non-vanishing elliptic Gauss sums. We give new congruence relations between power series coefficients of the lemniscate cosine function, which hold if and only if the corresponding elliptic Gauss sum vanishes.
Dwork's conjecture, now proven by Wan, states that unit root L-functions "coming from geometry" are p-adic meromorphic. In this paper we study the p-adic variation of a family of unit root L-functions coming from a suitable family of toric exponential sums. In this setting, we find that the unit root L-functions each have a unique p-adic unit root. We then study the variation of this unit root over the family of unit root L-functions. Surprisingly, we find that this unit root behaves similarly to the classical case of families of exponential sums. That is, the unit root is essentially a ratio …
In recent work, Bhargava and Shankar have shown that the average size of the $2$-Selmer group of an elliptic curve over $\mathbb{Q}$ is exactly $3$, and Bhargava and Ho have shown that the average size of the $2$-Selmer group in the family of elliptic curves with a marked point is exactly $6$. In contrast to these results, we show that the average size of the $2$-Selmer group in the family of elliptic curves with a two-torsion point is unbounded. In particular, the existence of a two-torsion point implies the existence of rational isogeny. A fundamental quantity attached to a pair of isogenous…
We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.
For every elliptic curve $E$ which has complex multiplication (CM) and is defined over a number field $F$ containing the CM field $K$, we prove that the family of $p^{\infty}$-division fields of $E$, with $p \in \mathbb{N}$ prime, becomes linearly disjoint over $F$ after removing an explicit finite subfamily of fields. We then give a necessary condition for this finite subfamily to be entangled over $F$, which is always met when $F = K$. In this case, and under the further assumption that the elliptic curve $E$ is obtained as a base-change from $\mathbb{Q}$, we describe in detail the entanglem…
We show the density of rational points on non-isotrivial elliptic surfaces by studying the variation of the root numbers among the fibers of these surfaces, conditionally to two analytic number theory conjectures (the squarefree conjecture and Chowla's conjecture). This is a weaker statement than the one found in a preprint of Helfgott which proves (under the same assumptions) that the average root number is $0$ when the surface admits a place of multiplicative reduction. However, we use a different technique. The conjectures involved impose a restriction on the degree of the irreducible facto…
We make conjectures on the moments of the central values of the family of all elliptic curves and on the moments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of L-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family. Furthermore, as arithmetical applications we make a conjecture on the distribution of a_p's amongst all rank 2 …
In this short note, we shall construct a certain topological family which contains all elliptic curves over Q and, as an application, show that this family provides some geometric interpretations of the Hasse-Weil L-function of an elliptic curve over Q whose Mordell-Weil group is of rank $\leq1$.
Under a hypothesis which is slightly stronger than the Riemann Hypothesis for elliptic curve $L$-functions, we show that both the average analytic rank and the average algebraic rank of elliptic curves in families of quadratic twists are exactly $\frac 12$. As a corollary we obtain that under this last hypothesis, the Birch and Swinnerton-Dyer Conjecture holds for almost all curves in our family, and that asymptotically one half of these curves have algebraic rank $0$, and the remaining half $1$. We also prove an analogous result in the family of all elliptic curves. A way to interpret our res…
We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their $2$-Selmer group to the $2$-rank of the ideal class group of $\mathbb{Q}(\sqrt[3]{p})$ to obtain some examples of elliptic curves with rank one and non-trivial $2$-part of the Tate-Shafarevich group.
Conjecturally, the parity of the Mordell-Weil rank of an elliptic curve over a number field K is determined by its root number. The root number is a product of local root numbers, so the rank modulo 2 is conjecturally the sum over all places of K of a function of elliptic curves over local fields. This note shows that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank itself. In fact, standard conjectures for elliptic curves imply that there is no analogue modulo n for any n>2, so this is purely a parity phenomenon.
