Noam D. Elkies, Mark Watkins
For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r=6) to over 100 (for r=10 and r=11). We describe our search methods, and tabulate, for each r=5,6,...,11, the five curves of lowest conductor, and (except for r=11) also the five of lowest absolute discriminant, that we found.
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Conductor distributions of elliptic curves
本文确定了有理椭圆曲线按高度排序时导子的显式分布，并给出了按导子枚举时导子远小于高度的曲线数量的最强上界。
We find all elliptic curves defined over $\mathbb{Q}$ that have a rational point of order $N$, $N\ge 4$, and whose conductor is of the form $p^aq^b$, where p, q are two distinct primes, a, b are two positive integers. In particular, we prove that Szpiro's conjecture holds for these elliptic curves.
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…
Given a non-CM elliptic curve E over Q, define the ``torsion conductor'' m_E to be the smallest positive integer so that the Galois representation on the torsion of E has image Pi^{-1}(Gal(Q(E[m_E])/Q), where Pi denotes the natural projection GL_2(\hat{Z}) onto GL_2(Z/m_E Z). We show that, uniformly for semi-stable non-CM elliptic curves E over Q, m_E is less than a constant times the 5th power of the conductor of E.
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…
Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor
本文证明了具有有理$2$-扭和素数幂导子的椭圆曲线的任意二次扭满足Watkins猜想，并对形如$y^2=x^3-dx$的椭圆曲线给出了同余数的下界。
Explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields
本文通过Weil公式，研究了数域上椭圆曲线和阿贝尔簇导数的显式下界，并证明了特定数域中不存在处处良好约减的阿贝尔簇。
On the Minimality of the Conductor for Elliptic Curve $L$-Functions
本文证明在解析秩界中，任何满足二次函数方程和解析延拓的算术不变量必须不小于导子，从而说明导子是此类框架下的最小不变量。
Elliptic curves with a rational 2-torsion point ordered by conductor and the boundedness of average rank
本文改进了按导子计数椭圆曲线的方法，将其应用于具有有理2-挠点的小族椭圆曲线，为无平方因子指标和有界Szpiro比率的曲线证明了计数定理，且假设条件比朴素推广更宽松。
The main result of this paper is to extend from $\Q$ to each of the nine imaginary quadratic fields of class number one a result of Serre (1987) and Mestre-Oesterlé (1989), namely that if $E$ is an elliptic curve of prime conductor then either $E$ or a $2$-, $3$- or $5$-isogenous curve has prime discriminant. For four of the nine fields, the theorem holds with no change, while for the remaining five fields the discriminant of a curve with prime conductor is either prime or the square of a prime. The proof is conditional in two ways: first that the curves are modular, so are associated to suita…
Given a parametrisation of an elliptic curve over Q by a Shimura curve, we show that the images of almost all Heegner points are of infinite order. For parametrisations of elliptic curves by modular curves this was proven earlier by Nekovar and Schappacher by a different method.
We propose a new model for counting $\mathbf{GL}_2(\mathbb{F}_p)$-number fields having the same local properties as the splitting field of the mod $p$-Galois representation associated with an elliptic curve over the rational numbers. We explain how this new model and Bhargava's local-to-global heuristics for counting $\mathbf{GL}_2(\mathbb{F}_p)$-number fields both shed light on the problem of estimating the number of elliptic curves over the rational numbers of squarefree conductor $N < X.$ The new model predicts the existence of significantly more $\mathbf{GL}_2(\mathbb{F}_p)$-number fields …
For a prime number $p$ we study the zeros modulo $p$ of divisor polynomials of rational elliptic curves $E$ of conductor $p$. Ono made the observation that these zeros of are often $j$-invariants of supersingular elliptic curves over $\overline{\mathbb{F}_p}$. We show that these supersingular zeros are in bijection with zeros modulo $p$ of an associated quaternionic modular form $v_E$. This allows us to prove that if the root number of $E$ is $-1$ then all supersingular $j$-invariants of elliptic curves defined over $\mathbb{F}_{p}$ are zeros of the corresponding divisor polynomial. If the roo…
Given an elliptic curve $E$ without complex multiplication defined over a number field $K$, consider the image of the Galois representation defined by letting Galois act on the torsion of $E$. Serre's open image theorem implies that there is a positive integer $m$ for which the Galois image is completely determined by its reduction modulo $m$. In this note, we prove a bound on the smallest such $m$ in terms of standard invariants associated with $E$. The bound is sharp and improves upon previous results.
