In this paper, we extend a theorem of Lu on the maximal order of a real quadratic field to a real quadratic order in general. We give a sufficient and necessary condition for a real quadratic order to have class number 1. As an application, we prove extensions of Yokoi's conjecture and Chowla's conjecture, which were proved by Biró.
We develop a graph-theoretic algorithm to compute the φ-Selmer group of the elliptic curve Eb:y2=x3+bx over Q(i), where b∈Z[i] and φ:Eb→E−4b is the degree 2 isogeny with kernel generated by (0,0). We associate to Eb a weighted, directed graph Gb, whose vertices are the odd Gaussian primes dividing b, and whose edge weights are determined by the quartic residue symbol between pairs of these primes. We then show that the local conditions defining the φ-Selmer group are encoded by a linear system over F2 involving a modified Laplacian of Gb.By applying our algorithm, we explicitly determine the φ-Selmer group of Eb when every odd prime divisor of b is inert, and we construct several infinite families of elliptic curves over Q(i) with Mordell–Weil rank 0.
For integers a,b≥2, putgn:=gcd(an−1,bn−1)(n≥1). We prove that (gn) satisfies a constant-coefficient linear recurrence if and only if a and b are multiplicatively dependent; the same conclusion holds if only a tail of (gn) is assumed to satisfy such a recurrence. More generally, if a and b are multiplicatively independent, then for every fixed shift N≥0, every integer sequence (Wn)n≥1 that satisfies a constant-coefficient linear recurrence and the pointwise divisibilitiesWn|an+N−1andWn|bn+N−1(n≥1) is eventually periodic. If (Wn) satisfies a recurrence with nonzero trailing coefficient, then it is periodic from the first term. In particular, every common integer linear divisibility-sequence factor of an−1 and bn−1, in the convention adopted here, is periodic. The proof combines the Bugeaud–Corvaja–Zannier bound with the structure of subexponential linear recurrences and a local valuation argument.
We classify Kummer–Witt Hom–Lie algebras attached to tamely ramified Kummer covers of degree n arising from twisted derivations on arithmetic schemes, in terms of the order of the twisting parameter. We obtain a trichotomy: the Hom–Lie algebra is either abelian, non-abelian but two-step solvable, or perfect. As an application, excluding the identity automorphism, we show that Larsson's conjecture holds precisely for even composite n and fails for odd composite n. We also give an operator-theoretic interpretation of the Hom–Lie bracket as an order-preserving twisted commutator, and relate its adjoint weights to the q-integers occurring in the one-variable q-de Rham differential.
Let K be a number field with class number 1. In this paper, we study the family of Severi-Brauer conics over K defined by equations z2-ax2-by2 = 0, where (a, b) E K & lowast;/(K & lowast;)2 & times; K & lowast;/(K & lowast;)2. We prove that, within this family, the density of the conics admitting at least one nontrivial K-rational point is zero. As an application, for an elliptic curve E/K satisfying E(K) D E[4], we show that the subset of H1(K, E)[2] consisting of classes with equal period and index has density zero. This is closely related to the question proposed in [3, Problem 2]. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
