Elliptic curves

Let L/K be a finite Galois extension of number fields with Galois group G = Gal(L/K). We introduce a new arithmetic invariant Φ L/K (a), defined for every nonzero ideal a ⊂ O K , by counting the G-orbits in the residue unit group (O L /aO L) ×. This Galois-Euler orbit totient function may be viewed as a symmetry-reduced analogue of Euler's totient function in algebraic number theory. We establish its fundamental properties rigorously: a Burnside-type orbit-counting formula, multiplicativity under coprime ideals, a resulting Euler product for the associated Dirichlet series, and a complete analysis of the local structure at primes that split, remain inert, or ramify in L. All statements are accompanied by complete proofs. Several explicit numerical examples in quadratic extensions are computed and verified.

The industrial 3D mesh model (3DMM) plays a significant part in engineering and computer aided designing field. Thus, protecting copyright of 3DMM is one of the major research problems that require significant attention. Further, the industries started outsourcing its 3DMM to cloud computing (CC) environment. For preserving privacy, the 3DMM are encrypted and stored on cloud computing environment. Thus, building efficient data masking of encrypted 3DMM is considered to be efficient solution for masking information of 3DMM. First, using the secret key, the original 3DMM is encrypted. Second without procuring any prior information of original 3DMM it is conceivable mask information on encrypted 3D mesh models. Third, the original 3DMM are reconstructed by extracting masked information. The existing masking methods are not efficient in providing high information masking capacity in reversible manner and are not robust. For overcoming research issues, this work models an efficient data masking (EDM) method that is reversible nature. Experiment outcome shows the EDM for 3DMM attain better performance in terms of peak signal-to-noise ratio (PSNR) and root mean squared error (RMSE) over existing data masking methods. Thus, the EDM model brings good tradeoffs between achieving high data masking capacity with good reconstruction quality of 3DMM.

Elliptic Curve Cryptosystems (ECC) have emerged as a powerful alternative to traditional public-key cryptosystems, offering equivalent security with significantly smaller key sizes. The efficiency of ECC, in terms of minimizing encryption time and enhancing computational performance, is strongly influenced by the number of point addition and doubling computations required in elliptic curve arithmetic. In this context, the study of arithmetic of points in elliptic curves plays a crucial role. This study emphasizes computations in projective coordinates of points on elliptic curves defined over the p-adic field Q p. A comparative study shows that arithmetic in projective coordinates reduces the number of operations required, thereby enhancing the efficiency relative to the affine coordinate system. The coordinate-level p-adic expansions of the arithmetic may be obtained by employing p-adic expansion techniques to the arithmetic of points on the elliptic curve E(Q p)(mod p n) in projective coordinates for n = 2, 3, ... In this paper, coordinate-level p-adic expansions of the arithmetic of points on E(Q p)(mod p 2) in projective coordinates are formulated, an algorithm for the computations is given illustrating the step-by-step process for computing point addition and doubling in E(Q p)(mod p 2) and its efficiency over the arithmetic of points on E(Q p)(mod p 2) in affine coordinates is described. This provides a systematic framework for performing elliptic curve arithmetic efficiently.

We propose a structural framework for understanding the arithmetic content of automorphic L-functions at their central points via what we call the General Central Singularity Formula (GCSF). Rather than focusing on special values or isolated derivatives, we reinterpret the Birch-Swinnerton-Dyer conjecture as arising from a universal singular expansion of the completed L-function at the central point. Our approach is based on a fixed relative trace formula (RTF) for GL2 (or a compact inner form), together with one-parameter families of admissible test functions satisfying prescribed moment vanishing conditions. Differentiation with respect to the deformation parameter isolates successive coefficients in the central expansion of the L-function on the spectral side, while rigorously annihilating all regular geometric contributions. We prove unconditionally that, after suitable spectral isolation and normalization, the r-th differentiated relative trace formula extracts exactly the r-th derivative of the completed L-function at the central point. On the geometric side, the same operation isolates a singular contribution of the trace formula, which canonically defines an arithmetic Chow class whose Arakelov self-pairing realizes this singular term. In the case of elliptic curves with complex multiplication, existing results in arithmetic intersection theory allow us to identify this height pairing explicitly, recovering the full Birch-Swinnerton-Dyer leading term formula, including the regulator, Tamagawa factors, and torsion terms. For general elliptic curves, the arithmetic identification of the singular geometric term is formulated conditionally, relying on standard conjectures and expected identities in the Kudla program, as well as on deformation and stability principles for relative trace formulas. From this perspective, the classical results of Waldspurger and Gross-Zagier appear as the cases of vanishing order 0 and 1, respectively, while higher-rank phenomena correspond to higher-order central singularities. The General Central Singularity Formula thus provides a unified conceptual framework in which central derivatives of automorphic L-functions and arithmetic invariants arise from a single geometric-spectral mechanism.

This report presents high-precision numerical evidence for a candidate elliptic curve over the rational field \mathbb{Q} with an analytic rank of 33. The curve was identified through a specialized search within the family of quadric surface intersections in \mathbb{P}^3. Using arbitrary-precision numerical verification (2000 dps), we document the stable vanishing of the L-series up to the 33rd order at the critical point s = 1, with a residual error of 10^{-1995}. This dataset provides 33 high-precision candidate generators, offering a significant benchmark for future algebraic reconstruction and the study of the Birch and Swinnerton-Dyer (BSD) Conjecture.

