Abstract Given a cubic curve C over a number field, we consider the K3 surface upper Y Subscript upper C Y C $Y_C$ constructed as the minimal desingularisation of the quotient of upper C times upper C C × C $C \times C$ by an automorphism of order 3. We relate the transcendental Brauer groups of upper Y Subscript upper C Y C $Y_C$ and upper C times upper C C × C $C \times C$ , allowing us to explicitly compute the former group in the case of a diagonal cubic curve defined over double struck upper Q Q $\mathbb{Q}$ . We obtain conjectural insight on the existence of Galois cubic points over double struck upper Q Q $\mathbb{Q}$ for everywhere locally soluble diagonal cubic curves.
Abstract We study the geometry of the complex hyperbolic triangle group normal upper Delta Subscript 4 comma normal infinity comma normal infinity semicolon normal infinity Δ 4 , ∞ , ∞ ; ∞ $\Delta_{4,\infty,\infty;\infty}$ acting on the complex hyperbolic plane bold upper H Subscript double struck upper C Superscript 2 H C 2 $\mathbf{H}^2_{\mathbb C}$ and show that the 3-manifold at infinity of normal upper Delta Subscript 4 comma normal infinity comma normal infinity semicolon normal infinity Δ 4 , ∞ , ∞ ; ∞ $\Delta_{4,\infty,\infty;\infty}$ is the complement of the chain link 8 Subscript 1 Superscript 4 8 1 4 $8^4_1$ in the 3-sphere. In particular, the four-cusped hyperbolic 3-manifold upper S cubed minus 8 Subscript 1 Superscript 4 S 3 − 8 1 4 $S^3-8^4_1$ admits a spherical CR-uniformisation. Since the ideal boundary of the Ford domain is an infinite-genus handlebody, we introduce a new technique to construct embedded topological disks that cap off the genus, ensuring their boundaries are compatible with the side-pairings. This surgical approach facilitates the determination of the fundamental group of the manifold at infinity and simplifies the computation compared to [ 14 ]. Based on this result and previous studies, we conjecture that the chain link upper C Subscript p C p $C_p$ serves as an ancestor of the 3-manifold at infinity for the critical complex hyperbolic triangle group normal upper Delta Subscript p comma q comma r semicolon normal infinity Δ p , q , r ; ∞ $\Delta_{p,q,r;\infty}$ , where 3 less than or equals p less than or equals 9 3 ≤ p ≤ 9 $3 \leq p \leq 9$ .
Abstract Let v ( k ) be the smallest integer larger than 1 that does not occur among the denominators in any identity of the form 1 = 1 n 1 + ⋯ + 1 n k , \begin{align*}1=\frac1{n_1}+\cdots+\frac1{n_k},\end{align*} where 1 less than or equals n 1 less than midline horizontal ellipsis less than n Subscript k 1 ≤ n 1 < ⋯ < n k $1 \leq n_1\lt\cdots\lt n_k$ are pairwise distinct integers. In their 1980 monograph, Erdős and Graham asked for quantitative estimates on the growth of v ( k ) and suggested the lower bound v left parenthesis k right parenthesis much greater than k factorial v ( k ) ≫ k ! $v(k)\gg k!$ . In this paper we give the first known improvement and show that there exists an absolute constant c greater than 0 c > 0 $c\gt0$ such that the inequality v ( k ) ≥ e c k 2 \begin{align*}v(k)\ge e^{c k^2}\end{align*} holds for all positive integers k .
Abstract In the homology cobordism group normal upper Theta Subscript double struck upper Z Superscript 3 Θ Z 3 $\Theta_\mathbb{Z}^3$ , it is not known if there are non-trivial linear dependences between Seifert fibered spheres. Based on involutive Heegaard Floer theory, Hendricks, Manolescu and Zemke introduced the local equivalence group German upper I I $\mathfrak{I}$ along with the homomorphism h colon normal upper Theta Subscript double struck upper Z Superscript 3 Baseline right arrow German upper I h : Θ Z 3 → I $h\;:\;\Theta_\mathbb{Z}^3 \rightarrow \mathfrak{I}$ . Using the work of Dai and Stoffregen, one can find non-trivial linear dependences between the images of Seifert fibered spheres under h . Therefore, it is interesting to ask if such dependences in German upper I I $\mathfrak{I}$ originate from normal upper Theta Subscript double struck upper Z Superscript 3 Θ Z 3 $\Theta_\mathbb{Z}^3$ . In this paper, by employing the r Subscript s r s $r_s$ -invariants from the filtered instanton Floer homology developed by Nozaki, Sato and Taniguchi, we provide certain conditions to guarantee that such relations are not realised even in the rational homology cobordism group. We also discuss the local equivalence class of the normal upper P normal i normal n left parenthesis 2 right parenthesis P i n ( 2 ) $\operatorname{Pin(2)}$ -equivariant Seiberg–Witten Floer stable homotopy type.
Abstract Let normal upper P normal upper G left parenthesis k minus 1 comma q right parenthesis P G ( k − 1 , q ) $\mathrm{PG}(k-1,q)$ be the left parenthesis k minus 1 right parenthesis ( k − 1 ) $(k-1)$ -dimensional projective space over the finite field double struck upper F Subscript q F q ${\mathbb F}_q$ . An arc in normal upper P normal upper G left parenthesis k minus 1 comma q right parenthesis P G ( k − 1 , q ) $\mathrm{PG}(k-1,q)$ is a set of points with the property that any k of them span the entire space. The notion of a pseudo-arc generalises that of an arc by replacing points with higher-dimensional subspaces. Constructions of pseudo-arcs can be obtained from arcs defined over extension fields; such pseudo-arcs are necessarily Desarguesian , in the sense that all their elements belong to a Desarguesian spread. In contrast, genuinely non-Desarguesian pseudo-arcs are far less understood and have previously been known only in a few sporadic cases. In this paper, we introduce a new infinite family of non-Desarguesian pseudo-arcs consisting of left parenthesis h minus 1 right parenthesis ( h − 1 ) $(h-1)$ -dimensional projective subspaces of normal upper P normal upper G left parenthesis h k minus 1 comma q right parenthesis P G ( h k − 1 , q ) $\mathrm{PG}(hk-1,q)$ based on the imaginary spaces of a normal rational curve. We determine the size of the constructed pseudo-arcs explicitly and show that, by adding suitable osculating spaces of a normal rational curve defined over a subgeometry, we obtain pseudo-arcs of size upper O left parenthesis q Superscript h Baseline right parenthesis O ( q h ) $O(q^h)$ . As q grows, these sizes have the same asymptotic order as the classical upper bound for pseudo-arcs established by J. A. Thas. We further investigate the interaction between these new pseudo-arcs and quadrics. While Desarguesian pseudo-arcs from normal rational curve are complete intersections of quadrics, we prove that the new pseudo-arcs are not contained in any quadric of the ambient projective space. Finally, we translate our geometric results into coding theory. We show that the new pseudo-arcs correspond precisely to recently introduced families of additive MDS codes arising from a polynomial framework. As a consequence of their non-Desarguesian nature, we prove that these codes are not equivalent to linear MDS codes. Moreover, the extension of pseudo-arcs via osculating spaces yields longer families of additive MDS codes.
