In this note, we define the notion of Tate–Shafarevich group and Selmer group of the Chow group of zero cycles of degree zero of an abelian variety defined over a global function field and prove that the Selmer group is finite.

In this note, we connect the n-torsions of the Picard group of an elliptic surface to the n-divisibility of the class group of torsion fields for a given integer n>1. We also connect the n-divisibility of the Selmer group to that of the class group of torsion fields.

We present two distinct families of imaginary biquadratic fields, each of which contains infinitely many members, with each member having large class groups. Construction of the first family involves elliptic curves and their quadratic twists, whereas to find the other family, we use a combination of elliptic and hyperelliptic curves. Two main results are used, one from Soleng [13] and the other from Banerjee and Hoque [3].

In this short note we prove that an involution on certain examples of surfaces of general type with pg=0, acts as identity on the Chow group of zero cycles of the relevant surface. In particular we consider examples of such surfaces when the quotient is an Enriques surface and show that the Bloch conjecture holds for such surfaces.

A secret image transmission technique has been put forward in this paper using (3,3) visual cryptographic shares generated from secret image that are fabricated into frames in Graphics Interchange Format (GIFs) to prevent any intruder from knowing the secret contained in the GIFs. The (3, 3) Visual Cryptography technique creates shares from a binary secret image. Using GIFs as communication hosts, each share has been embedded into the Least Significant Bit (LSB) of the pixels of any single meaningful frame of GIFs. The shares obtained from the corresponding meaningful frames of the GIFs, during decoding, are combined to create the authenticated image. The combination of visual cryptography (VC) technique and steganographic principles ensures not only the secure distribution of shares but also adds an extra layer of protection through the integration of the GIF format.

This paper examines unbiased Non-Data-Aided (NDA) Signal-to-Noise Ratio (SNR) estimation for hyper-cubic modulated signals in Additive White Rayleigh Noise (AWRN) channels. We investigate the Crame’r-Rao Lower Bound (CRLB) derivation, noting sensitivity to hyper-cubic constellation dimensions at low SNR. At higher SNR, we identify a unified behavior between multi-order square-QAM and hyper-cubic constellations, yielding a closed-form CRLB expression. Higher dimensions in hyper-cubic constellations increase the CRLB, mitigated by augmenting observations for improved precision. This study offers insights into optimizing SNR estimation precision across signal environments.

Let S be a certain affine algebraic surface over Q such that it admits a regular map to A2/Q. We show that any non-trivial torsion element in the Chow group CH1(S) can be pulled back to ideal classes of quadratic fields whose order can be made as large as possible. This gives an affirmative answer to a question analogous to one raised by Agboola and Pappas, in the case of certain affine algebraic surfaces. Spreading out S over Z and for a closed point P∈A2/Z, we show that the cardinality of a subgroup of the Picard group of the fiber SP remains unchanged when P varies over a Zariski open subset in A2/Z. We also show by constructing an element of odd order n≥3 in the class group of certain imaginary quadratic fields that the Picard group of SP has a subgroup isomorphic to Z/nZ.

Torsion Point Cryptography is an innovative encryption technique that employs the mathematical properties of elliptic curves to ensure maximum security. This research paper delves into the technical intricacies of this approach, elucidating how it operates and why it is so effective. The authors delve into various types of torsion points that can be employed in this cryptography method, including rational and non-rational points, and discuss the benefits and constraints associated with each. They also conduct a thorough analysis of the security implications of torsion point cryptography, comparing it to other commonly used cryptosystems and highlighting its potential vulnerabilities. In conclusion, this paper offers an exhaustive introduction to torsion point cryptography and its real-world applications for secure communication.

In this exposition we understand when the natural map from the two-fold self product of the Chow variety parametrizing codimension p cycles on a smooth projective variety X to the Chow group CH p(X) of degree zero cycles is surjective. We derive some consequences when the map is surjective.

Assume that we have a fibration of smooth projective varieties X → S over a surface S such that X is of dimension four and that the geometric generic fiber has finite-dimensional motive and the first étale cohomology of the geometric generic fiber with respect to l coefficients is zero and the second étale cohomology is spanned by divisors. We prove that then A3(X), the group of codimension three algebraically trivial cycles modulo rational equivalence, is dominated by finitely many copies of A0(S); this means that there exist finitely many correspondences Γi on S × X such that ςi Γi is surjective from A2(S) to A3(X).

With a homological Lefschetz conjecture in mind, we prove the injectivity of the pushforward morphism on low-dimensional rational Chow groups, induced by the closed embedding of an ample divisor, namely, the Theta divisor inside the Jacobian variety (Formula presented.). Here, C is a smooth irreducible complex projective curve.

Consider the Fano surface of a conic bundle embedded in P5. Let i denote the natural involution acting on this surface. In this note we provide an obstruction to the identity action of the involution on the group of algebraically trivial zero cycles modulo rational equivalence on the surface.

Let K be an uncountable algebraically closed field of characteristic , and let be a smooth projective connected variety of dimension , embedded into over . Let be a hyperplane section of , and let and be the groups of algebraically trivial algebraic cycles of codimension and modulo rational equivalence on and , respectively. Assume that, whenever is smooth, the group is regularly parametrized by an abelian variety and coincides with the subgroup of degree classes in the Chow group. We prove that the kernel of the push-forward homomorphism from is the union of a countable collection of shifts of a certain abelian subvariety inside . For a very general hyperplane section whose tangent space is the group of vanishing cycles.

In this paper, we study two topics. One is the divisibility problem of class groups of quadratic number fields and its connections to algebraic geometry. The other is the construction of Selmer group and Tate-Shafarevich group for an abelian variety defined over a number field.

In this paper, we generalize the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of SymmC into SymnC for m ≤ n, where C is a smooth projective curve, to symmetric powers of a smooth projective variety of higher dimension. We also prove the analog of this theorem for product of symmetric powers of smooth projective varieties. As an application we prove the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of self-product of theta divisor into the self-product of the Jacobian of a smooth projective curve.