This thesis presents a comprehensive research framework focused on the analytical resolution of the Birch and Swinnerton-Dyer (BSD) Conjecture. It introduces a proprietary high-precision computational protocol designed for the search, calculation, and verification of rational points on elliptic curves. The work establishes new analytical methodologies for evaluating L-series, Tate-Shafarevich groups, and rank stability, providing the theoretical foundations for identifying high-rank curves over \mathbb{Q}.

Notice: This document is a preprint (pre-review version) currently under formal peer review.

A single CRT-channel resonance amplitude β = 2φ = (N c-1)φ on the 840-state manifold of the Brahim Framework is shown to govern, with no free parameters, two independent open questions: (i) the v=1 to v=2 vibrational anomaly in actinium monouoride (AcF) reported by Athanasakis-Kaklamanakis et al. in Nature 2025, and (ii) the partition of the obstinate residue set MORDELL ⊂ (Z/840) * into Erdös-Straus R 0-forced and R 0-vulnerable classes. The bridge is the formula A(r) = β • L 4 • χ 5 (r) + L 5 • χ 7 (r) , χ p (r) = 1 r ≡ 4 (mod p) , with Lucas weights {L 2 , L 3 , L 4 , L 5 } = {3, 4, 7, 11}. In the AcF spectroscopy t (companion workbook), β is tted at 3.23 ± 0.16 cm-1 , matching 2φ = 3.236068 within 0.04σ, and yields χ 2 /dof = 0.90 across ve vibrational bandheads (standard Morse rejected at 193.9, cubic Morse rejected at 56.8). In the Erd®s-Straus context, the same formula partitions all six MORDELL residues into the four R 0-forced (A(r) > 0) and the two R 0-vulnerable (A(r) = 0), in exact agreement with empirical data from a lattice database covering 606,496 primes up to 9.06 × 10 6 (12,450 MORDELL primes; zero counterexamples). The empirical match is 6 of 6 by structure plus a 0.04σ value-level match across the two domains. The result is presented as cross-domain empirical structure on the 840-manifold; it does not extend the numerical bound of Salez (2014) on the 4/p Erdös-Straus conjecture.

Volume 7 of the Brahim Framework presents a comprehensive algebraic foundation that reduces the complex parameters of the Standard Model and other scientific domains to a single mathematical input: the **golden ratio**. Central to this theory is the multiplicative group GF(841)*, which provides the structural logic for a manifold of 840 states and explains patterns observed in previous volumes. The text rigorously derives the ten Brahim numbers through a precise six-step chain, verifying the results against a nine-point forensic protocol to ensure absolute exactness without free parameters.

Beyond physics, the framework establishes a semantic bridge, using category theory and the D-functor to map knowledge representation into this same coordinate space. The work concludes by offering eleven testable predictions across physics, biology, and information theory, shifting the framework from theoretical synthesis to an experimental programme subject to falsification. This volume essentially serves as the final mathematical architecture that unifies discrete number theory with the fundamental constants of nature.

Here are some research interests relevant to The Brahim Framework Volume 7.

1.  Mathematical Physics: The unification of fundamental physical constants and the reduction of the Standard Model's 25 parameters to a single algebraic root.

2.  Finite Field Theory: The structural application of Galois fields, specifically $GF(841)$, to model physical state manifolds.

3.  Group Theory: Exploring the action of the Klein four-group ($V_4$) and automorphism groups (such as the Frobenius automorphism) on physical states.

4.  Lie Algebras and Root Systems: Investigating the relationship between the $E_8$ exceptional Lie algebra and the 120-element "allowed set" derived from the Chinese Remainder Theorem.

5.  Information Theory: Studying Kolmogorov complexity and parametric compression to minimize the description length of physical and semantic systems.

6.  Computational Linguistics: Encoding semantic structures (Subject-Predicate-Object triples) into algebraic fields to study the geometry of meaning.

7.  Category Theory: Constructing strict injective functors (like the $D$-functor**) to map relationships between disparate categories such as semantics and $D$-space.

8.  Theoretical Biophysics: Deriving biological scaling laws, such as the Kleiber exponent, and investigating the redundancy of the genetic code through algebraic partitions.

9.  Number Theory: Analyzing the physical significance of **Lucas and Fibonacci sequences**, specifically their role in determining manifold dimensions and prime characteristics.

10. Algebraic Geometry: Modeling physical reality using $D$-space, branch lattices, and torus structures in complex space.

11. Quantum Chromodynamics (QCD): Deriving color charge counts and addressing the strong CP problem via the QCD vacuum angle $\theta_{QCD}$.

12. Cosmology: Providing an algebraic home for the cosmological constant ($\Lambda$) and modeling dark matter and dark energy fractions.

13. Popperian Epistemology: Developing rigorous falsification programmes with pre-registered predictions and specific kill criteria for theoretical frameworks.

14. Knowledge Representation: Utilizing the Chinese Remainder Theorem for semantic sharding and link prediction in large-scale knowledge graphs like Wikidata.

15. Formal Verification: Establishing nine-point protocols and dependency graphs to ensure the internal consistency of mathematical derivations.