We consider a complete discrete valuation field of characteristic p, with possibly non perfect residue field. Let V be a rank one continuous representation with finite local monodromy of its absolute Galois group. We will prove that the Arithmetic Swan conductor of V (defined after Kato in [Kat89] which fits in the more general theory of [AS02] and [AS06]) coincides with the Differential Swan conductor of the associated differential module $D^†(V)$ defined by Kedlaya in [Ked]. This construction is a generalization to the non perfect residue case of the Fontaine's formalism as presented in [Tsu…
In recent years, Rogers and Zudilin developed a method to write $L$-values attached to elliptic curves as periods. In order to apply this method to a broader collection of $L$-values, we study Eisenstein series and determine their Fourier series at cusps. Subsequently, we write the $L$-values of an elliptic curve of conductor 32 as an integral of Eisenstein series and evaluate the value at $k>1$ explicitly as a period. As a side result, we give simple integral expressions for the generating functions of $L(E,k)$ when even (or odd) $k$ runs over positive integers.
This paper introduces the concept of necklace binomial coefficients motivated by the enumeration of a special type of sequences. Several properties of these coefficients are described, including a connection between their roots and an elliptic curve. Further links are given to a physical model from quantum mechanical supersymmetry as well as properties of alkane molecules in chemistry.
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.
It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve $L$-functions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the orthogonal group. For test functions with restricted support, this is known to be the true for the one- and two-level densities of zeros within the families studied to date. However, for finite conductor Miller's experimental data reveal an interesting discrepancy from these limiting results. Here we use the L-functions ratios conjectures to calculate the 1-…
We compute the second moment in the family of quadratic Dirichlet $L$-functions with prime conductors over $\mathbb{F}_q[x]$ when the degree of the discriminant goes to infinity, obtaining one of the lower order terms. We also obtain an asymptotic formula with the leading order term for the mean value of the derivatives of $L$-functions associated to quadratic twists of a fixed elliptic curve over $\mathbb{F}_q(t)$ by monic irreducible polynomials, which allows us to show that there exists a monic irreducible polynomial such that the analytic rank of the corresponding twisted elliptic curve is…
Elliptic curves of conductor $2^m p$, quadratic twists, and Watkins' conjecture
本文针对在2处加性约化、至多两个奇素数外良好约化且含二阶有理点的椭圆曲线，证明了 Watkins 猜想的新情形，揭示模次数与 Mordell--Weil 秩之间的整除关系。
We prove Boyd's conjectures relating Mahler's measures and values of L-functions of elliptic curves in the cases when the corresponding elliptic curve has conductor 14.
Suppose $p$ is a prime of the form $u^2+64$ for some integer $u$, which we take to be 3 mod 4. Then there are two Neumann--Setzer elliptic curves $E_0$ and $E_1$ of prime conductor $p$, and both have Mordell--Weil group $\Z/2\Z$. There is a surjective map $X_0(p)\xrightarrowπ E_0$ that does not factor through any other elliptic curve (i.e., $π$ is optimal), where $X_0(p)$ is the modular curve of level $p$. Our main result is that the degree of $π$ is odd if and only if $u \con 3\pmod{8}$. We also prove the prime-conductor case of a conjecture of Glenn Stevens, namely that that if $E$ is an ell…
Gross vectors modulo 2 and elliptic curves of prime conductor
该研究证明了模2的格罗斯向量张成F_2^{S_p}，由此证明Kazalicki与Kohen的猜想，还得出素导体、根数为+1的正秩椭圆曲线的模次数被4整除、Watkins猜想对秩2此类曲线成立的结论。
Conductors and local newforms for the metaplectic group of rank 1
本文定义并计算了非阿基米德局部域上秩1亚辛群不可约真表示的导体，给出了局部新形式空间维数的显式公式，并证明了与局部theta对应的相容性。
In a recent preprint, F. Calegari has shown that for $\ell = 2, 3, 5$ and 7 there exist 2-dimensional surjective representations $ρ$ of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ coming from the $\ell$-torsion points of an elliptic curve defined over $\Q$, but not minimally, i.e., so that any elliptic curve giving rise to $ρ$ has prime-to-$\ell$ conductor greater than the (prime-to-$\ell$) conductor of $ρ$. In this brief note, we will show that the same is true for any prime $\ell >7$, concretely, we will show that for any such $\ell$ the elliptic curve $$E^\ell: \qquad Y^2 = X (X- 3^\ell ) …
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.
