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Denote by h = h (− p) the class number of the imaginary quadratic field ℚ (− p) with p prime. It is well known that h = 1 for p ∈ { 3 , 7 , 1 1 , 1 9 , 4 3 , 6 7 , 1 6 3 }. Recently, all the solutions of the Diophantine equation x 2 + p s = 4 y n with h = 1 were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen 97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation x 2 + p s = 2 r y n in unknown integers (x , y , s , r , n) , where s ≥ 0 , r ≥ 3 , n ≥ 3 , h ∈ { 1 , 2 , 3 } and gcd (x , y) = 1. To do this, we use the known results from the modularity of Galois representations associated with Frey–Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al. [ABSTRACT FROM AUTHOR]

For a quadratic field K without rationally defined complex multiplication, we prove that there exists of a prime p K depending only on K such that if d is a positive integer whose minimal prime divisor is greater than p K , then for any extension L / K of degree d and any elliptic curve E / K , we have E (L) tors = E (K) tors. By not assuming the GRH, this is a generalization of the results by Genao, and Gonález-Jiménez and Najman. [ABSTRACT FROM AUTHOR]

We study the problem of determining, given an integer k, the rational solutions to Ck : x3z+x2y2+y3z = kz4. For k≠0, the curve Ck has genus 3 and its Jacobian is isogenous to the product of three elliptic curves E1,k, E2,k, E3,k. We explicitly determine the rational points on Ck under the assumption that one of these elliptic curves has rank zero. We discuss the challenges involved in extending our result to handle all k ∈ ℚ. [ABSTRACT FROM AUTHOR]

Let F be the function field of a curve over an algebraically closed field with char (F) ≠ 2 , 3 , and let E / F be a non-isotrivial elliptic curve. Then for all finite extensions K / F and all non-torsion points P ∈ E (K) , the F -normalized canonical height of P is bounded below by h ˆ E (P) ≥ 1 10500 ⋅ h F (j E) 2 ⋅ [ K : F ] 2 . [ABSTRACT FROM AUTHOR]

Given a lattice L in the plane, we define the affiliated deep hole lattice H(L) to be spanned by a shortest vector of L and a deep hole of L contained in the triangle with sides corresponding to the shortest basis vectors. We study the geometric and arithmetic properties of deep hole lattices. In particular we investigate conditions on L under which H(L) is well-rounded and prove that H(L) is defined over the same field as L. For the period lattice corresponding to an isomorphism class of elliptic curves, we produce a finite sequence of deep hole lattices ending with a well-rounded lattice which corresponds to a point on the boundary arc of the fundamental strip under the action of SL 2 (Z) on the upper halfplane. In the case of CM elliptic curves, we prove that all elliptic curves generated by this sequence are isogenous to each other and produce bounds on the degree of isogeny. Finally, we produce a counting estimate for the planar lattices with a prescribed deep hole lattice. [ABSTRACT FROM AUTHOR]

Elliptic curve cryptography (ECC) is widely acknowledged as a method for implementing public key cryptography on devices with limited resources thanks to its use of small keys. A crucial and complex operation in ECC calculations is scalar point multiplication. To improve its execution time and computational complexity in low-power devices, such as embedded systems, several algorithms have been suggested for scalar point multiplication, with each featuring different techniques and mathematical formulas. In this research, we focused on combining some techniques to produce a scalar point multiplication algorithm for elliptic curves over finite fields. The employed methodology involved mathematical analysis to investigate commonly used point multiplication methods. The aim was to propose an efficient algorithm that combined the best computational techniques, resulting in lower computational requirements. The findings show that the proposed method can overcome certain implementation issues found in other multiplication algorithms. In certain scenarios, the proposed method offers a more efficient approach by reducing the number of point doubling and point addition operations on elliptic curves using the inverse of the targeted point. [ABSTRACT FROM AUTHOR]

