We study quantum invariants of projective varieties over number fields. Namely, an explicit formula for a functor $\mathscr{Q}$ on such varieties is proved. The case of abelian varieties with complex multiplication is treated in detail.

Admissible groups over number fields

A general approach to multistability analysis for fuzzy multidimensional-valued NNs with memristor.

Unit lattices of D4-quartic number fields with signature (2,1)

Corresponding Abelian extensions of integrally equivalent number fields

Approximation Theorems for Classifying Stacks over Number Fields

The study of special values of adjoint L-functions and congruence ideals is gradually becoming a classical theme in number theory, driven by the Bloch-Kato conjecture and generalisations of Wiles-Lenstra's numerical criterion. In this paper, we relate to the congruence ideals for cohomological cuspidal automorphic representations of over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint L-functions. For CM fields, using the existence of Galois representations, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of . This can be viewed as partial progress on the Bloch-Kato conjecture. The main technical ingredients are a careful study of the cohomology associated with the locally symmetric space of , its relation to automorphic representations, and the establishment of some algebraic properties of the congruence ideals. We anticipate that the methods developed here will find further applications in related problems, particularly in the study of congruence modules and their relation to the arithmetic of automorphic forms.

We provide an explicit formula for the dimension of the $*$-congruence orbits and bundles of Hermitian matrix pencils over the field of complex numbers. The formula is given in terms of the sizes of the canonical blocks in the Hermitian Kronecker canonical form of the pencils. This extends the formulas provided only for the generic orbits and bundles in a previous work.

In this paper, for a positive integer n, we compute the number of all n degree representations for a dihedral group G of order 2m, m ≥ 3. We evaluate dimensions of the spaces of invariant bilinear forms corresponding to each of the representation over the field of complex numbers ℂ (in fact over a number field consisting of a primitive 4mth root of unity). We also, assure that the same results hold equally good when considered over a field of characteristic ≡ 1 (mod 4m}. We explicitly discuss the existence of a non-degenerate invariant bilinear form.

Let f t f_t be a one-parameter family of rational maps defined over a number field K K . We show that for all t t outside of a set of natural density zero, every K K -rational preperiodic point of f t f_t is the specialization of some K ( T ) K(T) -rational preperiodic point of f f . Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of K K -rational preperiodic points of f f , giving some examples where this holds unconditionally. To illustrate the theory, we give new estimates on the average number of preperiodic points for the quadratic family f t ( z ) = z 2 + t f_t(z) = z^2 + t over the field of rational numbers.

In this paper, we study arithmetic progressions of squares over quadratic extensions of number fields. Using a method inspired by an approach of Mordell, we characterize such progressions as quadratic points on a genus [Formula: see text] curve. Specifically, we determine the set of [Formula: see text]-quadratic points on this curve under certain conditions on the base field [Formula: see text]. Our main results rely on the algebraic properties of specific elliptic curves after performing a base change to suitable number fields. As a consequence, we establish that, under appropriate assumptions, any non-elementary arithmetic progression of five or six squares properly defined over a quadratic extension of [Formula: see text] must be of a specific form. Moreover, we prove the non-existence of such progressions of length greater than six under these assumptions.

In this paper, a new criterion is given to determine the [Formula: see text]-rationality of some complex cubic number fields in terms of [Formula: see text]-divisibility of certain terms of a third-order recurrence sequence, several illustrated examples are constructed, the relations between generalized [Formula: see text]-conjecture and the [Formula: see text]-rationality are discussed, from which some explicit fields satisfying Greenberg’s Generalized Conjecture (GGC) are obtained.

Abstract The notion of $\theta $ -congruent numbers generalises the classical congruent number problem. A positive integer n is $\theta $ -congruent if it is the area of a rational triangle with an angle $\theta $ whose cosine is rational. Das and Saikia [‘On $\theta $ -congruent numbers over real number fields’, Bull. Aust. Math. Soc. 103 (2) (2021), 218–229] established criteria for numbers to be $\theta $ -congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $\theta $ -congruent and properly $\theta $ -congruent numbers. In this work, we provide complete answers to those questions. Indeed, we remove a technical assumption from their result on fields with degrees coprime to six, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree six and examine the exceptional cases $n=1, 2, 3$ and $6$ .

