Mod 2 representations of elliptic curves

Darmon points on elliptic curves over totally real fields

We show how to construct Darmon points on elliptic curves over totally real fields.

Symmetric diophantine systems and families of elliptic curves of high rank

While there has been considerable interest in the problem of finding elliptic curves of high rank over $\mathbb{Q}$, very few parametrized families of elliptic curves of generic rank $\geq 8$ have been published. In this paper we use solutions of certain symmetric diophantine systems to construct several parametrized families of elliptic curves with their generic ranks ranging from at least 8 to at least 12. Specific numerical values of the parameters yield elliptic curves with quite large coefficients and we could therefore determine the precise rank only in a few cases where the rank of the …

Elliptic nets and elliptic curves

An elliptic divisibility sequence is an integer recurrence sequence associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base fields. Suppose E is an elliptic curve over a field K, and P_1, ..., P_n are points on E defined over K. To this information we associate an n-dimensional array of values in K satisfying a nonlinear recurrence relation. Arrays satisfying this relation are called elliptic nets. We demonstrate an explicit bijection between the set of elliptic nets and the set …

A note on log-algebraicity on elliptic curves

We analyze log-algebraic power series identities for formal groups of elliptic curves over $\mathbb{Q}$ which arise from modular parametrizations. We further investigate applications to special values of elliptic curve $L$-functions.

Iterated integrals and unipotent periods on families of marked elliptic curves

We describe iterated integrals as unipotent periods on families of marked elliptic curves in terms of multiple zeta values and elliptic multiple zeta values.

Average analytic rank of elliptic curves with prescribed torsion

We show that average analytic rank of elliptic curves with prescribed torsion $G$ is bounded for every torsion group $G$ under GRH for elliptic curve $L$-functions.

On some remarkable congruences between two elliptic curves

We exhibit two non-isogenous rational elliptic curves with $17$-torsion subgroups isomorphic as Galois modules.

On the distribution of analytic ranks of elliptic curves

In this paper, under GRH for elliptic $L$-functions, we give an upper bound for the probability for an elliptic curve with analytic rank $\leq a$ for $a \geq 11$, and also give an upper bound of $n$-th moments of analytic ranks of elliptic curves. These are applications of counting elliptic curves with local conditions, for example, having good reduction at $p$.

A note on zeros of $L$-series of elliptic curves

In this note we study an analogy between a positive definite quadratic form for elliptic curves over finite fields and a positive definite quadratic form for elliptic curves over the rational number field. A question is posed of which an affirmative answer would imply the analogue of the Riemann hypothesis for elliptic curves over the rational number field.

Modularity of some elliptic curves over totally real fields

We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic curves over certain real quadratic fields are modular.

Positive-rank elliptic curves arising pythagorean triples

In the present paper, we introduce some new families of elliptic curves with positive rank arrising from Pythagorean triples.

On the Correspondence between Supersingular Elliptic Curves and maximal quaternionic Orders

We present a deterministic and explicit algorithm to compute the endomorphism rings of supersingular elliptic curves. As an example we compute the endomorphism rings of all supersingular elliptic curves defined over characteristic p=29,...,97.

On families of 13-congruent elliptic curves

We compute twists of the modular curve $X(13)$ that parametrise the elliptic curves 13-congruent to a given elliptic curve. Searching for rational points on these twists enables us to find non-trivial pairs of 13-congruent elliptic curves over ${\mathbb Q}$, i.e. pairs of non-isogenous elliptic curves over ${\mathbb Q}$ whose 13-torsion subgroups are isomorphic as Galois modules. We also find equations for the surfaces parametrising pairs of 13-congruent elliptic curves. There are two such surfaces, corresponding to 13-congruences that do, or do not, respect the Weil pairing. We write each as …

L-function and rational points on an elliptic curve via the classical number theory

L-function and rational points on an elliptic curve via the classical number theory.

The first p-jet space of an elliptic curve: global functions and lifts of Frobenius

We prove that there are no non-constant global functions and no lifts of Frobenius on the first $p$-jet space on an elliptic curve unless the elliptic curve itself has a lift of Frobenius.

Elliptic curves over $\mathbb{Q}_\infty$ are modular

We show that if p is a prime, then all elliptic curves defined over the cyclotomic Z_p extension of Q are modular.

Exponential Sums over Points of Elliptic Curves

We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on elliptic curves.

Typically bounding torsion on elliptic curves isogenous to rational $j$-invariant

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.

New Rank Records For Elliptic Curves Having Rational Torsion

We present rank-record breaking elliptic curves having torsion subgroups Z/2Z, Z/3Z, Z/4Z, Z/6Z, and Z/7Z.

Analogy between arithmetic of elliptic curves and conics

In this brief note we bring out the analogy between the arithmetic of elliptic curves and the Riemann zeta-function.