In 1985, Schoof devised an algorithm to compute zeta functions of elliptic curves over finite fields by directly computing the numerators of these rational functions modulo sufficiently many primes (see [9]). If E / K is an elliptic curve with nonconstant j -invariant defined over a function field K of characteristic p ≥ 5 , we know that its L -function L (T , E / K) is a polynomial in Z [ T ] (see [5, p. 11]). Inspired by Schoof, we study the reduction of L (T , E / K) modulo integers. We obtain three main results. Firstly, if E / K has non-trivial K -rational N -torsion for some positive integer N coprime with p , we extend a formula for L (T , E / K) mod N due to Hall (see [4, p. 133, Theorem 4]) to all quadratic twists E f / K with f ∈ K × ∖ K × 2. Secondly, without any condition on the 2-torsion subgroup of E (K) , we give a formula for the quotient modulo 2 of L -functions of any two quadratic twists of E / K. Thirdly, we use these results to compute the global root numbers of an infinite family of quadratic twists of an elliptic curve and, under some assumption, find in most cases the exact analytic rank of these twists. We also illustrate that in favourable situations, our second main result allows one to compute much more efficiently L (T , E f / K) mod 2 than an algorithm of Baig and Hall (see [1]). Finally, we use our formulas to compute some degree 2 L -functions directly. [ABSTRACT FROM AUTHOR]

The risks of misusing new and developing technology are increasing on a daily basis. In these technologies, electronic communication (e-communication) anonymously delivers sensitive information. It is a challenging task in e-communication to conceal the identity of the original users and preserve misuse risk protection (i.e., security and privacy) with low computations. On the other hand, anonymous communication may be detrimental. For security and privacy maintenance, we use the concept of Misuse Risk Protection (MRP) for low computations and High-Performance Computing (HPC). To overcome these challenges, the researchers use the concept of blind signature, blind singcryption and ID-based signcryption for e-communication environments. We analyzed blind and ID-based signcryption schemes and their security hardness, which is based on different Discrete Logarithm Problems (DLP) such as Elliptic Curve DLP (ECDLP), Hyperelliptic Curve DLP (HECDLP), and bilinear pairing etc. In this paper, we also investigated that the proposed schemes attain the desired MRP features, HPC (e.g., low computation and communication overheads) than the existing schemes, which makes the newly proposed scheme more suitable for low-power environments. Finally, all schemes with small key sizes provide low computation and communication overhead. Due to low computation and communication overheads, it is considered efficient. [ABSTRACT FROM AUTHOR]

The Manin constant c of an elliptic curve E over Q is the nonzero integer that scales the differential ωf​ determined by the normalized newform f associated to E into the pullback of a Néron differential under a minimal parametrization ϕ:X0​(N)Q​↠E. Manin conjectured that c=±1 for optimal parametrizations, and we prove that in general c∣deg(ϕ) under a minor assumption at 2 and 3 that is not needed for cube-free N or for parametrizations by X1​(N)Q​. Since c is supported at the additive reduction primes, which need not divide deg(ϕ), this improves the status of the Manin conjecture for many E. Our core result that gives this divisibility is the containment ωf​∈H0(X0​(N),Ω), which we establish by combining automorphic methods with techniques from arithmetic geometry; here the modular curve X0​(N) is considered over Z and Ω is its relative dualizing sheaf over Z. We reduce this containment to p-adic bounds on denominators of the Fourier expansions of f at all the cusps of X0​(N)C​ and then use the recent basic identity for the p-adic Whittaker newform to establish stronger bounds in the more general setup of newforms of weight k on X0​(N). To overcome obstacles at 2 and 3, we analyze nondihedral supercuspidal representations of GL2​(Q2​) and exhibit new cases in which X0​(N)Z​ has rational singularities. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over Q . We conjecture asymptotic estimates for the number of vanishings of L (E , 1 , χ) as χ varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on E. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Compared to earlier work by David, Fearnley and Kisilevsky that formulated analogous conjectures for characters of any odd prime order, in the composite order case, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. To do this, we introduce the notion of totally order ℓ characters to quantify how quickly the quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (resp., sextic) characters is much slower than in the sub-family of totally quartic (resp., sextic) characters. We provide a conceptual explanation for this phenomenon by observing that the full family of order ℓ twisted elliptic curve L-functions, with ℓ even and composite, is a mixed family with both unitary and orthogonal aspects. [ABSTRACT FROM AUTHOR]

