We consider special Lambert series as generating functions of divisor sums and determine their complete transseries expansion near rational roots of unity. Our methods also yield new insights into the Laurent expansions and modularity properties of iterated Eisenstein integrals that have recently attracted attention in the context of certain period integrals and string theory scattering amplitudes

Let $V$ be a strongly regular vertex operator algebra and let $\mathfrak{ch}_V$ be the space spanned by the characters of the irreducible $V$-modules. It is known that $\mathfrak{ch}_V$ is the space of solutions of a so-called modular linear differential equation (MLDE). In this paper we obtain a classification of those $V$ for which the corresponding MLDE is irreducible and monic of order $2$. It

In an important paper [8], Golyshev and Zagier introduce what we will refer to as Frobenius constants $\kappa_{\rho,n}$ associated to an ordinary linear differential operator $L$ with a reflection type singularity at $t = c$. For every other regular singularity $t = c^\prime$ and a homotopy class of paths $\gamma$ joining $c^\prime$ and $c$, constants $\kappa_{\rho,n} = \kappa_{\rho,n} (\gamma)$ describe

Recently, a universal formula for a non-holomorphic modular completion of the generating functions of refined BPS indices in various theories with $N = 2$ supersymmetry has been suggested. It expresses the completion through the holomorphic generating functions of lower ranks. Here we show that for $U(N)$ Vafa–Witten theory on Hirzebruch and del Pezzo surfaces this formula can be used to extract the

In “J. Moduli of $p$-divisible groups” [9], Scholze and Weinstein show that a certain diagram of perfectoid spaces is Cartesian. In this paper, we generalize their result. This generalization will be used in a forthcoming paper of ours (“E. Cycles in the cohomology of Rapoport–Zink towers coming from the Lubin–Tate tower” [6]) to compute certain non-trivial $p$-adic étale cohomology classes appearing

From the viewpoint of mirror symmetry, we revisit the hypergeometric system $E(3, 6)$ for a family of K3 surfaces. We construct a good resolution of the Baily–Borel–Satake compactification of its parameter space, which admits special boundary points (LCSLs) given by normal crossing divisors. We find local isomorphisms between the $E(3, 6)$ systems and the associated GKZ systems defined locally on the

A CHL model is the quotient of $\mathrm{K}3 \times E$ by an order $N$ automorphism which acts symplectically on the $\mathrm{K}3$ surface and acts by shifting by an $N$-torsion point on the elliptic curve $E$. We conjecture that the primitive Donaldson–Thomas partition function of elliptic CHL models is a Siegel modular form, namely the Borcherds lift of the corresponding twisted-twined elliptic genera

We study a K3 surface, which appears in the two-loop mixed electroweak-quantum chromodynamic virtual corrections to Drell–Yan scattering. A detailed analysis of the geometric Picard lattice is presented, computing its rank and discriminant in two independent ways: first using explicit divisors on the surface and then using an explicit elliptic fibration. We also study in detail the elliptic fibrations