Related papers: Modular forms of half-integral weights on SL(2,Z)

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 present a flavor model with the $S_3$ modular invariance in the framework of SU(5) GUT. The $S_3$ modular forms of weights $2$ and $4$ give the quark and lepton mass matrices with a common complex parameter, the modulus $\tau$. The GUT…

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…

Let f be a newform of weight 2k-2 and level 1. In this paper we provide evidence for the Bloch-Kato conjecture for modular forms. We demonstrate an implication that under suitable hypothesis if a prime divides the algebraic part of L(k,f),…

Two integral structures on the Q-vector space of modular forms of weight two on X_0(N) are compared at primes p exactly dividing N. When p=2 and N is divisible by a prime that is 3 mod 4, this comparison leads to an algorithm for computing…

We study the coherent cohomology of automorphic sheaves corresponding to Siegel modular forms $f$ of low weight on ${\rm GSp}(4)$ Shimura varieties. Inspired by the work of Prasanna--Venkatesh on singular cohomology of locally symmetric…

Let L >= 3. Using the moduli interpretation, we define certain elliptic modular forms of level Gamma(L) over any field k where 6L is invertible and k contains the Lth roots of unity. These forms generate a graded algebra R_L, which, over C,…

We consider the moduli space $\mathfrak{M}_{g,n}$ of Riemann surfaces of genus $g\ge0$ with $n\ge1$ ordered and directed marked points. For $d\ge 2g+n-1$ we show that $\mathfrak{M}_{g,n}$ is homotopy equivalent to a component of the…

Let $\ell\ge5$ be an odd prime and $j, s$ be positive integers. We study the distribution of the coefficients of integer and half-integral weight modular forms modulo odd positive integer $M$. As a consequence, we prove that for each…

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 give a new proof of the Quantum Unique Ergodicity conjecture for holomorphic integral weight modular forms on the upper half plane. The proof requires only partial results towards the Ramanujan conjecture and the shifted…

Let f be a cusp form of weight k+1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for…

We consider a mock modular form $M_{\Delta}(\tau)$ that arises naturally from Ramanujan's Delta-function. It is a weight $-10$ harmonic Maass form whose nonholomorphic part is the "period integral function'' of $\Delta(\tau)$. The Hecke…

We consider the genus of $20$ classes of unimodular Hermitian lattices of rank $12$ over the Eisenstein integers. This set is the domain for a certain space of algebraic modular forms. We find a basis of Hecke eigenforms, and guess global…

In this paper, we construct a novel class of simple modules for the $W$-algebra $W(2,2)$. Our approach involves taking tensor products of finitely many non-weight simple modules $\Omega(\lambda,\alpha,h)$ with an arbitrary simple restricted…

In this paper, we completely classify the rational weights $k$ for which the Kaneko-Zagier (KZ) differential equation admits a fundamental system of solutions consisting of modular forms for a principal congruence subgroup $\Gamma(N)$. By…

Recently, Allen et al. developed the Explicit Hypergeometric Modularity Method (EHMM) that establishes the modularity of a large class of hypergeometric Galois representations in dimensions two and three. Motivated by this framework, we…

In this article, for any Seifert fibered integral homology 3-sphere, we give explicit modular transformation formulas of homological blocks introduced by Gukov-Pei-Putrov-Vafa. Moreover, based on the modular transformation formulas, we have…

In this paper we prove a theorem about the coefficients in a block of a half integral weight modular form. We show that the result of Serre and Stark for weight 1/2 forms does not generalize to higher higher weights. Let f be a half…

Let $F$ be a totally real number field and $\mathfrak{o}$ the ring of integers of $F$. We study theta functions which are Hilbert modular forms of half-integral weight for the Hilbert modular group $\mathrm{SL}_2(\mathfrak{o})$. We obtain…