We present a Mordell–Weil sieve that can be used to compute points on certain bielliptic modular curves $X_0(N)$ over fixed quadratic fields. We study $X_0(N)(\mathbb {Q}(\sqrt {d}))$ for $N \in \{ 53,61,65,79,83,89,101,131 \}$ and ${\lvert d \rvert. [ABSTRACT FROM AUTHOR]

Numerous papers have studied the problem of determining upper bounds for the number of integer points on elliptic curves of the form y 2 = x 3 − m x , and quartic curves of the form X 2 − d Y 4 = k. Bounds for the number of integer solutions to such quartic equations typically depend on both of the coefficients d , k. The purpose of this paper is to examine more closely how the number of integer points on such quartic curves seems to depend almost entirely on the number of prime factors of k. This is done by focusing in on the problem of bounding the number of squares in certain recurrence sequences. In particular, using some arithmetic on elliptic curves, it is proved that infinitely many such sequences can have four squares, but that a fifth square remains elusive after extensive computation, suggesting that an absolute bound for the number of squares is more than likely, and also suggesting that the above assertion regarding the dependence on the number of prime factors of k. [ABSTRACT FROM AUTHOR]

In this note we study the Mahler measures of reciprocal polynomials P k = x + 1 / x + y + 1 / y + k for k = 4 ± 4 2 . We prove identities relating the Mahler measure of P 4 ± 4 2 to special values of the L -functions of weight 2 cusp forms of level 64. We also express the Beilinson regulator which is a 2 × 2 determinant of Mahler measures as special value of L -function of the elliptic curve defined by P 4 + 4 2 = 0. [ABSTRACT FROM AUTHOR]

In a paper from twenty years ago, Brown and Myers proved the clean and uniform result that if m > 1 is an integer, then every curve of the form y 2 = x 3 − x + m 2 has Mordell-Weil rank at least 2. In this article, we further our investigation into the more general family of curves, in which the polynomial x 3 − x is replaced by an arbitrary cubic which is squarefree and has three integral roots. In particular, a new strategy will unveil a subfamily of curves with surprisingly large average rank, and one startling example. [ABSTRACT FROM AUTHOR]

In this paper we introduce a local-refinement procedure to investigate stability data on an abelian category, and provide a sufficient and necessary condition for a stability data to be finest. We classify all the finest stability data for the categories of coherent sheaves over certain weighted projective curves, including the classical projective line, smooth elliptic curves and certain weighted projective lines. As applications, we obtain a classification of torsion pairs for these categories via stability data approach. As a by-product, a new proof for the classification of torsion pairs in any tube category is also provided. [ABSTRACT FROM AUTHOR]

The Fibonacci sequence is a well-known sequence of numbers with numerous applications in mathematics, computer science, and other fields. In recent years, there has been growing interest in studying Fibonacci-like sequences on elliptic curves. These sequences have a number of exciting properties and can be used to build new encryption systems. This paper presents a further generalization of the Fibonacci sequence defined on elliptic curves. We also describe an encryption system using this sequence which is based on the discrete logarithm problem on elliptic curves. [ABSTRACT FROM AUTHOR]

CSIDH is a post-quantum key-exchange scheme based on the action of ideal class groups on supersingular elliptic curves over prime fields. Its short keys and ciphertexts, together with its flexibility as a building block to construct complex cryptographic primitives, has motivated significant research on the efficiency of CSIDH and its resistance against side-channel attacks. In this work, some cutting-edge results from recent contributions are reviewed in a unified treatment, focusing on the mathematical ideas lying behind them rather than on cryptographic and low-level implementation techniques. In particular, we first describe ways to speed up the class-group-action evaluation, which range from the use of different models of elliptic curves to working with different ideal class groups. We then survey some constant-time variants of CSIDH, that make the time and memory consumption during the computation of a public/shared key independent of the secret key. Finally, we examine the computation of the ideal class action when the structure of the ideal class group is known, which is the case for a specific set of CSIDH parameters. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve defined over a finite field Fq, q = pd, p > 3, and a prime number l> 3 such that q ≡1 (mod l) and ‘ l| #E(Fq). In this paper we study the possible factorisation patterns over Fq[x] of the ‘lk-division polynomials associated to E with k≥ 2, extending the work of Verdure [6] for k = 1. [ABSTRACT FROM AUTHOR]