16. Algebraic Semantics: Mapping grammatical transformations, such as active-to-passive voice inversion, to multiplicative inversion in Galois fields.

17. Sporadic Groups: Investigating the divisibility relations between finite field manifolds and the Monster group.

18. Modular Forms and Curves: Examining the $j$-invariant of specific tori and its connection to the Langlands programme and the Riemann hypothesis.

19. Topological Invariants: Analyzing the mirror wall ($S = 214$) and its role as a rigid boundary in physical and mathematical manifolds.

20. Elliptic Curves: Mapping the arithmetic properties of thousands of elliptic curves into $D$-space to confirm additive identities related to the BSD conjecture.

We  prove there are infinitely many p = n^2 + 1. There are p odd number pairs that each sum to 4p. A precursor, p - 1, imposes primality when pi(4p) - pi(4p - 4) = 1. A new pair {1,s} increases primes by 1 while leaving total composits unchanged. That is because 1 is shared by both sets but prime s is new and the pair {1,s} is a new composite-prime C3 pair in p = P + C1 + C2 + C3 pairs.

The Brahim Framework presents a, parameter-free mathematical manifold designed to unify fundamental physical constants and subatomic particle properties through algebraic geometry and topology. Utilizing a discrete manifold of order 840, the system integrates the Golden Ratio, specialized integer sequences (such as the Keith sequence), and elliptic curve arithmetic to calculate an Effective Dimension for distinct physical states. This rigorous mathematical structure directly correlates with known quantum phenomena, generating precise mass predictions for hadrons and fermions while establishing strict algebraic mirrors for fundamental forces and CPT symmetry. By demonstrating a direct, computable mapping between abstract mathematical symmetries, including Galois field GF(841) configurations and primary topological nodes designated as Brahim Numbers and experimental Standard Model physics, this dataset offers a comprehensive, testable, and zero-free-parameter model for exploring the underlying mathematical architecture of the universe using 12 forms of D(x):

D(x) = −log_φ(x) the definition
x = φ^(−D) exponential inverse
x · φ^D = 1 unit identity
x · D' = const scaling law
D(x) = surprisal_φ(x) information-theoretic (base-φ surprisal)
D(x) = −ln(x) · k, k = 1/ln(φ) natural log form with golden constant
D = arccosh(...) hyperbolic representation
D = λ₁(M) leading eigenvalue of the measurement operator
D = CF depth continued fraction depth
D + D⁻¹ = 0  chiral fold (the mirror identity D(x) + D(1/x) = 0)
d(ln x) + ln(φ) · dD = 0 conservation law (differential form)
H(x, D) = ln(x) + ln(φ)·D = 0 Hamiltonian constraint surface

Forms 1-10 are notational rewrites. Form 11 is where it becomes physics: a conserved current. The E(x) = 2π invariant is the circumference of the orbit in the conjugate coordinate space. We note the observation of 12 forms, 12 Lucas numbers summing to 840, 12 = 3 × 4 = Nc × Nst.

In this paper, we present four novel criteria for determining the irreducibility of
polynomials over the field of rational numbers Q. Moving beyond classical single
prime divisibility conditions, our approach focuses on the global arithmetic and
structural properties of the polynomial’s entire coefficient set. We first establish ir
reducibility for polynomials whose coefficients form sequences of consecutive primes
or arithmetic progressions of primes. We then extend these concepts to a mul
tiplicative framework, proving the irreducibility of polynomials constructed from
pairwise coprime bases raised to exponents that form an arithmetic progression.
By leveraging the Rational Root Theorem, complex root bounding, and reciprocal
polynomial transformations, we provide rigorous proofs for these new families of
irreducible polynomials, offering deterministic tools with potential applications in
algebraic number theory, cryptography, and computational algebra

As we all know that today's most popular research area is cloud computing. It is because cloud computing reduces the computing cost and at the same it increases the scalability and flexibility of computing resources. The cloud computing comprises of the network of the machines which are designated as the servers. In cloud computing the configurable resources such as the applications, servers, networks are shared and it enables the convenient and on demand access to these resources. In addition to this, these resources can be easily assigned and released without much effort, the service provider is not involved while doing this [10]. The cloud computing provides the service by the means of huge number of the computing resources. The primary and the most serious issues which are needed to be addressed and solved are like privacy of the data stored on cloud and the control on data retrieval. Even though it's high advantages, the organizations fears to adopt Cloud because of the issues related to its security and the challenges which are associated in implementing it. In this paper we present the security model using Elliptic Curve Cryptography algorithm to address the above said issues of the cloud.

RN44 established the structural form of Wall 3: the correct operation is a Connes-style distributional quotient trace over spatial scaling orbits, governed by the geometric carrier pole at s = 1. The remaining task is no longer structural discovery, but theorem-safe analytic packaging of the residue prescription. This note isolates that analytic remainder. It defines the residue object directly in the Mellin domain, separates the identity singularity from the prime-power terms, and records the exact conditions under which the Wall 3 prescription is well-defined. The point is not to discover a new mechanism, but to state precisely what must be checked to convert the structural Wall 3 crossing into a fully packaged theorem.

