Related papers: On generalized modular forms with a cuspidal divis…

We describe the complex multiplication (CM) values of modular functions for $\Gamma_0(N)$ whose divisor is given by a linear combination of Heegner divisors in terms of special cycles on the stack of CM elliptic curves. In particular, our…

In [B-G1] and [B-G2], Borisov and Gunnells constructed for each level (N > 1) and for each weight (k > 1) a modular symbol with values in $Sk(\Gamma_1(N))$ using products of Eisenstein series. In this paper we generalize this result by…

We prove that a Siegel cusp form of degree 2 for the full modular group is determined by its set of Fourier coefficients a(S) with 4 det(S) ranging over odd squarefree integers. As a key step to our result, we also prove that a classical…

Let $g$ be a Hecke cusp form of half-integral weight, level $4$ and belonging to Kohnen's plus subspace. Let $c(n)$ denote the $n$th Fourier coefficient of $g$, normalized so that $c(n)$ is real for all $n \geq 1$. A theorem of Waldspurger…

The notion of formal Siegel modular forms for an arithmetic subgroup $\Gamma$ of the symplectic group of genus $n$ is a generalization of symmetric formal Fourier-Jacobi series. Assuming an upper bound on the affine covering number of the…

Let $\Gamma$ be a group and $r_n(\Gamma)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function $\calz_\Gamma(s) = \suml^\infty_{n=1} r_n(\Gamma)n^{-s}$. When…

We find bases for the spaces $M_2\Big(\Gamma_0(24),\Big(\frac{d}{\cdot}\Big)\Big)$ ($d=1,8,12, 24$) of modular forms. We determine the Fourier coefficients of all $35$ theta products $\varphi[a_1,a_2,a_3,a_4](z)$ in these spaces. We then…

We attempt to compute expressions in terms of the Dedekind eta function for all weight 2 new cusp forms with level up to 100, using methods of Allen et. al. In cases where no expression exists, we raise the level instead of the weight,…

Let $\Gamma$ be a cocompact, discrete, and irreducible subgroup of $\mathrm{PSL}_{2}(\mathbb{R})^{n}$. Let $\nu$ be a unitary character of $\Gamma$. For $k\in1\slash 2\,\mathbb{Z}$, let $\sknu$ denote the complex vector space of cusp forms…

Let $N$ be a square-free integer such that the arithmetic group $\Gamma_0(N)^+$ has genus zero; there are $44$ such groups. Let $j_N$ denote the associated Hauptmodul normalized to have residue equal to one and constant term equal to zero…

We prove the following version of the Campana's orbifold conjecture: Let $X$ be a complex non-singular projective variety of dimension $n$. Let $D_1,\ldots,D_{n+1}$ be $\mathbb Z$-linearly independent effective divisors in ${\rm Div}(X)$…

We study functions $f$ on $\mathbb Q$ which statisfy a ``quantum modularity'' relation of the shape $$ f(x+1)=f(x), \qquad f(x) - |x|^{-k} f(-1/x) = h(x) $$ where $h:\mathbb R_{\neq 0} \to \mathbb C$ is a function satisfying various…

We prove that vector-valued Siegel cusp forms for $\Gamma_0^n(N)$ with certain nebentypus are determined by their fundamental Fourier coefficients with discriminants coprime to the level $N$, assuming $N$ is odd and square-free. In the case…

We prove various converse theorems for automorphic forms on \Gamma_0(N), each assuming fewer twisted functional equations than the last. We show that no twisting at all is needed for holomorphic modular forms in the case that N is 18, 20,…

We further develop a model unifying general relativity with quantum mechanics proposed in our earlier papers (J. Math. Phys. 38, 5840 (1998); 41, 5168 (2000)). The model is based on a noncommutative algebra $A$ defined on a groupoid $\Gamma…

In this paper we apply results from the theory of congruences of modular forms (control of reducible primes, level-lowering), the modularity of elliptic curves and Q-curves, and a couple of Frey curves of Fermat-Goldbach type, to show the…

There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group PSL(2,Z) including the following statements: The ring of holomorphic modular forms is generated by the holomorphic…

We generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra $A_0$, any faithful normal state $\varphi_0$ and any discrete group $\Gamma$, the associated Bernoulli crossed product von…

We construct a categorification of the modular data associated with every family of unipotent characters of the spetsial complex reflection group $G(d,1,n)$. The construction of the category follows the decomposition of the Fourier matrix…

The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the…