Related papers: Difference of modular functions and their CM value…

In 1960 Schwinger [J. Schwinger, Proc.Natl.Acad.Sci. 46 (1960) 570- 579] proposed the algorithm for factorization of unitary operators in the finite M dimensional Hilbert space according to a coprime decomposition of M. Using a special…

Generating functions $h_r(\tau)$ of D4-D2-D0 BPS indices, appearing in Calabi-Yau compactifications of type IIA string theory and identical to rank 0 Donaldson-Thomas invariants, are known to be higher depth mock modular forms satisfying a…

We consider incomplete exponential sums in several variables of the form S(f,n,m) = \frac{1}{2^n} \sum_{x_1 \in \{-1,1\}} ... \sum_{x_n \in \{-1,1\}} x_1 ... x_n e^{2\pi i f(x)/p}, where m>1 is odd and f is a polynomial of degree d with…

We present a fast algorithm for modular exponentiation when the factorization of the modulus is known. Let $a,n,m$ be positive integers and suppose $m$ factors canonically as $\prod_{i=1}^k p_i^{e_i}$. Choose integer parameters $t_i\in [1,…

We improve a result of Lau and Zhao on the variance of Fourier coefficients of primitive cuspidal modular forms for SL2(Z) in arithmetic progressions. This is achieved by using bounds on the first moment of Rankin-Selberg L-functions in the…

We describe a relation between the invariants of $n$ ordered points in $P^d$ and of points contained in a union of linear subspaces $P^{d1}\cup P^{d2} \subset P^d$. This yields an attaching map for GIT quotients parameterizing point…

We generalize the factorization method for inverse medium scattering using a particular factorization of the difference of two far field operators. Whilst the factorization method been used so far mainly to identify the shape of a…

We consider bivariate polynomials over the skew field of quaternions, where the indeterminates commute with all coefficients and with each other. We analyze existence of univariate factorizations, that is, factorizations with univariate…

The degeneracies of $1/4$ BPS states with unit torsion in heterotic string theory compactified on a six-torus are given in terms of the Fourier coefficients of the reciprocal of the Igusa cusp Siegel modular form $\Phi_{10}$ of weight $10$.…

We establish power saving asymptotics for the sum of the divisor function along a binary quartic form, improving on work of Daniel. The proof involves an application of a recent two dimensional delta method due to Li, Rydin-Myerson, and…

We define a new equivariant (with respect to a finite group $G$ action) version of the Poincar\'e series of a multi-index filtration as an element of the power series ring ${\widetilde{A}}(G)[[t_1, \ldots, t_r]]$ for a certain modification…

I find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{ns}(p^k)$ associated to the normalizer of a non-split Cartan subgroup of level $p^k$ where $p\not=2,3$ is a prime. The Cuspidal…

We revisit the well known Bohr-Sommerfeld quantization rule (BS) for a 1-D Pseudo-differential self-adjoint Hamiltonian within the algebraic and microlocal framework of Helffer and Sj\"ostrand; BS holds precisely when the Gram matrix…

A formalism is presented to evaluate the Sivers function in constituent quark models. A non-relativistic reduction of the scheme is performed and applied to the Isgur-Karl model. The sign for the $u$ and $d$ flavor contributions that we…

We prove that the Buchweitz-Greuel-Schreyer Conjecture on the minimal rank of a matrix factorization holds for a generic polynomial of given degree and strength. The proof introduces a notion of the secondary strength of a polynomial, and…

This paper extends the dual calculus with inductive types and coinductive types. The paper first introduces a non-deterministic dual calculus with inductive and coinductive types. Besides the same duality of the original dual calculus, it…

We give congruences modulo powers of 2 for the Fourier coefficients of certain level 2 modular functions with poles only at 0, answering a question posed by Andersen and the first author. The congruences involve a modulus that depends on…

We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in $(0,1]$ ordered by denominators. We show convergence to a stable law…

We prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight -2 harmonic weak Maass forms to spaces of…

In this paper we consider weakly holomorphic modular forms (i.e. those meromorphic modular forms for which poles only possibly occur at the cusps) of weight $2-k\in 2\Z$ for the full modular group $\SL_2(\Z)$. The space has a distinguished…