The security of modern public-key cryptography is fundamentally predicated on the presumed intractability of the Elliptic Curve Discrete Logarithm Problem (ECDLP). This paper introduces a theoretical attack framework that reduces ECDLP to an lookup operation. By mapping the cyclic group of an elliptic curve into a procedurally generated hyperbolic lattice space (), we bypass sequential traversal (steps) in favor of spatial coordinate resolution. Utilizing the Symmetric Hyperbolic Domain Chain Compression Protocol (SHD-CCP)-which relies on a 64-bit cubic architecture comprising a quaternion core, Dynamic Scaling Memory (DSMEM), and Halo/Ghost boundary regions-we construct a "Quadrance Lattice Reduction." Furthermore, we theorize that elliptic curve orders are topologically tiled by prime triplets , serving as deterministic generators for a 3-nested memory lookup. The successful formulation of this closed-form oracle implies the simultaneous collapse of ECC, RSA, and Diffie-Hellman, exposing deep structural vulnerabilities in cryptographic hardness assumptions and establishing a constructive pathway toward proving .

Katz and Sarnak conjectured a correspondence between the n-level density statistics of zeros from families of L-functions with eigenvalues from random matrix ensembles. In many cases the sums of smooth test functions, whose Fourier transforms are finitely supported, over scaled zeros in a family converge to an integral of the test function against a density W n,G depending on the symmetry G of the family (unitary, symplectic or orthogonal). This integral bounds the average order of vanishing at the central point of the corresponding family of L-functions. We can obtain better estimates on this vanishing in two ways. The first is to do more number theory, and prove results for larger n and greater support; the second is to do functional analysis and obtain better test functions to minimize the resulting integrals. We pursue the latter here when n = 2, minimizing 1 Φ(0, 0) R 2 W 2,G (x, y)Φ(x, y)dxdy over test functions Φ : R 2 → [0, ∞) with compactly supported Fourier transform. We study a restricted version of this optimization problem, imposing that our test functions take the form φ(x)ψ(y) for some fixed admissible ψ(y) and supp φ ⊆ [-1, 1]. Extending results from the 1-level case, namely the functional analytic arguments of Iwaniec, Luo and Sarnak and the differential equations method introduced by Freeman and Miller, we explicitly solve for the optimal φ for appropriately chosen fixed test function ψ. The solution allows us to deduce strong estimates for the proportion of newforms of rank 0 or 2 in the case of SO(even), rank 1 or 3 in the case of SO(odd), and rank at most 2 for O, Sp, and U; our estimates are a significant strengthening of the best known estimates obtained with the 1-level density. We conclude by discussing further improvements on estimates by the method of iteration.

We explore the effect of zeros at the central point on nearby zeros of elliptic curve L-functions, especially for one-parameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman's Specialization Theorem, for t sufficiently large the L-function of each curve Et in the family has r zeros (called the family zeros) at the central point. We observe experimentally a repulsion of the zeros near the central point, and the repulsion increases with r. There is greater repulsion in the subset of curves of rank r + 2 than in the subset of curves of rank r in a rank r family. For curves with comparable conductors, the behavior of rank 2 curves in a rank 0 one-parameter family over Q is statistically different from that of rank 2 curves from a rank 2 family. Unlike excess rank calculations, the repulsion decreases markedly as the conductors increase, and we conjecture that the r family zeros do not repel in the limit. Finally, the differences between adjacent normalized zeros near the central point are statistically independent of the repulsion, family rank and rank of the curves in the subset. Specifically, the differences between adjacent normalized zeros are statistically equal for all curves investigated with rank 0, 2 or 4 and comparable conductors from one-parameter families of rank 0 or 2 over Q. (1.4) 3 We normalize all L-functions to have functional equation s → 1s, and thus central point is at s = 1 2 . 4 The group of rational function solutions (x(T ), y(T )) ∈ Q(T ) 2 to y 2 = x 3 + A(T )x + B(T ) is isomorphic to Z r ⊕ T, where T is the torsion part and r is the rank.

We generalize a construction of families of moderate rank elliptic curves over Q to number fields K/Q. The construction, originally due to Scott Arms, Álvaro Lozano-Robledo and Steven J. Miller, invokes a theorem of Rosen and Silverman to show that computing the rank of these curves can be done by controlling the average of the traces of Frobenius; the construction for number fields proceeds in essentially the same way. One novelty of this method is that we can construct families of moderate rank without having to explicitly determine points and calculating determinants of height matrices.

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves, for the first we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently Yekutieli discussed a connection between these two problems, and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. In [JMMM] we generalized these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a given z to x 2 + Dy 2 = z 2 when D is 1 or 2 modulo 4 and the class group of Q[ √ -D] is a free Z 2 module, which always happens if the class number is at most 2. In this paper we discuss the main results of [JMMM] using some concrete examples in the case of D = 105.

AbstractTextExtending recent work of others, we provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato–Tate law. We present two methods of proof. Both use the framework of Murty and Sinha (2009) [MS]; the first involves only knowledge of the moments of the Fourier coefficients of the L-functions and combinatorics, and saves a logarithm, while the second requires a Sato–Tate law. Our purpose is to illustrate how the caliber of the result depends on the error terms of the inputs and what combinatorics must be done.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=faW2iDpe5IE

AbstractTextThe Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell–Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s,E). Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a family. Mestre (1986) [Mes] showed the first zero occurs by O(1/loglogNE), where NE is the conductor of E, though we expect the correct scale to study the zeros near the central point is the significantly smaller 1/logNE. We significantly improve on Mestre's result by averaging over a one-parameter family of elliptic curves E over Q(T). We assume GRH, Tate's conjecture if E is not a rational surface, and either the ABC or the Square-Free Sieve Conjecture if the discriminant has an irreducible polynomial factor of degree at least 4. We find non-trivial upper and lower bounds ...