On The Rank Of Congruent Elliptic Curves

In this paper, $p$ and $q$ are two different odd primes. First, We construct the congruent elliptic curves corresponding to $p$, $2p$, $pq$, and $2pq,$ then, in the cases of congruent numbers, we determine the rank of the corresponding congruent elliptic curves.

Extending Elliptic Curve Chabauty to higher genus curves

We give a generalization of the method of "Elliptic Curve Chabauty" to higher genus curves and their Jacobians. This method can sometimes be used in conjunction with covering techniques and a modified version of the Mordell-Weil sieve to provide a complete solution to the problem of determining the set of rational points of an algebraic curve $Y$.

Counting elliptic curves with an isogeny of degree three

We count by height the number of elliptic curves over the rationals that possess an isogeny of degree three.

Elliptic curves of unbounded rank and Chebyshev's bias

We establish a conditional equivalence between quantitative unboundedness of the analytic rank of elliptic curves over $\mathbb Q$ and the existence of highly biased elliptic curve prime number races. We show that conditionally on a Riemann Hypothesis and on a hypothesis on the multiplicity of the zeros of $L(E,s)$, large analytic ranks translate into an extreme Chebyshev bias. Conversely, we show under a certain linear independence hypothesis on zeros of $L(E,s)$ that if highly biased elliptic curve prime number races do exist, then the Riemann Hypothesis holds for infinitely many elliptic cu…

Families of elliptic curves ordered by conductor

In this article, we study the family of elliptic curves $E/\mathbb{Q}$, having good reduction at $2$ and $3$, and whose $j$-invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the ratio $Δ(E)/C(E)$ is squarefree; and second, the set of elliptic curves $E$ such that $Δ(E)/C(E)$ is bounded by a small power $(<3/4)$ of $C(E)$. Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these familie…

Computing the trace of an algebraic point on an elliptic curve

We present a simple and efficient algorithm to compute the sum of the algebraic conjugates of a point on an elliptic curve.

The even parity Goldfeld conjecture: congruent number elliptic curves

In 1979 Goldfeld conjectured: 50\% of the quadratic twists of an elliptic curve defined over the rationals have analytic rank zero. In this expository article we present a few recent developments towards the conjecture, especially its first instance - the congruent number elliptic curves.

A heuristic for boundedness of ranks of elliptic curves

We present a heuristic that suggests that ranks of elliptic curves over the rationals are bounded. In fact, it suggests that there are only finitely many elliptic curves of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields.

Local diophantine properties of modular curves of $\cal{D}$-elliptic sheaves

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.

Isogenies over quadratic fields of elliptic curves with rational $j$-invariant

We determine the possible degrees of cyclic isogenies defined over quadratic fields for non-CM elliptic curves with rational $j$-invariant.

Non-existence of elliptic curves with everywhere good reduction over some real quadratic fields

We prove the non-existence of elliptic curves having good reduction everywhere over some real quadratic fields.

Ranks of elliptic curves in families of quadratic twists

We show that the unboundedness of the ranks of the quadratic twists of an elliptic curve is equivalent to the divergence of certain infinite series.

On positive proportion of rank zero twists of elliptic curves over Q

Extending the idea of \cite{dab2} and using the 2-descent method, we provide three general families of elliptic curves over Q such that a positive proportion of prime-twists of such elliptic curves have rank zero simultaneously.

On the topological aspects of arithmetic elliptic curves

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 $\leq1$.

Elliptic curves with complex multiplication and $Λ$-structures

This thesis examines the relationship between elliptic curves with complex multiplication and Lambda structures. Our main result is to show that the moduli stack of elliptic curves with complex multiplication, and the universal elliptic curve with complex multiplication over it, both admit Lambda structures and that the structure morphism is a Lambda morphism. This implies that elliptic curves with complex multiplication can be canonically lifted to the Witt vectors of the base (these are big and global Witt vectors). We also show that elliptic curves with complex multiplication of Shimura typ…

A positive proportion of elliptic curves over $\mathbb{Q}$ have rank one

We prove that, when all elliptic curves over $\mathbb{Q}$ are ordered by naive height, a positive proportion have both algebraic and analytic rank one. It follows that the average rank and the average analytic rank of elliptic curves are both strictly positive.

Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization

In this paper, we present several methods for construction of elliptic curves with large torsion group and positive rank over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve factorization method (ECM).

Numerical modular symbols for elliptic curves

We present a detailed analysis of how to implement the computation of modular symbols for a given elliptic curve by using numerical approximations. This method turns out to be more efficient than current implementation as the conductor of the curve increases.

