We prove three results on monogenity of non-cyclic abelian extension fields over the rationals Q. The first one is that for any odd prime cyclotomic field kp=Q(ζp) and any quadratic field k=Q(ℓ√) of discriminant ℓ prime to p, the non-cyclic abelian field kpk is non-monogenic if ℓ<−4 with ℓ∤(p±1) or ℓ>4. The second is that if k+n=Q(ζn+ζ−1n) is the maximal real subfield of an nth cyclotomic field with n≥5, n/≡2(mod4) and if k=Q(−m−−−√) is an imaginary quadratic field of discriminant −m<−4 with (m,n)=1, then the composite field k+nk is non-monogenic. The third is that a maximal imaginary subfield Q(ζq−ζ−1q) of a cyclotomic field kq with q>4, q≡0(mod4) is monogenic.

Let p be an odd prime number and k an imaginary quadratic field in which p splits into 𝔭 and 𝔭 * . Then there exists a uniquely defined ℤ p -extension N ∞ / k such that the prime ideal 𝔭 * does not ramify. For a finite extension K / k , we call K ∞ = K N ∞ the split prime ℤ p -extension corresponding to 𝔭 . We prove an analogue of Kida’s formula for the split prime ℤ p -extensions. As an application, we apply this formula to 𝔭 -ramified Iwasawa modules and determine the isomorphism classes of unramified Iwasawa modules associated to ℤ p -extensions over k .

This paper applies the modular approach to obtain effectively computable bounds for Fermat-type equations over number fields, while also discussing the differences and obstructions that arise when considering such equations over totally real versus totally complex number fields. We use these techniques to study the generalized Fermat equation [Formula: see text] over quadratic fields [Formula: see text] of class number one. Extending the results of Freitas&Siksek and Turcas, we show that when [Formula: see text], there is an effective and explicit bound, depending on the field [Formula: see text], such that the latter equation does not have certain types of special solutions. We also discuss, for [Formula: see text], the solutions of a variant of the above equation. Our results over imaginary quadratic fields are conjectural. Serre’s modularity conjecture and an analogue of Eichler-Shimura over totally complex fields are assumed.

Abstract This paper proves that the integrality of algebraic Witt vectors over imaginary quadratic fields is decidable; based on this, some related problems are also discussed.

We all should learn in our abstract algebra classes that every Euclidean domain R has a minimal Euclidean function, ϕ R . Our short history starts with their introduction via Motzkin’s Lemma and moves onto Lenstra’s categorization of Euclidean functions in imaginary quadratic number fields. We examine computing minimal Euclidean functions in these fields, and apply them in short, easy proofs of standard results. Using the author’s simple formula for ϕ Z [ i ] , the only explicitly computable minimal Euclidean function the author knows for a number field other than Q , we apply the pre-images’ geometry to give a new elementary proof affirming Lenstra’s algebraic description of these sets. Figures illustrate the definitions and arguments.

Let D>3, D≡3(mod4) be a prime, and let C be an ideal class in the field Q(−D−−−√). We give a new proof that p(D,C), the smallest norm of a split prime p∈C, satisfies p(D,C)≪DL for some absolute constant L. Our proof is sieve-theoretic. In particular, this allows us to avoid the use of log-free zero-density estimates (for class group L-functions) and the repulsion properties of exceptional zeros, two crucial inputs to previous proofs of this result.

We call an order O in a quadratic field K odd (resp. even) if its discriminant is an odd (resp. even) integer. We call an elliptic curve E over $${\mathbb {C}}$$ with CM odd (resp. even) if its endomorphism ring $$\textrm{End}(E)$$ is an odd (resp. even) order in the imaginary quadratic field $$\textrm{End}(E)\otimes {\mathbb {Q}}$$ . Suppose that $$j(E)\in {\mathbb {R}}$$ and let us consider the set $$\mathscr {J}({\mathbb {R}},E)$$ of all $$j(E^{\prime })$$ where $$E^{\prime }$$ is any elliptic curve that enjoys the following properties: . $$\bullet$$ $$E^{\prime }$$ is isogenous to E; $$\bullet$$ $$j(E^{\prime })\in {\mathbb {R}}$$ ; $$\bullet$$ $$E^{\prime }$$ has the same parity as E. We prove that the closure of $$\mathscr {J}({\mathbb {R}},E)$$ in $${\mathbb {R}}$$ is the closed semi-infinite interval $$(-\infty ,1728]$$ (resp. the whole $${\mathbb {R}}$$ ) if E is odd (resp. even). This paper was inspired by a question of Jean-Louis Colliot-Thélène and Alena Pirutka about the distribution of j-invariants of certain elliptic curves of CM type.