On Galois groups over tamely ramified cyclotomic extensions of algebraic number fields

On the residues and Euler–Kronecker constants of cyclic number fields

The electronic Schrödinger equation describes the motion of a finite number of electrons under Coulomb interaction forces in a field of a finite number of clamped nuclei. It is proved that its solutions for negative eigenvalues, below the essential spectrum, lie in the spectral Barron spaces $\mathcal{B}^s$ for $s<1$. The example of the hydrogen ground state shows that this result cannot be improved.

Let g ( x ) ∈ Z [ x ] be a monic irreducible polynomial of degree n. We say that g ( x ) is monogenic if, for a root θ of g ( x ) , the set { 1 , θ , … , θ n − 1 } forms an integral basis of the ring of integers Z K of the number field K = Q ( θ ) . Consider f ( x ) = x n + a x 2 + dx + b ∈ Z [ x ] with d 2 = 4 ab , and g ( x ) = x m + c ∈ Z [ x ] , where n ≥ 3 and m ≥ 1 , such that ( f ° g ) ( x ) = ( x m + c ) n + a ( x m + c ) 2 + d ( x m + c ) + b is irreducible over Q . In this study, we establish necessary and sufficient conditions involving a, b, c, d, m, n for the polynomial ( f ° g ) ( x ) to be monogenic. Additionally, we examine the nature of solutions to specific differential equations, and present a class of monogenic polynomials with non-square-free discriminants as an application.

Let K/Q be a Galois extension of number fields. We study the ideal classes of primes p of K of residue degree bigger than 1 in the class group of K. In particular, we explore those extensions K/Q for which there exists an integer f>1 such that the ideal classes of primes p of K of residue degree f generate the full class group of K. We show that there are many such fields. Then we use this approach to obtain information on the class group of K, like the rank of the ℓ-torsion subgroup of the class group, factors of the class number, fields with class group of certain exponents, and even structure of the class group in some cases. Moreover, such f can be used to construct annihilators of the class groups. In fact, for any extension K/F (even non-abelian), if the class group of K is generated by primes of relative degree f for the extension K/F and the class group of F is trivial, this method can be used to construct ‘relative’ annihilators.

Abstract A Fuchsian group $\Gamma $ has a modular embedding if its adjoint trace field is a totally real number field and every unbounded Galois conjugate $\Gamma ^\sigma $ comes equipped with a holomorphic (or conjugate holomorphic) map ${\phi ^\sigma : \mathbb{B}^{1} \to \mathbb{B}^{1}}$ intertwining the actions of $\Gamma $ and $\Gamma ^\sigma $ on the Poincaré disk $\mathbb{B}^{1}$. This paper provides the first cocompact nonarithmetic Fuchsian groups with a modular embedding that are not commensurable with a triangle group. The main result, proved using period domains, is that any immersed totally geodesic complex curve on a complex hyperbolic $2$-orbifold has a modular embedding. Another consequence is arithmeticity of totally geodesic curves on finite-volume complex hyperbolic surfaces that are commensurable with quotients of $\mathbb{B}^{1}$ by the group generated by reflections in quadrilaterals satisfying certain angle conditions.

Natural convective in enclosures are very important topics in thermal engineering because they find versatile industrial applications. An internal circular cylinder's vertical position and heat source on fluid flow and heat transfer in a triangular cavity are investigated. Numerical simulations were carried out to analyze variations in the average Nusselt number, streamline topology, temperature distribution, and velocity fields by using ANSYS Fluent. The results show that the Nusselt number rises from approximately 0.91–0.94 at lower positions (Y = 0.1–0.3) to a maximum of about 0.97 near Y = 0.4 driven by intensified thermal gradients and buoyancy-induced circulation. Within the upper-to-mid region (Y = 0.2–0.4) the formation of large adjacent vortices enhances macro-scale mixing, resulting in nearly a 4% improvement in heat transfer relative to the reference case. At mid-level positions (Y = 0.4–0.6) quasi-steady symmetric circulations are sustained, maintaining effective convection with Nu values of 0.95–0.97. In contrast, at higher locations (Y = 0.7–0.9), the weakening of vortex strength leads to flow stagnation and localized deterioration in heat transfer, reducing Nu to about 0.90–0.92. Overall, the findings underscore the critical importance of internal component placement in improving natural cooling performance, and further suggest that the most efficient thermal behavior is achieved when the cylinder and heat source are positioned within 0.2 < Y < 0.4, offering practical guidance for optimizing the thermal design of triangular enclosures.