Related papers: Difference of modular functions and their CM value…
Since the theorems of Schur and van der Waerden, numerous partition regularity results have been proved for linear equations, but progress has been scarce for non-linear ones, the hardest case being equations in three variables. We prove…
We construct isomorphisms between spaces of vector-valued modular forms for the dual Weil representation and certain spaces of scalar-valued modular forms in the case that the underlying finite quadratic module $A$ has order $p$ or $2p$,…
In his Inventiones papers in 1995 and 1998, Borcherds constructed holomorphic automorphic forms $\Psi(F)$ with product expansions on bounded domains $D$ associated to rational quadratic spaces $V$ of signature (n,2). The input $F$ for his…
Quaternionic modular forms on $\mathsf{G}_2$ carry a surprisingly rich arithmetic structure. For example, they have a theory of Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. For quaternionic…
We prove the equivalence of a class of generalised Schur partition functions $\mathcal Z_G(q;\alpha)$ of 4d $\mathcal N=2$ superconformal gauge theories to contour integral representations of vector-valued modular forms of the type that…
Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix…
Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of…
In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular…
The basic disentanglement theorem established by the present authors states that estimates on a weighted geometric mean over (convex) families of functions can be disentangled into quantitatively linked estimates on each family separately.…
We prove that a Schur function of rectangular shape $(M^n)$ whose variables are specialized to $x_1,x_1^{-1},...,x_n,x_n^{-1}$ factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at…
In this note, using entirely algebraic or elementary methods, we determine a new asymptotic lower bound for the number of odd values of one of the most important modular functions in number theory, the Klein $j$-function. Namely, we show…
We prove a factorization formula for the point-to-point partition function associated with a model of directed polymers on the space-time lattice $\mathbb{Z}^{d+1}$, subject to an i.i.d. random potential and in the regime of weak disorder.…
Exponential Puiseux semirings are additive submonoids of $\qq_{\geq 0}$ generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we…
In this paper we study local and global well-posedness of the following Cauchy problem: $$ \bigg \{ \begin{array}{rl} i\partial_t\Psi+\frac{1}{2}\Delta_{x}\Psi = A_0\Psi +\alpha |\Psi|^{\gamma-1}\Psi, & (t,x)\in\R\times\R,\\…
Recently, Bruinier and Ono classified cusp forms $f(z) := \sum_{n=0}^{\infty} a_f(n)q ^n \in S_{\lambda+1/2}(\Gamma_0(N),\chi)\cap \mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this…
Recently, \cite{Cao:2025hio} demonstrated the $2$-split for form factor under specific kinematic constraints. This factorization is analogous to that observed in scattering amplitudes. A key consequence of this structure is the presence of…
We study the coefficients of a natural basis for the space of weak harmonic Maass forms of weight $5/2$ on the full modular group. The non-holomorphic part of the first element of this infinite basis encodes the values of the partition…
We twist Zagier's function $f_{k,D}$ by a sign function and a genus character. Assuming weight $0 < k \equiv 2 \pmod{4}$, and letting $D$ be a positive non-square discriminant, we prove that the obstruction to modularity caused by the sign…
We prove some Sawyer-type characterizations for multilinear fractional maximal function for the upper triangle case. We also provide some two-weight norm estimates for this operator. As one of the main tools, we use an extension of the…
Using the notion of higher-order Fourier dimension introduced in \cite{M2} (which was a sort of psuedorandomness condition stemming from the Gowers norms of Additive Combinatorics), we prove a maximal theorem and corresponding…