Family Of Elliptic Curves Research Articles

Hadamard products and BPS networks

We study examples of fourth-order Picard-Fuchs operators that are Hadamard products of two second-order Picard-Fuchs operators. Each second-order Picard-Fuchs operator is associated with a family of elliptic curves, and the Hadamard product computes period integrals on the fibred product of the two elliptic surfaces. We construct 3-cycles on this geometry as the union of 2-cycles in the fibre over contours on the base. We then use the special Lagrangian condition to constrain the contours on the base. This leads to a construction that is reminiscent of spectral networks and exponential networks that have previously appeared in string theory literature.

Transformations for Appell series over finite fields and traces of Frobenius for elliptic curves

Using properties of certain character sums over finite fields, we here obtain some transformations for finite field Appell series F2⁎ and F4⁎ in terms of F34-Gaussian hypergeometric series. These motivate us to find a finite field analog for certain transformations satisfied by the classical Appell series F2 and F4. We also deduce an expression for the finite field Appell series F4⁎ in terms of F23-Gaussian hypergeometric series. Consequently, we investigate connections of trace of Frobenius endomorphism for certain families of elliptic curves and the finite field Appell series F4⁎.

Elliptic curves and continued fractions

It is proved that the rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. As an illustration, we consider a family of elliptic curves with complex multiplication.

Constant root number on integer fibres of elliptic surfaces

Rizzo showed that the family of elliptic curves W(t):y2=x3+tx2−(t+3)x+1, a well-known example of Washington, has root number W(W(t))=−1 for all t∈Z[11]. In this paper we generalise this example and identify the families of small degree on which this phenomenon happens. Motivated by results from [2] and [5], we study in detail the two families Fs(t):y2=x3+3tx2+3sx+st and Lw,s,v(t):wy2=x3+3(t2+v)x2+3sx+s(t2+v) and describe necessary and sufficient conditions for which subfamilies of Fs(t) have constant root number on integer fibres. We further prove similar but partial results on Lw,s,v(t). Our results give examples of subfamilies for which there is rank elevation at integer fibres.

Generalized Tresca yield condition as a family of elliptic curves with application to mode I crack problems

Generalized Tresca yield condition as a family of elliptic curves with application to mode I crack problems

On quartic Diophantine equations with trivial solutions in the Gaussian integers

We show that the quartic Diophantine equations $ax^4+by^4=cz^2$ has only trivial solution in the Gaussian integers for some particular choices of $a,b$ and $c$. Our strategy is by elliptic curves method. In fact, we exhibit two null-rank corresponding families of elliptic curves over Gaussian field. We also determine the torsion groups of both families.

On the rational solutions of y^2 =x^3 + k^{6n+3}

We consider a family of elliptic curves E(k^{6n+3}) : y^2 = x^3 + k^{6n+3} for some integers k and n ≥ 0 and prove that their rank is zero and the torsion part is isomorphic to Z_2. This is an extension of a recent work of Wu and Qin [14].

Large families of elliptic curves ordered by conductor

In this paper 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 quotient $\Delta (E)/C(E)$ of the discriminant divided by the conductor is squarefree; and second, the set of elliptic curves $E$ such that the Szpiro quotient$\beta _E:=\log |\Delta (E)|/\log (C(E))$ is less than $7/4$. 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 families, when ordered by conductor. Moreover, we prove that the average size of the $2$-Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$. The key new ingredients necessary for the proofs are ‘uniformity estimates’, namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes.

Classical Iwasawa theory and infinite descent on a family of abelian varieties

For primes \(q \equiv 7 \ \mathrm {mod}\ 16\), the present manuscript shows that elementary methods enable one to prove surprisingly strong results about the Iwasawa theory of the Gross family of elliptic curves with complex multiplication by the ring of integers of the field \(K = {\mathbb {Q}}(\sqrt{-q})\), which are in perfect accord with the predictions of the conjecture of Birch and Swinnerton-Dyer. We also prove some interesting phenomena related to a classical conjecture of Greenberg, and give a new proof of an old theorem of Hasse.

Eighth power moment of Kloosterman sum, supercharacters, and elliptic curves

We here use a supercharacter theory of $${\mathbb {F}}_p^2$$ to express the eighth power moment of Kloosterman sum in terms of traces of Frobenius endomorphism of certain families of elliptic curves.

Elliptic Curve Cryptography Point Multiplication Core for Hardware Security Module

In today's technology, a sheer number of Internet of Things applications use hardware security modules for secure communications. The widely used algorithms in security modules, for example, digital signatures and key agreement, are based upon elliptic curve cryptography (ECC). A core operation used in ECC is the point multiplication, which is computationally expensive for many Internet of things applications. In many IoT applications, such as intelligent transportation systems and distributed control systems, thousands of safety messages need to be signed and verified within a very short time-frame. Considerable research has been conducted in the design of a fast elliptic curve arithmetic on finite fields using residue number systems (RNS). In this article, we propose an RNS-based ECC core hardware for the two families of elliptic curves that are short Weierstras and twisted Edwards curves. Specifically, we present RNS implementations for SECP256K1 and ED25519 standard curves. We propose an RNS hardware architecture supporting fast elliptic curve point-addition (ECPA), point-doubling (ECPD), and point-tripling (ECPT). We implemented different ECC point multiplication algorithms on the Xilinx FPGA platform. The test results confirm that the performance of our fully RNS ECC point multiplication is better than the fastest ECC point multiplication cores in the literature.