We prove that the (logarithmic) Mahler measure $m(P)$ of $P(x,y)=x+1/x+y+1/y+3$ is equal to the $L$-value $2L'(E,0)$ attached to the elliptic curve $E:P(x,y)=0$ of conductor 21. In order to do this we investigate the measure of a more general Laurent polynomial $P_{a,b,c}(x,y)=a(x+1/x)+b(y+1/y)+c$ and show that the wanted quantity $m(P)$ is related to a "half-Mahler" measure of $\tilde P(x,y)=P_{\sqrt{7},1,3}(x,y)$. In the finale we use the modular parametrization of the elliptic curve $\tilde P(x,y)=0$, again of conductor 21, due to Ramanujan and the Mellit--Brunault formula for the regulator…
We extend the algorithm of Darmon-Green and Darmon-Pollack for computing p-adic Darmon points on elliptic curves to the case of composite conductor. We also extend the algorithm of Darmon-Logan for computing ATR Darmon points to treat curves of nontrivial conductor. Both cases involve an algorithmic decomposition into elementary matrices in congruence subgroups Γ(N) for ideals N in certain rings of S-integers. We use these extensions to provide additional evidence in support of the conjectures on the rationality of Darmon points.
Let $E$ be a rational elliptic curve and let $p$ be an odd prime of additive reduction. Let $K$ be an imaginary quadratic field and fix a positive integer $c$ prime to the conductor of $E$. The main goal of the present article is to define an anticyclotomic $p$-adic $L$-function $Ł$ attached to $E/K$ when $E/\QQ_p$ attains semistable reduction over an abelian extension. We prove that $Ł$ satisfies the expected interpolation properties; namely, we show that if $χ$ is an anticyclotomic character of conductor $cp^n$ then $χ(Ł)$ is equal (up to explicit constants) to $L(E,χ,1)$ or $L'(E,χ,1)$.
On the torsion growth in quadratic number fields for elliptic curves defined over the rationals
本文研究了在二次数域中椭圆曲线 torsion 子群的增长问题，通过分析导数与素数之间的关系，揭示了有理数域上椭圆曲线的 torsion 增长特性。
Ramification of Tate modules for rank $2$ Drinfeld modules
本文研究秩2 Drinfeld模分点生成的函数域扩张的分歧，定义其导子作为椭圆曲线导子的类似物，通过导子计算在特定受限情形下证明了Szpiro猜想的类似结论。
Let $E$ be an optimal elliptic curve defined over $\mathbb{Q}$. The critical subgroup of $E$ is defined by Mazur and Swinnerton-Dyer as the subgroup of $E(\mathbb{Q})$ generated by traces of branch points under a modular parametrization of $E$. We prove that for all rank two elliptic curves with conductor smaller than 1000, the critical subgroup is torsion. First, we define a family of critical polynomials attached to $E$ and describe two algorithms to compute such polynomials. We then give a sufficient condition for the critical subgroup to be torsion in terms of the factorization of critical…
We prove the conjectural relations between Mahler measures and $L$-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for $L$-values of CM elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family $(1+X)(1+Y)(X+Y)-αXY$, $α\in\mathbb R$.
Let E be an elliptic curve defined over the rationals and let N be its conductor. Assume N is prime. In this paper we give numerical evidence that suggests some conjectures on the 2-divisibility of certain sums of Heenger points on E of discriminant D dividing 4N. One of these conjectures suggests a possible link between the parity of the eigenvalue a_A(2) and the parity of the Shafarevich-Tate group of certain elliptic curves A of square conductor.
On periods of Elliptic curves
研究在素数p处具有分裂乘性约化的椭圆曲线E的L-不变量，通过Darmon引入的自守p-adic周期描述其驯顺部分，并利用改进的例外零点猜想的扭曲版本证明改进L-不变量的等式，在导子恰好为p且自守周期与导子1的最优嵌入相关时，借助de-Shalit的工作无条件证明该等式。
We characterize the possible reductions of $j$-invariants of elliptic curves which admit complex multiplication by an order $\mathcal{O}$ where the curve itself is defined over $\mathbb{Z}_p$. In particular, we show that the distribution of these $j$-invariants depends on which primes divide the discriminant and conductor of the order.
