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 G be a multiplicatively written finite group of order n. The Gao constant of the group G, denoted E(G), is defined as the smallest positive integer ℓ with the following property: for any given sequence g1⋅…⋅gℓ over G, there exist n distinct integers i1,…,in∈{1,…,ℓ} such that the product of gi1,…,gin, in some order, is the identity element of G. This constant originates from the celebrated additive theorem proved by Erdős, Ginzburg and Ziv in 1961, which amounts to proving E(G)≤2|G|−1 holds in case that G is abelian. It is also well-known that E(G)=2|G|−1 holds for all finite cyclic groups. In 2010, Gao and Li [J. Pure Appl. Algebra] conjectured that E(G)≤3|G|2 for every finite non-cyclic group G. In this paper, we confirm the conjecture for all non-cyclic groups G whose order is not divisible by four, and we also characterize the groups achieving the equality E(G)=3|G|2 as those with a cyclic subgroup of index two.
In this paper, we establish average bounds forSX(f,gj;α,β)=∑nλf(n)λgj(n)e(αnβ)ϕ(nX) over the spectral family, where f is a fixed Hecke cusp form (Maass or holomorphic) for SL2(Z), and {gj} is an orthonormal basis of Hecke–Maass cusp forms with spectral parameters tj, 0<β≤1 and α≠0 are fixed real numbers, ϕ is a smooth function with compact support (1,2). We derive estimates that improve upon the resonance barrier on average. This work provides insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec, Luo, and Sarnak.
