# The web of modularity : arithmetic of the coefficients of modular forms and q-series

@inproceedings{Ono2003TheWO, title={The web of modularity : arithmetic of the coefficients of modular forms and q-series}, author={Ken Ono}, year={2003} }

Basic facts Integer weight modular forms Half-integral weight modular forms Product expansions of modular forms on $\mathrm{SL}_2(\mathbb{Z})$ Partitions Weierstrass points on modular curves Traces of singular moduli and class equations Class numbers of quadratic fields Central values of modular $L$-functions and applications Basic hypergeometric generating functions for $L$-values Gaussian hypergeometric functions Bibliography Index.

## 685 Citations

On Fourier coefficients of modular forms of half integral weight at squarefree integers

- MathematicsMathematische Zeitschrift
- 2018

We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the…

Bounds for Siegel Modular Forms of genus 2 modulo $p$

- Mathematics
- 2011

Sturm obtained the bounds for the number of the first Fourier coefficients of elliptic modular form $f$ to determine vanishing of $f$ modulo a prime $p$. In this paper, we study analogues of Sturm's…

A modularity criterion for Klein forms, with an application to modular forms of level $13$

- Mathematics
- 2010

Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series

- Mathematics
- 2007

Zagier [23] proved that the generating functions for the traces of level 1 singular moduli are weight 3/2 modular forms. He also obtained generalizations for “twisted traces”, and for traces of…

Fourier Coefficients of Modular Forms of Half-Integral Weight in Arithmetic Progressions

- Mathematics
- 2020

We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progression. We prove a result analogous to previous work of Fouvry-GangulyKowalski-Michel and Kowalski-Ricotta in…

Modular forms of half-integral weight with few non-vanishing coefficients modulo ℓ

- Mathematics
- 2008

Bruinier and Ono classified cusp forms of half-integral weight F(z):=∞Σn=0a(n)q n ∈ S λ+1 2 (Γ 0 (N), Χ )∩Z[[q]] whose Fourier coefficients are not well distributed for modulo odd primes l. Ahlgren…

Linear relations among the Fourier coefficients of modular forms on groups Γ0(N) of genus zero and their applications

- Mathematics
- 2007

Singular moduli of higher level and special cycles

- Mathematics
- 2015

We describe the complex multiplication (CM) values of modular functions for $$\Gamma _0(N)$$Γ0(N) whose divisor is given by a linear combination of Heegner divisors in terms of special cycles on the…

Sturm type theorem for Siegel modular forms of genus 2 modulo p

- Mathematics
- 2013

Suppose that f is an elliptic modular form with integral coefficients. Sturm obtained in (14) bounds for a nonnegative integer n such that every Fourier coefficient of f vanishes modulo a prime p if…

Congruences for the coefficients of weakly holomorphic modular forms

- Mathematics
- 2006

Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomenon is completely…