Let a 15 (n) , a 20 (n) and a 24 (n) be defined by q ∏ k = 1 ∞ (1 - q k) (1 - q 3 k) (1 - q 5 k) (1 - q 15 k) = ∑ n = 1 ∞ a 15 (n) q n , q ∏ k = 1 ∞ (1 - q 2 k) 2 (1 - q 10 k) 2 = ∑ n = 1 ∞ a 20 (n) q n , q ∏ k = 1 ∞ (1 - q 2 k) (1 - q 4 k) (1 - q 6 k) (1 - q 12 k) = ∑ n = 1 ∞ a 24 (n) q n (| q | < 1) , and let p > 3 be a prime. In this paper, for p ≡ 3 (mod 4) we reveal the connection between a 20 (p) and residue-counts of x 4 - 4 x 2 + 4 x modulo p as x runs over 0 , 1 , … , p - 1 , and the connection between a 24 (p) and residue-counts of x 3 + c / x modulo p as x runs over 1 , 2 , … , p - 1 , where c is an integer not divisible by p. We also deduce the congruences for a 15 (p) , a 24 (p) modulo 16 and a 20 (p) modulo 4, and pose some analogous conjectures. [ABSTRACT FROM AUTHOR]

We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs generated by three polynomials, we relate the distribution of leaves to the Sato-Tate distribution of Frobenius traces of elliptic curves. We also present extensive numerical results which we hope may shed some light on the distribution of leaves for larger families of polynomials. [ABSTRACT FROM AUTHOR]

Orthogonal families of bicircular quartics are naturally viewed as pairs of singular foliations of C ^ by vertical and horizontal trajectories of a non-vanishing quadratic differential. Yet the identification of these trajectories with real quartics in C P 2 is subtle. Here, we give an efficient, geometric argument in the course of updating the classical theory of confocal families in the modern language of quadratic differentials and the Edwards normal form for elliptic curves. In particular, we define a parameterized Edwards transformation, providing explicit birational equivalence between each curve in a confocal family and a fixed curve in normal form. [ABSTRACT FROM AUTHOR]

We introduce a new generalization of -congruent numbers by defining a rational -parallelogram envelope. It turns out that the existence of a rational -parallelogram envelope associated to an n is connected to the property of n being a -congruent number. Then, we study some problems related to the rational -parallelogram envelopes, using the arithmetic of algebraic curves. Our results generalize the recent work of Ochiai, where only the case = π / 2 was considered. Moreover, we answer open questions raised by Ochiai in the more general setting of an arbitrary Pythagorean angle. [ABSTRACT FROM AUTHOR]

Securing digital images against modern attacks is a hot research topic in the field of cryptography. Several encryption schemes have been proposed, but either they do not achieve high security or they are computationally costly for any image, and thus may not be suitable for real-time applications. We design an image encryption scheme by formulating an optimization problem with an objective function that captures both entropy and correlation. We select a core of the plain-image to compute a parameter that optimizes the objection function over the core using artificial bee colony (ABC) algorithm. Furthermore, we propose a new efficient random number generator over isomorphic elliptic curves. This generator together with the parameter is then used to generate cipher-images. The strength of the proposed random number generator and encryption scheme is analyzed by applying several standard tests. Detailed experimental results reveal that our scheme achieved entropy, correlation, number of pixel change rate and unified average changing intensity close to the theoretically optimal values, and is up to 5 × 104 times faster than the existing schemes. [ABSTRACT FROM AUTHOR]

