Related papers: Zeros of weakly holomorphic modular forms of level…

We investigate the modularity of formal Fourier--Jacobi series by establishing cohomological vanishing results for line bundles defined on compactifications of $\mathcal{A}_g$. Working over $\mathbb{C}$, we show that the minimal…

Values of quaternionic modular forms are related to twisted central $L$-values via periods and a theorem of Waldspurger. In particular, certain twisted $L$-values must be non-vanishing for forms with no zeroes. Here we study, theoretically…

We establish the existence of many holomorphic Hecke eigenforms $f$ of large weight $k$ for the full modular group, for which the least positive integer $n_f$ such that $\lambda_f(n_f)<0$ satisfies $n_f \ge (\log k)^{1-o(1)}.$ This is…

We find modular transformations of normalized characters for the following $W$-algebras: (a) $W^{min}_k(\frak{g})$, where $\frak{g}=D_n \, (n \geq 4)$, or $E_6$, $E_7$, $E_8$, and $k$ is a negative integer $\geq -2$, or $\geq…

In this paper we investigate the Fourier coefficients of harmonic Maass forms of negative half-integral weight. We relate the algebraicity of these coefficients to the algebraicity of the coefficients of certain canonical meromorphic…

Lehner's 1949 results on the $j$-invariant showed high divisibility of the function's coefficients by the primes $p\in\{2,3,5,7\}$. Expanding his results, we examine a canonical basis for the space of level $p$ modular functions holomorphic…

We prove the following statement about any Siegel modular form $F$ of degree $n$ and arbitrary odd level $N$ on the group $\Gamma_{0}^{(n)}(N)$. Let $A(F,T)$ denote the Fourier coefficients of $F$ and write $T=(T(i,j))$. Suppose that $F$…

In this paper, we investigate cycle integrals of weakly holomorphic modular forms. We show that these integrals coincide with the cycle integrals of classical cusp forms. We use these results to define a Shintani lift from integral weight…

We compare modular forms of characteristic $p>0$ (i.e. Drinfeld's modular forms) and automorphic forms. We prove that spaces of these modular forms (which are of characteristic $p$) can be described by function spaces of characteristic…

We present a dimension formula for spaces of vector-valued modular forms of integer weight in case the associated multiplier system has finite image, and discuss the weight distribution of the module generators of holomorphic and cusp…

An eta-quotient of level $N$ is a modular form of the shape $f(z) = \prod_{\delta | N} \eta(\delta z)^{r_{\delta}}$. We study the problem of determining levels $N$ for which the graded ring of holomorphic modular forms for $\Gamma_{0}(N)$…

For each prime $\ell$, let $|\cdot|_\ell$ be an extension to $\bar \Q$ of the usual $\ell$-adic absolute value on $\Q$. Suppose $g(z) = \sum_{n=0}^\infty c(n)q^n \in M_{k+\half}(N)$ is an eigenform whose Fourier coefficients are algebraic…

Universal algebra and clone theory have proven to be a useful tool in the study of constraint satisfaction problems since the complexity, up to logspace reductions, is determined by the set of polymorphisms of the constraint language. For…

To every $k$-dimensional modular invariant vector space we associate a modular form on $SL(2,\mathbb{Z})$ of weight $2k$. We explore number theoretic properties of this form and find a sufficient condition for its vanishing which yields…

We study low-lying zeros of $L$-functions attached to holomorphic cusp forms of level $1$ and large weight. In this family, the Katz--Sarnak heuristic with orthogonal symmetry type was established in the work of Iwaniec, Luo and Sarnak for…

In this paper, we develop several general techniques to investigate modular invariants of conformal field theories whose algebras of the holomorphic and anti-holomorphic sectors are different. As an application, we find all such…

Given a primitive, non-CM, holomorphic cusp form $f$ with normalized Fourier coefficients $a(n)$ and given an interval $I\subset [-2, 2]$, we study the least prime $p$ such that $a(p)\in I$ . This can be viewed as a modular form analogue of…

We study pullbacks of modular forms of weight 1 from the modular curve X(4) to the modular curve X(4p), where p is an odd prime. We find the extent to which such modular forms separate points on X(4p). Our main result is that these modular…

Let M_0^R be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space H^4…

Let $M(1)$ be the vertex operator algebra with the Virasoro element $\omega$ associated to the Heisenberg algebra of rank $1$ and let $M(1)^{+}$ be the subalgebra of $M(1)$ consisting of the fixed points of an automorphism of $M(1)$ of…