On pairs of 17-congruent elliptic curves

We compute explicit equations for the surfaces Z(17,1) and Z(17,3) parametrising pairs of $17$-congruent elliptic curves. We find that each is a double cover of the same elliptic K3-surface. We use these equations to exhibit the first non-trivial example of a pair of symplectically 17-congruent elliptic curves over the rationals. We also compute the corresponding genus 2 curve whose Jacobian has a $(17,17)$-splitting.

Mod p representations on elliptic curves

Modular Galois representations into GL_2(F_p) with cyclotomic determinant arise from elliptic curves for p = 2,3,5. We show (by constructing explicit examples) that such elliptic curves cannot be chosen to have conductor as small as possible at all primes other than p. Our proof involves finding all elliptic curves of conductor 85779, a custom computation carried out for us by Cremona. This leads to a counterexample to a conjecture of Lario and Rio. For p > 5, we construct irreducible representations with cyclotomic determinant that do not arise from any elliptic curve over Q.

Elliptic Curves over Real Quadratic Fields are Modular

We prove that all elliptic curves defined over real quadratic fields are modular.

Selmer Groups of Elliptic Curves with Complex Multiplication

This paper contains some results regarding the Iwasawa module structure of Selmer groups of elliptic curves with complex multiplication.

数域上椭圆曲线解析秩矩的上界

Upper bounds for moments of analytic ranks of elliptic curves over number fields

研究数域上椭圆曲线解析秩矩的上界，通过证明条件上界及解析秩大于给定界的椭圆曲线密度的条件界来进行研究。

Elementary background in elliptic curves

This paper gives additional background in algebraic geometry as an accompaniment to the article, ``Formal Groups, Elliptic Curves, and some Theorems of Couveignes'' [arXiv:math.NT/9708215]. Section 1 discusses the addition law on elliptic curves, and Sections 2 and 3 explain about function fields, uniformizers, and power series expansions with respect to a uniformizer.

On the discriminant of elliptic curves with non-trivial torsion

For those elliptic curves defined over the rational with non--trivial torsion subgroup, we find a tight relationship between the torsion subgroup itself and a Galois group naturally arising from the curve.

More constructing pairing-friendly elliptic curves for cryptography

The problem of constructing elliptic curves suitable for pairing applications has received a lot of attention. To solve this, we propose a variant algorithm of a known method by Brezing and Weng. We produce new families of parameters using our algorithm for pairing-friendly elliptic curves of embedding degree 8, and we actually compute some explicit curves as numerical examples.

Local solubility and height bounds for coverings of elliptic curves

We study genus one curves that arise as 2-, 3- and 4-coverings of elliptic curves. We describe efficient algorithms for testing local solubility and modify the classical formulae for the covering maps so that they work in all characteristics. These ingredients are then combined to give explicit bounds relating the height of a rational point on one of the covering curves to the height of its image on the elliptic curve. We use our results to improve the existing methods for searching for rational points on elliptic curves.

Frobenius lifts and elliptic curves with complex multiplication

We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for elliptic curves of Shimura type, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.

Regulators of elliptic curves

We study the regulator of the Mordell-Weil group of elliptic curves over number fields, functions fields of characteristic zero or function fields of characteristic $p>0$. We prove a new Northcott property for the regulator of elliptic curves of rank at least $4$ defined over a number field. In the function field case, we obtain that non-trivial families of elliptic curves with bounded regulators and bounded ranks are limited families. We conclude by asking new questions.

On the cyclic torsion of elliptic curves over cubic number fields (III)

This is the third part of a series of papers discussing the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=39$, we show that $\mathbb{Z}/N\mathbb{Z}$ is not a subgroup of $E(K)_{tor}$ for any elliptic curve $E$ over a cubic number field $K$.

A Simple Method for High-Rank Families of Elliptic Curves with Specified Torsion

We describe a simple, but effective, method for deriving families of elliptic curves, with high rank, all of whose members have the same torsion subgroup structure.

An oft cited letter from Tate to Serre on computing local heights on elliptic curves

A 1979 letter from John Tate to Jean-Pierre Serre in which the author describes an efficient and elegant algorithm for computing local heights on elliptic curves.

Descent for the punctured universal elliptic curve, and the average number of integral points on elliptic curves

We show that the average number of integral points on elliptic curves, counted modulo the natural involution on a punctured elliptic curve, is bounded from above by $2.1 \times 10^8$. To prove it, we design a descent map, whose prototype goes at least back to Mordell, which associates a pair of binary forms to an integral point on an elliptic curve. Other ingredients of the proof include the upper bounds for the number of solutions of a Thue equation by Evertse and Akhtari-Okazaki, and the estimation of the number of binary quartic forms by Bhargava-Shankar. Our method applies to $S$-integral …

Elliptic Curves over Totally Real Cubic Fields are Modular

We prove that all elliptic curves defined over totally real cubic fields are modular. This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to Thorne and to Kalyanswamy.