The cryptographic system is the foundation of blockchain security mechanisms. This paper constructs a blockchain security mechanism based on the Chinese national cryptographic SM2 algorithm, involving the generation of public–private key pairs and the signing of block information using key pairs. However, the original SM2 algorithm has a high computational complexity, which cannot meet the requirements of low on-chain delay in blockchain applications. To solve this problem, this paper optimizes the SM2 digital signature algorithm. The scalar multiplication operation in the digital signature algorithm has a large computational complexity, which affects the efficiency of a digital signature. To reduce its computational complexity, this paper improves the traditional window non-adjacent form (w-NAF) algorithm for the scalar multiplication operation and proposes a low computational complexity Fw-NAF algorithm based on the prime number precomputation window, which greatly reduces its computational complexity. [ABSTRACT FROM AUTHOR]

In this paper, we present the new signature scheme based on elliptic curve (EC) and factoring (FAC) problems using chaotic maps (CMs). The newly developed scheme requires only minimal and low-complexity computations, which makes it very efficient. We compare our scheme with the other schemes with respect to signature generation cost, signature size, and verification cost and show that our scheme is superior to the other schemes. To the best of our knowledge, this is the first time a signature scheme based on EC and FAC problems using CMs has been proposed. [ABSTRACT FROM AUTHOR]

Recently, the second author and Qin numerically verified some Mahler measure identities of genus 2 and 3 polynomial families. In this paper, we use the elliptic regulator to prove an identity involving shifted Mahler measure for Boyd's family. [ABSTRACT FROM AUTHOR]

Recent progress in the theory of Heron triangles and their elliptic curves led to new families of congruent number elliptic curves with rank at least two. Based on these results, we derive a parametric family of congruent number elliptic curves with rank at least three. It turns out that this family is isomorphic to a family which was recently discovered by the third-named author, however the new approach is simpler, more flexible and gives new insight. In particular, it provides in addition three formulae for congruent numbers. [ABSTRACT FROM AUTHOR]

Firstly, we present a more explicit formulation of the complete system D (N) of representatives of Manin's symbols over Q , which was initially given by Shimura. Then, we establish a bijection between D (M) × D (N) and D (M N) for (M , N) = 1 , which reveals a recursive structure between Manin's symbols of different levels. Based on Manin's complete system Π (N) of representatives of cusps on X 0 (N) and Cremona's characterization of the equivalence between cusps, we establish a bijection between a subset C (N) of D (N) and Π (N) , and then establish a bijection between C (M) × C (N) and C (M N) for (M , N) = 1 . We also provide a recursive structure for elliptical points on X 0 (N) . Based on these recursive structures, we obtain recursive algorithms for constructing Manin symbols over Q , cusps, and elliptical points on X 0 (N) . This may give rise to more efficient algorithms for modular elliptic curves. As direct corollaries of these recursive structures, we present a recursive version of the genus formula and prove constructively formulas of the numbers of D (N) , cusps, and elliptic points on X 0 (N) . [ABSTRACT FROM AUTHOR]

A proof is given of several conjectures from a recent paper of Nakaya concerning the supersingular polynomial s s p (N ∗) (X) for the Fricke group Γ 0 ∗ (N) , for N ∈ { 2 , 3 , 5 , 7 }. One of these conjectures gives a formula for the square of s s p (N ∗) (X) (mod p) in terms of a certain resultant, and the other relates the primes p for which s s p (N ∗) (X) splits into linear factors (mod p) to the orders of certain sporadic simple groups. [ABSTRACT FROM AUTHOR]

