Related papers: Multiplicative independence of modular functions
We generalize results of Mingo and Nica on graded independence from the context of $\mathbb Z_2$--graded (Fermionic) noncommutative probability spaces to that of $\mathbb Z_n$--graded noncommutative probability spaces. We show that for $q$…
We prove a new general multiplicity estimate applicable to sets of functions without any assumption on algebraic independence. The multiplicity estimates are commonly used in determining measures of algebraic independence of values of…
In his striking 1995 paper, Borcherds found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant $-d$ evaluated at the modular $j$-function.…
We describe a general method for giving $p$-adic interpretations of $G$-functions arising from degenerating periods of smooth projective algebraic varieties. Using this, we are able to implement a strategy due to Andr\'e for bounding…
In an important paper, Zagier proved that certain half-integral weight modular forms are generating functions for traces of polynomials in the $j$-function. It turns out that Zagier's work makes it possible to algorithmically compute…
In order to better understand the structure of classical rings of invariants for binary forms, Dixmier proposed, as a conjectural homogeneous system of parameters, an explicit collection of invariants previously studied by Hilbert. We…
The aim of this paper is two-fold: Firstly, we prove Toda's $\chi$-independence conjecture for Gopakumar--Vafa invariants of arbitrary local curves. Secondly, following Davison's work, we introduce the BPS cohomology for moduli spaces of…
We prove that $\delta$-derivations of a simple finite-dimensional Lie algebra over a field of characteristic zero, with values in a finite-dimensional module, are either inner derivations, or, in the case of adjoint module, multiplications…
We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogues of Fourier-Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla's…
In their 2015 paper, Mertens and Rolen prove that for a certain level 6 "almost holomorphic" modular function $P$, the degree of $P(\tau)$ over $\mathbb{Q}$ for quadratic $\tau$ is as large as expected, settling a conjecture of Bruinier and…
We prove an automorphic analogue of Deligne's conjecture for symmetric fourth $L$-functions of Hilbert modular forms. We extend the result of Morimoto based on generalization and refinement of the results of Grobner and Lin to cohomological…
We show that for every finite set of prime numbers S, there are at most finitely many singular moduli that are S-units. The key new ingredient is that for every prime number p, singular moduli are p-adically disperse. We prove analogous…
We study multiplicative dependence between terms of the $k$-generalized Pell sequence $(P_n^{(k)})_{n\ge 2-k}$, defined by the linear recurrence \[ P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}, \] with initial…
The goal of this paper is to construct infinite dimensional Lie algebras using infinite product identities, and to use these Lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex…
Singular Gelfand-Tsetlin modules of index 2 are modules whose tableaux bases may have singular pairs but no singular triples of entries on each row. In this paper we construct singular Gelfand-Tsetlin modules for arbitrary singular…
We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a…
Higher Green functions are real-valued functions of two variables on the upper half plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, but instead of the usual equation…
For any square-free integer $N$ such that the "moonshine group" $\Gamma_0(N)^+$ has genus zero, the Monstrous Moonshine Conjectures relate the Hauptmoduli of $\Gamma_0(N)^+$ to certain McKay-Thompson series associated to the representation…
Recently, Todorov and Wilf independently realized that de Branges' original proof of the Bieberbach and Milin conjectures and the proof that was later given by Weinstein deal with the same special function system that de Branges had…
The main result of this paper is a generalization of Popa's free independence result for subalgebras of ultraproduct ${\rm II_1}$ factors [Po95] to the framework of ultraproduct von Neumann algebras $(M^\omega, \varphi^\omega)$ where $(M,…