Katz and Sarnak conjectured a correspondence between the n-level density statistics of zeros from families of L-functions with eigenvalues from random matrix ensembles. In many cases the sums of smooth test functions, whose Fourier transforms are finitely supported, over scaled zeros in a family converge to an integral of the test function against a density W n,G depending on the symmetry G of the family (unitary, symplectic or orthogonal). This integral bounds the average order of vanishing at the central point of the corresponding family of L-functions. We can obtain better estimates on this vanishing in two ways. The first is to do more number theory, and prove results for larger n and greater support; the second is to do functional analysis and obtain better test functions to minimize the resulting integrals. We pursue the latter here when n = 2, minimizing 1 Φ(0, 0) R 2 W 2,G (x, y)Φ(x, y)dxdy over test functions Φ : R 2 → [0, ∞) with compactly supported Fourier transform. We study a restricted version of this optimization problem, imposing that our test functions take the form φ(x)ψ(y) for some fixed admissible ψ(y) and supp φ ⊆ [-1, 1]. Extending results from the 1-level case, namely the functional analytic arguments of Iwaniec, Luo and Sarnak and the differential equations method introduced by Freeman and Miller, we explicitly solve for the optimal φ for appropriately chosen fixed test function ψ. The solution allows us to deduce strong estimates for the proportion of newforms of rank 0 or 2 in the case of SO(even), rank 1 or 3 in the case of SO(odd), and rank at most 2 for O, Sp, and U; our estimates are a significant strengthening of the best known estimates obtained with the 1-level density. As a representative example, the previous best 1-level analysis yields a lower bound of 0.7839 for vanishing to order at most 2 for the SO(even) family of cuspidal newforms, and our 2-level work improves this to 0.952694. We conclude by discussing further improvements on estimates by the method of iteration.

Let X : y 2 = f (x) be a hyperelliptic curve over Q(T ) of genus g ≥ 1. Assume that the jacobian of X over Q(T ) has no subvariety defined over Q. Denote by Xt the specialization of X to an integer T = t, let a Xt (p) be its trace of Frobenius, and A X ,r (p) = 1 p p t=1 a Xt (p) r its r-th moment. The first moment is related to the rank of the jacobian J X (Q(T )) by a generalization of a conjecture of Nagao: Generalizing a result of S. Arms, Á. Lozano-Robledo, and S.J. Miller, we compute first moments for various families resulting in infinitely many hyperelliptic curves over Q(T ) having jacobian of moderately large rank 4g + 2, where g is the genus; by Silverman's specialization theorem, this yields hyperelliptic curves over Q with large rank jacobian. Note that Shioda has the best record in this directon: he constructed hyperelliptic curves of genus g with jacobian of rank 4g + 7. In the case when X is an elliptic curve, Michel proved p • A X ,2 = p 2 + O p 3/2 . For the families studied, we observe the same second moment expansion. Furthermore, we observe the largest lower order term that does not average to zero is on average negative, a bias first noted by S.J. Miller in the elliptic curve case. We prove this bias for a number of families of hyperelliptic curves.

Katz and Sarnak conjectured a correspondence between the n-level density statistics of zeros from families of L-functions with eigenvalues from random matrix ensembles. In many cases the sums of smooth test functions, whose Fourier transforms are finitely supported, over scaled zeros in a family converge to an integral of the test function against a density W n,G depending on the symmetry G of the family (unitary, symplectic or orthogonal). This integral bounds the average order of vanishing at the central point of the corresponding family of L-functions. We can obtain better estimates on this vanishing in two ways. The first is to do more number theory, and prove results for larger n and greater support; the second is to do functional analysis and obtain better test functions to minimize the resulting integrals. We pursue the latter here when n = 2, minimizing 1 Φ(0, 0) R 2 W 2,G (x, y)Φ(x, y)dxdy over test functions Φ : R 2 → [0, ∞) with compactly supported Fourier transform. We study a restricted version of this optimization problem, imposing that our test functions take the form φ(x)ψ(y) for some fixed admissible ψ(y) and supp φ ⊆ [-1, 1]. Extending results from the 1-level case, namely the functional analytic arguments of Iwaniec, Luo and Sarnak and the differential equations method introduced by Freeman and Miller, we explicitly solve for the optimal φ for appropriately chosen fixed test function ψ. The solution allows us to deduce strong estimates for the proportion of newforms of rank 0 or 2 in the case of SO(even), rank 1 or 3 in the case of SO(odd), and rank at most 2 for O, Sp, and U; our estimates are a significant strengthening of the best known estimates obtained with the 1-level density. We conclude by discussing further improvements on estimates by the method of iteration.

We generalize a construction of families of moderate rank elliptic curves over Q to number fields K/Q. The construction, originally due to Scott Arms, Álvaro Lozano-Robledo and Steven J. Miller, invokes a theorem of Rosen and Silverman to show that computing the rank of these curves can be done by controlling the average of the traces of Frobenius; the construction for number fields proceeds in essentially the same way. One novelty of this method is that we can construct families of moderate rank without having to explicitly determine points and calculating determinants of height matrices.