We show that there are infinitely many primes p for which the cubic twist y 2 = x 3 + 16 p 2 (respectively the quartic twist y 2 = x 3 − p x) has positive Mordell-Weil rank by playing off the structure of the Q -rational torsion points again the projective geometry of these twists. A key ingredient to both proofs are sieve results of Heath-Brown and Moroz, and Friedlander and Iwaniec, on primes represented by cubic and quartic polynomials. As a consequence we show that there are infinitely many primes congruent to 1 (mod 9) and respectively − 1 (mod 9) that can be written as a sum of two rational cubes. We show that there are infinitely many primes p for which the cubic twist y 2 = x 3 + 16 p 2 (respectively the quartic twist y 2 = x 3 − p x) has positive Mordell-Weil rank by playing off the structure of the Q -rational torsion points again the projective geometry of these twists. A key ingredient to both proofs are sieve results of Heath-Brown and Moroz, and Friedlander and Iwaniec, on primes represented by cubic and quartic polynomials. As a consequence we show that there are infinitely many primes congruent to 1 (mod 9) and respectively − 1 (mod 9) that can be written as a sum of two rational cubes. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over \mathbb {Q}. Then, we show that the average analytic rank of E over cyclic extensions of degree l over \mathbb {Q} with l a prime not equal to 2, is at most 2+r_\mathbb {Q}(E), where r_\mathbb {Q}(E) is the analytic rank of the elliptic curve E over \mathbb {Q}. This bound is independent of the degree l. Using a recent result of Bhargava, Taniguchi and Thorne [Improved error estimates for the Davenport–Heilbronn theorems, arxiv.org/abs/2107.12819, 2021], we obtain a non-trivial upper bound on the average analytic rank of E over S_3-fields. [ABSTRACT FROM AUTHOR]

We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non‐Galois extensions whose Galois closure has a Galois group permutation‐isomorphic to a prescribed group G (in short, "G‐extensions"). In particular, for alternating groups and (an infinite family of) projective linear groups G, we show that most elliptic curves over (for example) Q$\mathbb {Q}$ gain rank over infinitely many G‐extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion, which allows to deduce that "many" elliptic curves gain rank over infinitely many G‐extensions, conditional on the parity conjecture and on the existence of geometric Galois realizations with group G and certain local properties. [ABSTRACT FROM AUTHOR]

Let E / Q be an optimal elliptic curve of analytic rank zero. It follows from the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero that the order of the torsion subgroup of E / Q divides the product of the order of the Shafarevich–Tate group of E / Q , the (global) Tamagawa number of E / Q , and the Tamagawa number of E / Q at infinity. This consequence of the Birch and Swinnerton-Dyer conjecture was noticed by Agashe and Stein in 2005. In this paper, we prove this divisibility statement unconditionally in many cases, including the case where the curve E / Q is semi-stable. [ABSTRACT FROM AUTHOR]

Let A be an n -by- n matrix. The ternary form F A (t , x , y) = det ⁡ (t I n + x ℜ (A) + y ℑ (A)) characterizes the numerical range of A. For n = 4 , we assume that the ternary form F A (t , x , y) is irreducible and the complex projective algebraic curve F A (t , x , y) = 0 is elliptic. We prove that the number of analytic curves composing the real algebraic curve of F A (t , x , y) = 0 is 2. This result is applied to show that the j -invariant of the elliptic curve F A (t , x , y) = 0 is greater than or equal to 1. [ABSTRACT FROM AUTHOR]

The group of rational points on an elliptic curve over Q is always a finitely generated Abelian group, hence isomorphic to Zr × G with G a finite Abelian group. Here, r is the rank of the elliptic curve. In this paper, we determine sufficient conditions that need to be set on the prime numbers p and q so that the elliptic curve E : y² = x³ − 3pqx over Q would possess a rank zero or one. Specifically, we verify that if distinct primes p and q satisfy the congruence p ≡ q ≡ 5 (mod 24), then E has rank zero. Furthermore, if p ≡ 5 (mod 12) is considered instead of a modulus of 24, then E has rank zero or one. Lastly, for primes of the form p = 24k + 17 and q = 24ℓ + 5, where 9k + 3ℓ + 7 is a perfect square, we show that E has rank one. [ABSTRACT FROM AUTHOR]

Let E / ℚ be an elliptic curve. Let ℚ G denote the compositum of all fields with Galois group G over ℚ. We classify the possible torsion subgroups of E over ℚ D n for n not divisible by 3 or 4, where D n is the dihedral group on a regular n -gon. We also obtain upper bounds for the torsion of E over ℚ G for various semi-direct products G = q (ζ p) ⋊ ℤ / p ℤ by way of classifications of torsion over ℚ ℤ / p q ℤ . [ABSTRACT FROM AUTHOR]