Non‐vanishing theorems for central L‐values of some elliptic curves with complex multiplication

The paper uses Iwasawa theory at the prime $p=2$ to prove non-vanishing theorems for the value at $s=1$ of the complex $L$-series of certain quadratic twists of the Gross family of elliptic curves with complex multiplication by the field $K = \BQ(\sqrt{-q})$, where $q$ is any prime $\equiv 7 \mod 8$. Our results establish some broad generalizations of the non-vanishing theorem first proven by D. Rohrlich using complex analytic methods. Such non-vanishing theorems are important because it is known that they imply the finiteness of the Mordell-Weil group and the Tate-Shafarevich group of the corresponding elliptic curves over the Hilbert class field of $K$. It is essential for the proofs to study the Iwasawa theory of the higher dimensional abelian variety with complex multiplication which is obtained by taking the restriction of scalars to $K$ of the particular elliptic curve with complex multiplication introduced by Gross.

Maps from K-trivial varieties and connectedness problems

In this paper we study varieties covered by rational or elliptic curves. First, we show that images of Calabi-Yau or irreducible symplectic varieties under rational maps are almost always rationally connected. Second, we investigate elliptically connected and elliptically chain connected varieties, and varieties swept out by a family of elliptic curves. Among other things, we show that Calabi-Yau or hyperk\"ahler manifolds which are covered by a family of elliptic curves contain uniruled divisors and that elliptically chain connected varieties of dimension at least two contain a rational curve, and so do K-trivial varieties with finite fundamental group which are covered by elliptic curves.

Аналитические и теоретико-числовые свойства двумерных сигма-функций

Обзор посвящен классическим и современным задачам, связанным с целой функцией $\sigma({\bf u};\lambda)$, которая определяется семейством неособых алгебраических кривых рода 2, где ${\bf u}= (u_1,u_3)$, $\lambda=(\lambda_4, \lambda_6, \lambda_8, \lambda_{10})$. Эта функция является аналогом сигма-функции Вейерштрасса $\sigma({{u}};g_2,g_3)$ семейства эллиптических кривых. Логарифмические производные порядка 2 и выше функции ${\sigma({\mathbf{u}};\lambda)}$ порождают поле гиперэллиптических функций от ${\mathbf{u}} = (u_1,u_3)$ на якобианах кривых с фиксированным значением вектора параметров $\lambda$. Мы рассматриваем три ряда Гурвица $\sigma({\mathbf{u}};\lambda)=\sum_{m,n\ge0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}$, $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}$ и $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!}$. Обзор посвящен теоретико-числовым свойствам функций $a_{m,n}(\lambda)$, $\xi_k(u_1;\lambda)$ и $\mu_k(u_3;\lambda)$. Он включает самые последние результаты, доказательства которых использует тот фундаментальный факт, что функция $\sigma ({\mathbf{u}};\lambda)$ определяется системой четырех уравнений теплопроводности в неголономном репере шестимерного пространства.

A New Family of Elliptic Curves with Unbounded Rank

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 $d$.

On the rank of elliptic curves arising from Pythagorean quadruplets

By a Pythagorean quadruplet $(a,b,c,d)$, we mean an integer solution to the quadratic equation $a^2 + b^2 = c^2 + d^2$. We use this notion to construct infinite families of elliptic curves of higher rank as far as possible. Furthermore, we give particular examples of rank eight.

Orders of points in families of elliptic curves

Orders of points in families of elliptic curves

Explicit examples of Hurwitz Frobenius manifolds in genus one

We provide three new solutions to the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations by computing examples of prepotentials of Hurwitz Frobenius manifolds, their deformations, and real doubles. By applying the general formalism to two families of elliptic curves, we obtain solutions to the WDVV system depending on four and eight variables. We also correct mistakes in previously known solutions in dimensions four and six associated with the same families of curves.

Computing the optimal Ate pairing over elliptic curves with embedding degrees 54 and 48 at the 256-bit security level

Due to recent advances in the computation of finite fields discrete logarithms, the Barreto-Lynn-Scott family of elliptic curves of embedding degree 48 became suitable for instantiating pairing-based cryptography at the 256-bit security level. Observing the uncertainty around determining the constants that govern the best approach for computing discrete logarithms, Scott and Guillevic consider pairing-friendly elliptic curves of embedding degree higher than 50, and discovered a new family of elliptic curves with embedding degree 54. This work aims at investigating the theoretical and practical cost of both the Miller algorithm and the final exponentiation in the computation of the optimal ate pairing on the two aforementioned curves. Both our theoretical results, based on the operation counts of base-field operations, and our experimental observations collected from a real implementation, confirm that BLS48 curves remain the faster curve in the computation of the optimal ate pairing at the 256-bit security level.

Short average distribution of a prime counting function over families of elliptic curves

Let E be an elliptic curve defined over Q and let N be a positive integer. Then, ME(N) counts the number of primes p such that the group Ep(Fp) is of order N. For a given positive integer ℓ, we study the probability of the event {ME(N)=ℓ}. In an earlier joint work with Balasubramanian, we showed that ME(N) follows the Poisson distribution when an average is taken over a family of elliptic curves with parameters A and B where A,B≥Nℓ+12(log⁡N)1+γ and AB>N3(ℓ+1)2(log⁡N)2+γ for a positive integer ℓ and any γ>0. In this paper, we show that for sufficiently large N, the same result holds even if we take A and B in the range exp⁡(Nϵ220(ℓ+1))≥A,B>Nϵ and AB>N3(ℓ+1)2(log⁡N)3+2γ(log⁡log⁡N)ℓ+8 for any ϵ>0.