This paper records three structural outputs of the SCT programme, with the operator ledger corrected explicitly.
First, the density theorems: the winding-sector decomposition of the critical carrier with centrifugal potentials V_m = m² W(δ) produces a raw eigenvalue count N_raw(E) ~ E² log E / (4ω) after the even-sector normalization. The carrier is therefore dimensionally saturated relative to the Riemann–von Mangoldt counting N(T) ~ T log T / (2π) (Kill #67). If an adelic quotient by ℚ* supplies the required dimensional reduction, the post-quotient density fixes the Klein coupling k = π/8 and gives the identity 4αk = ln 2.
Second, the auxiliary chiral obstruction: the paper defines a flat-principal non-self-adjoint chiral normal-form operator
H_{χ,k}^{(m)} = -∂_δ² + m²/D + im(2kδ/D)∂_δ - im k(1 + k²δ²)/D²
on L²(I, dδ). This operator is associated with the SCT shear data, but it is not the full self-adjoint Friedrichs carrier sector of Papers 38–39. For the chiral admissible finite-energy class, the corrected Frobenius analysis excludes the singular branch, all boundary terms vanish, and the Wronskian identity gives
0 = -im ∫ (A' - 2B) |ψ|² dδ, with A' - 2B = 4k(1 + k²δ²) / D² > 0,
which excludes real eigenvalues of the auxiliary operator for m ≠ 0.
Third, the honest boundary: the programme has proved the Gaussian prime-side shape, the density and metric-correction statements, the dimensional saturation obstruction, and a genuine model-level chiral exclusion theorem. It has not proved that the auxiliary chiral normal form is the full geometric carrier sector, has not identified individual carrier eigenvalues with zero ordinates, and does not assert a conditional proof of RH in this paper.

Version: v3 (April 2026)
What changed: The dimensional reduction factor in the post-quotient density matching theorem (Theorem B) was corrected from √2·E to 4√2·E. This is Kill #76 in the programme's falsification ledger.
The error: Version 2 applied the reduction factor √2·E to the even-sector counting N_raw^even(E) rather than to the full carrier counting N_raw(E). Since N_raw = 2·N_raw^even (even + odd sectors) and the even-sector restriction itself introduces a factor of 2, the net effect was a factor-of-4 undercount in the reduction factor.
The correction: The corrected theorem applies the full reduction factor 4√2·E to the full carrier N_raw(E):
N_post(E) = N_raw(E) / (4√2·E) = E·log(E) / (16k)
Setting N_post = E·log(E)/(2π) gives k = π/8, unchanged from v2.
What is unaffected: The value k = π/8, the identity 4αk = ln 2, the conditional critical-line theorem (Theorem 10.3), the spectral exclusion results (RN11), Kill #65 (eigenvalue mismatch), and Kill #67 (dimensional saturation) are all unchanged. The correction affects only the numerical coefficient of the reduction factor, not its functional form or any downstream theorem.
Verification: The corrected arithmetic has been verified computationally: π/(k√2) = π/((π/8)√2) = 8/√2 = 4√2 ≈ 5.657, matching the stated ratio exactly.

It is well-known that the Theorem of Pythagoras 𝐚 𝟐 + 𝐛 𝟐 = 𝐜 𝟐 can be illustrated in the plane by:  A 3-sided plane figure (namely a right-angled triangle)  Three 4-sided plane figures (namely three squares) It would seem logical, therefore, that the 3-dimensional equivalent of the Theorem of Pythagoras 𝐚 𝟑 + 𝐛 𝟑 + 𝐜 𝟑 = 𝐝 𝟑 would be illustrated in 3D space by:  A 4-faced solid (namely a tetrahedron)  Four 6-faced solids (namely four cubes)

The original Transverse Trace Conjecture (TTC) of the SCT programme is formulated on the full coupled spectral space. We show that the combined results of the critical-slice characterisation (Paper 5), the functional-equation isometry (Paper 33), and the winding-confinement analysis (RN7) single out a unique invariant subspace — the critical arithmetic sector — on which the trace identity should be stated. We prove that this sector is an exact reducing subspace for both the geometric spectral operator and the prime translation action, yielding a canonical self-adjoint restricted operator with real discrete spectrum.

The Restricted Transverse Trace Conjecture (RTTC) is stated precisely on this sector. We prove that RTTC implies the identification of the restricted spectrum with the Riemann zero ordinates, and hence implies the Riemann Hypothesis. We decompose RTTC into a ladder of seven sub-conjectures and identify the single hardest sub-piece: the exact extraction of the prime-power coefficients from the noncommutative trace on the critical sector.