We study the moduli stack of degree 0 semistable G-bundles on an irreducible curve E of arithmetic genus 1, where G is a connected reductive group in arbitrary characteristic. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups H of G (the E-pseudo-Levi subgroups), where each stratum is computed in terms of H-bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan–Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where E has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra \mathfrak {g} (respectively, algebraic group G). We also provide a Tannakian description of these moduli stacks and use it to show that if E is not a supersingular elliptic curve, the moduli of framed unipotent bundles on E are equivariantly isomorphic to the unipotent cone in G. Finally, we classify the E-pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples. [ABSTRACT FROM AUTHOR]

This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form p = 8k +1, receives attention. [ABSTRACT FROM AUTHOR]

In this paper, we work over C . Criterions for an abelian surface to be products of elliptic curves are known. We give a new criterion from the view point of the Galois cover. [ABSTRACT FROM AUTHOR]

Given any positive integer n, there exist triangles, called Heron triangles, with rational sides whose area is n. Assuming the finiteness of the Shafarevich–Tate group, we construct a family of infinitely many Heronian elliptic curves of rank 1 arising from Heron triangles of a certain type. We then explicitly produce a separate family of infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and 3. [ABSTRACT FROM AUTHOR]

In this paper, as an analog of the number field case, for an elliptic curve E defined over the algebraic numbers and for any subfield F of algebraic numbers, we say that E has the Northcott property over F if there are at most finitely many F -rational points on E of uniformly bounded height, and we say that E has the property (P) over F if for any infinite subset S of F -rational points on E , f (S) = S for an F -endomorphism f of E implies that f is an automorphism. We establish some criteria for both properties and provide typical examples. We also show that the Northcott property implies the property (P), but the converse is not true. [ABSTRACT FROM AUTHOR]

In the present era of digital communication, secure data transfer is a challenging task in the case of open networks. Low-key-strength encryption techniques incur enormous security threats. Therefore, efficient cryptosystems are highly necessary for the fast and secure transmission of multimedia data. In this article, cryptanalysis is performed on an existing encryption scheme designed using elliptic curve cryptography (ECC) and a Hill cipher. The work shows that the scheme is vulnerable to brute force attacks and lacks both Shannon's primitive operations of cryptography and Kerckchoff's principle. To circumvent these limitations, an efficient modification to the existing scheme is proposed using an affine Hill cipher in combination with ECC and a 3D chaotic map. The efficiency of the modified scheme is demonstrated through experimental results and numerical simulations. [ABSTRACT FROM AUTHOR]

Most of the existing smart grid authentication schemes suffer from high computational costs and complex authentication processes, which are not suitable for resource-constrained smart devices in the smart grid, while some lightweight schemes suffer from various security vulnerabilities, none of which can achieve the required trade-off between efficiency and security. To solve these problems, this paper designed an enhanced provably secure smart grid lightweight anonymous authentication scheme based on elliptic curve encryption algorithm. Introducing auxiliary validator to get rid of the reliance on the power supplier in the authentication phase and to achieve mutual authentication between the gateway and the smart meter under the condition of protecting the true identity of the smart meter. In addition, the scheme can trace and revoke the identity of malicious smart meters through pseudo identity. Through the security analysis under the random oracle model and simulation tool ProVerif verifies that the proposed scheme has high security properties. Performance analysis shows that the proposed scheme can meet the requirements for security and efficiency in smart grid environment. [ABSTRACT FROM AUTHOR]

In recent decades there has been an increasing interest in Elliptic curve cryptography (ECC) and, especially, the Elliptic Curve Digital Signature Algorithm (ECDSA) in practice. The rather recent developments of emergent technologies, such as blockchain and the Internet of Things (IoT), have motivated researchers and developers to construct new cryptographic hardware accelerators for ECDSA. Different types of optimizations (either platform dependent or algorithmic) were presented in the literature. In this context, we turn our attention to ECC and propose a new method for generating ECDSA moduli with a predetermined portion that allows one to double the speed of Barrett's algorithm. Moreover, we take advantage of the advancements in the Artificial Intelligence (AI) field and bring forward an AI-based approach that enhances Schoof's algorithm for finding the number of points on an elliptic curve in terms of implementation efficiency. Our results represent algorithmic speed-ups exceeding the current paradigm as we are also preoccupied by other particular security environments meeting the needs of governmental organizations. [ABSTRACT FROM AUTHOR]