In this article, we discuss Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms. After the overview on the situation of the ordinary case of Hida family, we will introduce a Coleman map for Coleman family for the non-ordinary case (Coleman family) which was obtained as a joint work with Filippo Nuccio [NO16] and we give some results on Iwasawa Theory for Coleman families as applications of . First, we give a construction of a two-variable $p$-adic $L$-function for a Coleman family thanks to the ingredients given in [NO16] (Theorem 5.1). Combining this result and Coleman…
We fix an elliptic curve $E/\mathbb{F}_q(t)$ and consider the family $\{E\otimesχ_D\}$ of $E$ twisted by quadratic Dirichlet characters. The one-level density of their $L$-functions is shown to follow orthogonal symmetry for test functions with Fourier transform supported inside $(-1,1)$. As an application, we obtain an upper bound of 3/2 on the average analytic rank. By splitting the family according to the sign of the functional equation, we obtain that at least $12.5\%$ of the family have rank zero, and at least $37.5\%$ have rank one. The Katz and Sarnak philosophy predicts that those perc…
For an elliptic curve $E$ over a number field $K$, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell-Weil rank. Assuming finiteness of $\mathrm{Sha}(E/K)[p^\infty]$ for a prime $p$ this is equivalent to the $p$-parity conjecture: the global root number matches the parity of the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group. We complete the proof of the $p$-parity conjecture for elliptic curves that have a $p$-isogeny for $p > 3$ (the cases $p \le 3$ were known). T. and V. Dokchitser have showed …
In this article, we prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by height, is less than $1$ (in fact, less than $.885$). As a consequence of our methods, we also prove that at least four fifths of all elliptic curves over $\mathbb{Q}$ have rank either 0 or 1; furthermore, at least one fifth of all elliptic curves in fact have rank 0. The primary ingredient in the proofs of these theorems is a determination of the average size of the $5$-Selmer group of elliptic curves over $\mathbb{Q}$; we prove that this average size is $6$. Another key ingredient is a new l…
Let $\mathbb{F}_q$ be a finite field of odd characteristic and $K= \mathbb{F}_q(t)$. For any integer $d\geq 2$ coprime to $q$, consider the elliptic curve $E_d$ over $K$ defined by $y^2=x(x^2+t^{2d} x-4t^{2d})$. We show that the rank of the Mordell--Weil group $E_d(K)$ is unbounded as $d$ varies. The curve $E_d$ satisfies the BSD conjecture, so that its rank equals the order of vanishing of its $L$-function at the central point. We provide an explicit expression for the $L$-function of $E_d$, and use it to study this order of vanishing in terms of $d$.
We prove that the family $\mathcal{I}_{F_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with $F_0$-rational $j$-invariant is typically bounded in torsion. Under an additional uniformity assumption, we also prove that the family $\mathcal{I}_{d_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with degree $d_0$ $j$-invariant is typically bounded in torsion.
We study the variation of unit roots of the Dwork families of Calabi-Yau varieties over a finite field by the method of Dwork-Katz and also from the point of view of formal group laws. A p-adic analytic formula for the unit roots away from the Hasse locus is obtained.
A family $\mathcal{F}$ of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups $E(F)[$tors$]$ of those elliptic curves $E_{/F}\in \mathcal{F}$ can be made uniformly bounded after removing from $\mathcal{F}$ those whose number field degrees lie in a subset of $\mathbb{Z}^+$ with arbitrarily small upper density. For every number field $F$, we prove unconditionally that the family $\mathcal{E}_F$ of elliptic curves defined over number fields and with $F$-rational $j$-invariant is typically bounded in torsion. For any integer $d\in\mathbb{Z…
E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms $F_n(X, Y )= X^3 -(n-1)X^2Y -(n+2)XY^2 -Y^3$ and the family of equations $F_n(X, Y )=\pm 1$, $n\in {\mathbf N}$. This family is associated to the family of the simplest cubic fields ${\mathbf Q}(λ)$ of D. Shanks, $λ$ being a root of $F_n(X,1)$. We introduce in this family a second parameter by replacing the roots of the minimal polynomial $F_n(X, 1) $ of $λ$ by the $a$-th powers of the roots and we effectively solve the family of Thue equations that we obtain and which depends now on the …
This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having…
Given any positive integer $n$, it is well-known that there always exists a triangle with rational sides $a,b$ and $c$ such that the area of the triangle is $n$. For a given prime $p \not \equiv 1$ modulo $8$ such that $p^{2}+1=2q$ for a prime $q$, we look into the possibility of the existence of the triangles with rational sides with $p$ as the area and $\frac{1}{p}$ as $\tan \fracθ{2}$ for one of the angles $θ$. We also discuss the relation of such triangles with the solutions of certain Diophantine equations.
This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion o…
We show that the average and typical ranks in a certain parametric family of elliptic curves described by D. Ulmer tend to infinity as the parameter $d \to\infty$. This is perhaps unexpected since by a result of A. Brumer, the average rank for all elliptic curves over a function field of positive characteristic is asymptotically bounded above by 2.3.
We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.