We present a rigorous proof of the Riemann Hypothesis (RH) within the Ducci Unified Spectral Theory (DUST) framework. The proof proceeds in three fully independent stages. Stage I constructs the Asher-Ducci operator A on a separable Hilbert space H built from the prime-periodic lattice L 210 and establishes its fundamental properties from first principles. Stage II proves the Q = 1 Self-Adjointness Theorem: the Q-parameterised extension A(Q) is self-adjoint if and only if Q = 1, equivalently Re(s) = 1 2. This is the ignition condition, a structural necessity forced by the anti-Hermitian nature of the duality operator D away from Q = 1. Stage III connects the spectral theory of A to the Riemann xi-function via a heat-trace identity and Mellin transform, proving that every nontrivial zero of ζ(s) must be an eigenvalue of A(Q) Q=1 , which forces Re(s) = 1 2. Each logical gap is explicitly labelled as a Conjecture when it remains unproven; the main theorem is stated with full clarity about which steps are theorems and which are conditional. MSC 2020: 11M26 (Riemann zeta function), 47B25 (Self-adjoint operators), 11N05 (Distribution of primes), 47A10 (Spectrum, resolvent).

We present a rigorous proof of the Riemann Hypothesis (RH) within the Ducci Unified Spectral Theory (DUST) framework. The proof proceeds in three fully independent stages. Stage I constructs the Asher-Ducci operator A on a separable Hilbert space H built from the prime-periodic lattice L 210 and establishes its fundamental properties from first principles. Stage II proves the Q = 1 Self-Adjointness Theorem: the Q-parameterised extension A(Q) is self-adjoint if and only if Q = 1, equivalently Re(s) = 1 2. This is the ignition condition, a structural necessity forced by the anti-Hermitian nature of the duality operator D away from Q = 1. Stage III connects the spectral theory of A to the Riemann xi-function via a heat-trace identity and Mellin transform, proving that every nontrivial zero of ζ(s) must be an eigenvalue of A(Q) Q=1 , which forces Re(s) = 1 2. Each logical gap is explicitly labelled as a Conjecture when it remains unproven; the main theorem is stated with full clarity about which steps are theorems and which are conditional. MSC 2020: 11M26 (Riemann zeta function), 47B25 (Self-adjoint operators), 11N05 (Distribution of primes), 47A10 (Spectrum, resolvent).

Many encryption strategies have been applied to ensure data confidentiality and improve cloud security. The most recent cryptosystems are based on homomorphic (HE), attribute-based (ABE), and hybrid encryption. However, most of them suffer from numerous drawbacks: high time consumption, encrypted message size, and some vulnerabilities. Hence, a secure method is highly required to get a satisfying security level while keeping the computational complexity reduced. This paper outlines a novel technique that combines elliptic curve cryptography (ECC) and blockchain technology. The data is first encoded using the elliptic curve integrated encryption scheme, then signed using signed using the elliptic curve digital signature algorithm, and finally confirmed by the blockchain network before being stored in the cloud. The performance evaluation results prove that the proposed system is highly resistant to man-in-the-middle and replay attacks and performs better than a set of existing solutions in terms of cryptography cost, encryption/decryption time, and algortithm complexity.

Among the various arithmetic operations required in implementing public key cryptographic algorithms, the elliptic curve point multiplication has probably received the maximum attention from the research community in the last decade. Many methods for efficient and secure implementation of point multiplication have been proposed. The efficiency of these methods mainly depends on the representation one uses for the scalar multiplier. In the current work we propose an efficient algorithm based on the so-called double-base number system. We introduce the new concept of double-base chains which, if manipulated with care, can significantly reduce the complexity of scalar multiplication on elliptic curves. Besides we have adopted some other measures to further reduce the operation count. Our algorithm compares favorably against classical and other similar approaches.

We present a large-scale computational study of prime valuations appearing in exponential sums of the form N = A x + B y , under the coprimality condition gcd(A, B) = 1 and exponents x, y ≥ 3. The experiments investigate whether such sums can exhibit valuation patterns compatible with perfect cubes. Across thousands of detected primitive primes, valuation residues modulo 3 show a strongly biased distribution, with no observed instances satisfying the synchronization required for cubic perfect powers. These findings support the hypothesis of a structural 3-adic obstruction governing coprime exponential sums.

This work presents a complete spectral and dynamical analysis of elliptic operators with a rank-one global coupling. Such operators arise naturally in constrained variational problems, where a single global degree of freedom induces nonlocal effects while leaving local force-based dynamics unchanged.

We derive the exact spectral structure of the operator, showing that the rank-one coupling affects only a single global mode while preserving self-adjointness and compactness of the resolvent. A codimension-one spectral phase transition is identified, characterized by the emergence of a zero eigenvalue and a controlled singularity of the resolvent at a critical coupling strength.

The analysis is extended to nonlinear settings via Lyapunov–Schmidt reduction, revealing that the global mode governs the bifurcation structure while higher modes remain slaved. A dynamical extension using center manifold theory establishes the effective low-dimensional dynamics near the critical regime. Finally, the framework is generalized to compact Riemannian manifolds, highlighting the role of geometry and topology in the global spectral behavior.

The results provide a rigorous and self-contained classification of operators with a single global degree of freedom and are of independent interest in spectral theory, nonlocal elliptic equations, and constrained variational analysis.

In this work, we are going to study the elliptic curves over the ring , precisely we will establish the short exact sequence which is split, and we will define the group extension Ea,b(A3) of by Ker(π3) which seems to be beneficial and interesting in ECC.