We prove certain divisibility results for the 2-part of the class number of quadratic fields Q (- D) and Q (D) , where D = p q is a congruent number with two prime factors p and q satisfying p ≡ q ≡ 5 (mod 8) . [ABSTRACT FROM AUTHOR]

It is well-known that cryptography is a branch of secrecy in science and mathematics, which usually preserves the confidentiality and authenticity of the information, where its growth is parallel with the rapid evolution of the internet and communication. As one of the prominent public key cryptosystems, the Elliptic Curve Cryptosystem (ECC) offers efficiency and complex mathematical operations with a smaller bit compared to other types of public key schemes. Throughout the evolution of cryptography, ElGamal Elliptic Curve Cryptosystem (ElGamal ECC) revolved from ElGamal public key scheme for user efficiency and privacy. In this study, an improved method will be introduced using ElGamal ECC as the foundation with the incorporation of the Bézier curve coefficient matrix, where the ElGamal ECC value is considered as the control point of the Bézier curve during the encryption and decryption processes. The proposed method is designed to develop a robust ciphertext system algorithm for better efficiency and to increase the level of protection in ElGamal ECC. In this paper, the performance of the proposed method is compared with the normal ElGamal ECC. The results of this study show that the proposed method offers no significant difference in terms of the implementation time during the encryption and decryption process. However, it does offer extra layers of protection when operated with complex mathematical operations. [ABSTRACT FROM AUTHOR]

A family 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 ∈ F can be made uniformly bounded after removing from F those whose number field degrees lie in a subset of Z + with arbitrarily small upper density. For every number field F , we prove unconditionally that the family E F of elliptic curves defined over number fields and with F -rational j -invariant is typically bounded in torsion. For any integer d ∈ Z + , we also strengthen a result on typically bounding torsion for the family E d of elliptic curves defined over number fields and with degree d j -invariant. [ABSTRACT FROM AUTHOR]

A congruent number is a positive integer which can be represented as the area of a right triangle such that all of its side lengths are rational numbers. The problem determining whether a given number is congruent is usually studied by computing the Mordell-Weil rank of the corresponding elliptic curve. The Monsky matrix gives a way to compute efficiently the 2-Selmer rank, thereby gives an upper bound for the Mordell-Weil rank. In this paper, by using Monsky's matrix, we present new families of non-congruent numbers such that all of their odd prime factors are of the form 8 k + 3. Our result generalizes previous works of Reinholz–Spearman–Yang [12] and Cheng–Guo [3]. [ABSTRACT FROM AUTHOR]

Let p be a prime number and E p denote the elliptic curve y 2 = x 3 + p x. It is known that for p which is congruent to 1 , 9 modulo 16, the rank of E p over Q is equal to 0 , 2. Under the condition that the Birch and Swinnerton-Dyer conjecture is true, we give a necessary and sufficient condition that the rank is 2 in terms of the constant term of some polynomial that is defined by a recurrence formula. [ABSTRACT FROM AUTHOR]

The Birch and Swinnerton–Dyer conjecture famously predicts that the rank of an elliptic curve can be computed from its L -function. In this article, we consider a weaker version of this conjecture called the parity conjecture and prove the following. Let E1 and E2 be two elliptic curves defined over a number field K whose 2-torsion groups are isomorphic as Galois modules. Assuming finiteness of the Shafarevich–Tate groups of E1 and E2 , we show that the Birch and Swinnerton-Dyer conjecture correctly predicts the parity of the rank of E1×E2. Using this result, we complete the proof of the p -parity conjecture for elliptic curves over totally real fields. [ABSTRACT FROM AUTHOR]