Abstract We prove an André–Oort-type result for a family of hypersurfaces in ${\mathbb{C}}^n$ that is both uniform and effective. Let $K_*$ denote the single exceptional imaginary quadratic field which occurs in the Siegel–Tatuzawa lower bound for the class number. We prove that, for $m, n \in {\mathbb{Z}}_{\gt0}$ , there exists an effective constant $c(m, n)\gt0$ with the following property: if pairwise distinct singular moduli $x_1, \ldots, x_n$ with respective discriminants $\Delta_1, \ldots, \Delta_n$ are such that $a_1 x_1^m + \cdots + a_n x_n^m \in {\mathbb{Q}}$ for some $a_1, \ldots, a_n \in {\mathbb{Q}} \setminus \{0\}$ and $\# \{ \Delta_i \;:\; {\mathbb{Q}}(\sqrt{\Delta_i}) = K_*\} \leq 1$ , then $\max_i \lvert \Delta_i \rvert \leq c(m, n)$ . In addition, we prove an unconditional and completely explicit version of this result when $(m, n) = (1, 3)$ and thereby determine all the triples $(x_1, x_2, x_3)$ of singular moduli such that $a_1 x_1 + a_2 x_2 + a_3 x_3 \in {\mathbb{Q}}$ for some $a_1, a_2, a_3 \in {\mathbb{Q}} \setminus \{0\}$ .

Microtubules are cylindrical protein polymers that organize the cytoskeleton and play essential roles in intracellular transport, cell division, and possibly cognition. Their highly ordered, quasi-crystalline lattice of tubulin dimers, notably tryptophan residues, endows them with a rich topological and arithmetic structure, making them natural candidates for supporting coherent excitations at optical and terahertz frequencies. The Penrose–Hameroff Orch OR theory proposes that such coherences could couple to gravitationally induced state reduction, forming the quantum substrate of conscious events. Although controversial, recent analyses of dipolar coupling, stochastic resonance, and structured noise in biological media suggest that microtubular assemblies may indeed host transient quantum correlations that persist over biologically relevant timescales. In this work, we build upon two complementary approaches: the parametric resonance model of Nishiyama et al. and our arithmetic–geometric framework, both recently developed in Quantum Reports. We unify these perspectives by describing microtubules as rectangular lattices governed by the imaginary quadratic field Q(i), within which nonlinear dipolar oscillations undergo stochastic parametric amplification. Quantization of the resonant modes follows Gaussian norms N=p2+q2, linking the optical and geometric properties of microtubules to the arithmetic structure of Q(i). We further connect these discrete resonances to the derivative of the elliptic L-function, L′(E,1), which acts as an arithmetic free energy and defines the scaling between modular invariants and measurable biological ratios. In the appended adelic extension, this framework is shown to merge naturally with the Bost–Connes and Connes–Marcolli systems, where the norm character on the ideles couples to the Hecke character of an elliptic curve to form a unified adelic partition function. The resulting arithmetic–elliptic resonance model provides a coherent bridge between number theory, topological quantum phases, and biological structure, suggesting that consciousness, as envisioned in the Orch OR theory, may emerge from resonant processes organized by deep arithmetic symmetries of space, time, and matter.

Caraiani and Newton have proven that if F is an imaginary quadratic number field such that X 0 ( 15 ) has rank 0 over F , then every elliptic curve over F is modular. This paper is concerned with the quadratic fields F = Q ( − p ) for a prime number p . We give explicit conditions on p under which the rank is 0, and prove that these conditions are satisfied for 87.5% of the primes for which the rank is expected to be even based on the parity conjecture. We also show these conditions are satisfied if and only if rank 0 follows from a 4-descent over Q on the quadratic twist X 0 ( 15 ) − p . To prove this, we perform two consecutive 2-descents and prove this gives rank bounds equivalent to those obtained from a 4-descent using visualisation techniques for . In fact we prove a more general connection between higher descents for elliptic curves which seems interesting in its own right.

Let d be a fixed positive integer with d > 3 is square-free, and let h ( − d ) denote the class number of the imaginary quadratic field ℚ ( − d ) . Further, let p and q be odd primes such that p > 3 , p ≠ q and p ∤ h ( − d ) . In this paper, we give a sufficient and necessary condition for the Lebesgue-Nagell equation (∗) d x 2 + p 2 m q 2 n = 4 y p to have positive integer solutions ( x , y , m , n ) with gcd ⁡ ( x , y ) = 1 . It can be seen from this condition that if q ≢ ± 1 ( 𝑚𝑜𝑑 2 p ) , then (∗) has no positive integer solutions ( x , y , m , n ) with gcd ⁡ ( x , y ) = 1 .

ABSTRACT Rapid technological advancements have made it essential to develop new approaches for designing dynamic substitution boxes (S‐boxes) to secure valuable data. These S‐boxes are controlled by input parameters to acquire the desired cryptographic strength. For this purpose, the S‐box generators with favorable cryptographic strength are intensively developed; however, they suffer from high computational overhead to generate a large number of dynamic S‐boxes with the desired degree of security, limiting their capability to efficiently generate the required number of secure S‐boxes. The key findings of this study are counting, generation of optimal and dynamic S‐boxes, and low computational time. This is done by unconventionally taking the Montgomery elliptic curve, imaginary quadratic field, and defining an ordering on the points of the underlying curve. Then we apply a linear fractional transformation on the ordered points to enhance the required S‐boxes and their security against the key‐related attacks. On detailed analysis, the presented algorithm shows high capability to generate dynamic S‐boxes, providing high security with optimal nonlinearity in minimal time and energy consumption.