The Birch and Swinnerton-Dyer (BSD) Conjecture stands as one of the most profound and challenging problems in modern number theory, representing a bridge between the algebraic properties of elliptic curves and the analytic behavior of their associated L-functions. First proposed in the 1960s, it has resisted all conventional attempts at resolution, remaining a centerpiece of the Clay Mathematics Institute's Millennium Prize Problems. In this comprehensive and exhaustive report, we present a definitive resolution of the BSD Conjecture within the E8-Holographic Framework

Implementation and viability of Pairing-based cryptographic protocol for wireless sensor network is a challenging task to research community. Recently we have proposed an efficient One-pass Key Authentication protocol for wireless sensor network in random oracle model where we have presented pairing algorithm which is suited to implement for our protocol. The main difficulty of pairing based protocol is the incredible hardware and software constraint. In this paper, we suggest practical deployments of pairing on sensor nodes of the proposed protocol. These include the hardware architecture and various micro pairing implementation issues on wireless sensor networks. In this paper, we present the practical deployment of Micro-pairings on sensor nodes and hardware architecture of our proposed protocol.

This article proposes the technique to reduce the computation cost of scalar multiplication (SM) on elliptic curve for low processor constraints devices. Also the algorithm using the proposed technique secure against Side Channel Attack. There is no such algorithm is found to solve ECDLP in subexponential time. Equivalent strength of security is provided by ECC with shorter key size as compared to other Public key Cryptosystem. Therefore, ECC has acquired broad attention to research and development especially for low processor devices. Scalar multiplication on ECC operates on point doubling and adding. The secret the key stream consist of bit is '0' or '1'. While SM operate, doubling of points is followed by addition of points. This is a serious threat for low processor devices. For implementation, it is a challenge to develop algorithms that are not only efficient, but low cost.

Cryptosystems may be classified as either public key or private key. The system developed in this paper combines the protocols of these two types of cryptography. Techniques and characteristics of previously established cryptosystems are employed as well to produce a three-pass system that is more secure than these earlier systems. Each transmission in the correspondence protocol is at least triply encrypted. The system is also both polygraphic and polyalphabetic in nature. Furthermore, a digital signature enhances the system security by enabling the recipient of an encrypted message to authenticate the identity of the sender. Finally, a greater computational efficiency is achieved than exists in similar previously established systems due to recent results in number theory.

We give a general framework for uniform, constant-time oneand two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the xline or Kummer surface, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper "signed" output back on the curve or Jacobian. This extends the work of López and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to twodimensional scalar multiplication. Our results show that many existing fast pseudomultiplication implementations (hitherto limited to applications in Diffie-Hellman key exchange) can be wrapped with simple and efficient pre-and post-computations to yield competitive full scalar multiplication algorithms, ready for use in more general discrete logarithmbased cryptosystems, including signature schemes. This is especially interesting for genus 2, where Kummer surfaces can outperform comparable elliptic curve systems. As an example, we construct an instance of the Schnorr signature scheme driven by Kummer surface arithmetic. Input : Positive β-bit integers m = β-1 i=0 mi2 i and n = β-1 i=0 ni2 i with m β-1 and n β-1 not both zero, and elements 1, 0) : (M, (E, O)) ← (xQ, xDBLADD(xQ, xP , x⊖)) ; case (0, 1, 1) : (O, (E, M)) ← (x⊕, xDBLADD(xQ, x⊕, xP )) ; case (1, 0, 0) : (O, (E, M)) ← (x⊕, xDBLADD(xP , x⊕, xQ)) ; case (1, 0, 1) : (M, (E, O)) ← (xP , xDBLADD(xP , xQ, x⊖)) ; case (1, 1, 0) : (O, (E, M)) ← (x⊕, xDBLADD(x⊕, xP , xQ)) ; case (1, 1, 1) : (O, (E, M)) ← (x⊕, xDBLADD(x⊕, xQ, xP )) ; endsw for i ← β -2 down to 0 do

We derive a new formula for computing arbitrary odd-degree isogenies between elliptic curves in Montgomery form. The formula lends itself to a simple and compact algorithm that can efficiently compute any low odd-degree isogenies inside the supersingular isogeny Diffie-Hellman (SIDH) key exchange protocol. Our implementation of this algorithm shows that, beyond the commonly used 3-isogenies, there is a moderate degradation in relative performance of (2d + 1)-isogenies as d grows, but that larger values of d can now be used in practical SIDH implementations. We further show that the proposed algorithm can be used to both compute isogenies of curves and evaluate isogenies at points, unifying the two main types of functions needed for isogeny-based public-key cryptography. Together, these results open the door for practical SIDH on a much wider class of curves, and allow for simplified SIDH implementations that only need to call one general-purpose function inside the fundamental computation of the large degree secret isogenies. As an additional contribution, we also give new explicit formulas for 3-and 4-isogenies, and show that these give immediate speedups when substituted into pre-existing SIDH libraries.

Barreto-Lynn-Scott (BLS) curves are a stand-out candidate for implementing high-security pairings. This paper shows that particular choices of the pairing-friendly search parameter give rise to four subfamilies of BLS curves, all of which offer highly efficient and implementationfriendly pairing instantiations. Curves from these particular subfamilies are defined over prime fields that support very efficient towering options for the full extension field. The coefficients for a specific curve and its correct twist are automatically determined without any computational effort. The choice of an extremely sparse search parameter is immediately reflected by a highly efficient optimal ate Miller loop and final exponentiation. As a resource for implementors, we give a list with examples of implementation-friendly BLS curves through several high-security levels.