We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in the Stein-Watkins database has rank at least as large as predicted by the conjecture of Birch and Swinnerton-Dyer.
We continue work of Gekeler and others on elliptic curves over ${\mathbb F}_q(T)$ with conductor $\infty\cdot{\mathfrak n}$ where ${\mathfrak n}\in{\mathbb F}_q[T]$ has degree 3. Because of the Frobenius isogeny there are infinitely many curves in each isogeny class, and we discuss in particular which of these curves is the strong Weil curve with respect to the uniformization by the Drinfeld modular curve $X_0({\mathfrak n})$. As a corollary we obtain that the strong Weil curve $E/{\mathbb F}_q(T)$ always gives a rational elliptic surface over $\bar{{\mathbb F}_q}$.
We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we have proved the formula for 16714 of the 16725 such curves of conductor less than 5000.
The modular degree m_E of an elliptic curve E/Q is the minimal degree of any surjective morphism X_0(N) -> E, where N is the conductor of E. We give a necessarily set of criteria for m_E to be odd. Specializing to N prime our results imply a conjecture of Mark Watkins. As a technical tool we also prove a certain multiplicity one result for p=2 that may be of independent interest.
We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve $E/\Fp_q(C)$ over a function field over a finite field that have rank $\geq 2$, and for their average rank. The main tools are constructions and results of Katz and uniform versions of the Chebotarev density theorem for varieties over finite fields. Moreover, we conditionally derive a bound in some cases where the degree of the conductor is unbounded.
In this paper, we will study modular Abelian varieties with odd congruence numbers by examining the cuspidal subgroup of $J_0(N)$. We will show that the conductor of such Abelian varieties must be of a special type. For example, if $N$ is the conductor of an absolutely simple modular Abelian variety with an odd congruence number, then $N$ has at most two prime divisors, and if $N$ is odd, then $N=p^α$ or $N=pq$ for some prime $p$ and $q$. In the second half of this paper, we will focus on modular elliptic curves with odd modular degree. Our results, combined with the work of Agashe, Ribet, and…
We describe a tabulation of (conjecturally) modular elliptic curves over the field Q(sqrt(5)) up to the first curve of rank 2. Using an efficient implementation of an algorithm of Lassina Dembele, we computed tables of Hilbert modular forms of weight (2,2) over Q(sqrt(5)), and via a variety of methods we constructed corresponding elliptic curves, including (again, conjecturally) all elliptic curves over Q(sqrt(5)) that have conductor with norm less than or equal to 1831.
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 …
In the present paper, we introduce some new families of elliptic curves with positive rank arrising from Pythagorean triples.
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.
Let $K$ be a number field and $E/K$ be an elliptic curve with no $2$-torsion points. In the present article we give lower and upper bounds for the $2$-Selmer rank of $E$ in terms of the $2$-torsion of a narrow class group of a certain cubic extension of $K$ attached to $E$. As an application, we prove (under mild hypotheses) that a positive proportion of prime conductor quadratic twists of $E$ have the same $2$-Selmer group.
We present rank-record breaking elliptic curves having torsion subgroups Z/2Z, Z/3Z, Z/4Z, Z/6Z, and Z/7Z.
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.
We prove that the first two coefficients in the series expansion around $s=1$ of the $p$-adic $L$-function of an elliptic curve over $\mathbb{Q}$ are related by a formula involving the conductor of the curve. This is analogous to a recent result of Wuthrich for the classical $L$-function, which makes use of the functional equation. We present a few other consequences for the $p$-adic $L$-function and a generalisation to the base-change to an abelian number field.
In this note, we consider an l-isogeny descent on a pair of elliptic curves over Q. We assume that l > 3 is a prime. The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finite- dimensional F_l-vector spaces defined in terms of the splitting fields of the kernels of the two isogenies. We give examples of proving the l-part of the Birch and Swinnerton-Dyer conjectural formula for certain curves of small conductor.
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.
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 $p$.
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.
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。
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 the family of elliptic curves over \Q given by the equation E_{p}: Y^2=(X-p)^3+X^3+(X+p)^3 where p is a prime number, is studied. It is shown that the maximal rank of the elliptic curves is at most 3 and some conditions under which we have rank(E_p(\Q))=0 or rank(E_p(\Q))=1 or rank(E_p(\Q))>=2 are given. Moreover It is shown that in the family there are elliptic curves with rank 0,1,2 and 3.
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.
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.
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