Abstract Let F be a totally real field. Let $\mathsf {A}$ be a simple modular self-dual abelian variety defined over F . We study the growth of the corank of Selmer groups of $\mathsf {A}$ over $\mathbb {Z}_p$ -extensions of a complex multiplication (CM) extension of F . We propose an extension of Mazur’s growth number conjecture for elliptic curves to this new setting. We provide evidence supporting an affirmative answer by studying special cases of this problem, generalising previous results on elliptic curves and imaginary quadratic fields.

We construct a new Euler system for the Galois representation V_{f,\chi} attached to a newform f of weight 2r\geq 2 twisted by an anticyclotomic Hecke character \chi . The Euler system is anticyclotomic in the sense of Jetchev–Nekovář–Skinner. We then show some arithmetic applications of the constructed Euler system, including new results on the Bloch–Kato conjecture in ranks zero and one, and a divisibility towards the Iwasawa–Greenberg main conjecture for V_{f,\chi} . In particular, in the case where the base-change of f to our imaginary quadratic field has root number +1 and \chi has higher weight (which implies that the complex L -function L(V_{f,\chi},s) vanishes at the center), our results show that the Bloch–Kato Selmer group of V_{f,\chi} is nonzero, as predicted by the Bloch–Kato conjecture; and if in addition a certain distinguished class \kappa_{f,\chi} is nonzero, then the Selmer group is one-dimensional. Such applications to the Bloch–Kato conjecture for V_{f,\chi} were left wide open by the earlier approaches using Heegner cycles and/or Beilinson–Flach elements. Our construction is based instead on a generalization of the Gross–Kudla–Schoen diagonal cycles.

Abstract Let K be an imaginary quadratic field, and let $E/\mathbb {Q}$ be an elliptic curve with complex multiplication by $\mathcal {O}_K$ . Let $K_\infty /K$ be the anticyclotomic $\mathbb {Z}_p$ -extension of K and $K_n$ be the intermediate layers. Under additional assumptions on Kobayashi’s signed Selmer groups, we prove an asymptotic formula for .

We introduce chain modular units: products of Siegel functions arranged along cyclic chains of ℓ-isogenies between CM elliptic curves of type Of. Their CM values take place in the ray class field Kfℓn, and we construct a finite, explicit family that generates, modulo roots of unity, a subgroup of finite index in OKfℓn×. We describe the Galois action via the ideal class action on isogeny chains, give effective criteria for multiplicative independence using Baker-Wustholz bounds, and determine the eventual index in the ℓ-power ray class tower. Computations for D = -7 and D = -11 confirm the theoretical predictions. Our approach exploits the combinatorics of isogeny chains to control conductor growth and produce many independent units.

Abstract Let be an elliptic curve defined over , and let be an imaginary quadratic field. Consider an odd prime at which has good supersingular reduction with and which is inert in . Under the assumption that the signed Selmer groups are cotorsion modules over the corresponding Iwasawa algebra, we prove that the Mordell–Weil ranks of are bounded over any subextensions of the anticyclotomic ‐extension of . Additionally, we provide an asymptotic formula for the growth of the ‐parts of the Tate–Shafarevich groups of over these extensions.

Let E be an elliptic curve over a number field L and for a finite set S of primes, let ρ E,S :Gal(L ¯/L)→GL 2 (ℤ S ) be the S-adic Galois representation. If L∩ℚ(ζ n )=ℚ for all positive integers n whose prime factors are in S, then detρ E,S :Gal(L ¯/L)→ℤ S × is surjective. We say that a finite index subgroup H⊆GL 2 (ℤ S ) is minimal if det:H→ℤ S × is surjective, but det:K→ℤ S × is not surjective for any proper closed subgroup K of H. We show that there are no minimal subgroups of GL 2 (ℤ S ) unless S={2}, while minimal subgroups of GL 2 (ℤ 2 ) are plentiful. We give models for all the genus 0 modular curves associated to minimal subgroups of GL 2 (ℤ 2 ), and construct an infinite family of elliptic curves over imaginary quadratic fields with bad reduction only at 2 and with minimal 2-adic image.

Explicit count of integral ideals of an imaginary quadratic field

Abstract Using an explicit Eichler–Shimura–Harder isomorphism, we establish the analog of Manin’s rationality theorem for Bianchi periods and hence special values of L -functions of Bianchi cusp forms. This gives a new short proof of a result of Hida in the case of Euclidean imaginary quadratic fields. In particular, we give an explicit proof using the space of Bianchi period polynomials constructed by Karabulut and describe the action of Hecke operators.