We consider the family of complex algebraic $\mathrm{K}3$ surfaces $𝒳$ with Picard lattice containing the unimodular lattice $H\oplus E_7(-1)\oplus E_7(-1)$. The surface $𝒳$ admits a birational model isomorphic to a quartic hypersurface that generalizes the Inose quartic. We prove that a general member of this family admits exactly four inequivalent Jacobian elliptic fibrations and construct explicit pencils for them.

In Cogdell et al., LMS Lecture Notes Series 459, 393–427 (2020), the authors proved a type of Kronecker’s limit formula associated to any divisor $D$ on any smooth Kähler manifold $X$, assuming that $D$ is smooth in codimension one. In the present article, it is shown how the aforementioned analogue of Kronecker’s limit formula applies to reprove and generalize Weil reciprocity. More precisely, we extend Weil reciprocity to (suitably normalized) meromorphic modular forms of even weight on a smooth, compact Riemann surface, and present a variant of Weil reciprocity for a class of harmonic functions with special types of singularities on a finite volume quotient of a symmetric space or a compact, smooth projective Kähler variety. We also prove an integral version of Weil reciprocity for a compact, smooth projective Kähler variety.

We define a reflective numerical semigroup of genus $g$ as a numerical semigroup that has a certain reflective symmetry when viewed within $\mathbb{Z}$ as an array with $g$ columns. Equivalently, a reflective numerical semigroup has one gap in each residue class modulo $g$. In this paper, we give an explicit description for all reflective numerical semigroups. With this, we can describe the reflective members of well-known families of numerical semigroups as well as obtain formulas for the number of reflective numerical semigroups of a given genus or given Frobenius number.

A system of $4$ nonlinearly-coupled Ordinary Differential Equations has been recently introduced to investigate the evolution of human respiratory virus epidemics. In this paper we prove that some explicit solutions of that system can be obtained by algebraic operations, provided the parameters of the model satisfy certain constraints.