Related papers: Note on Fourier expansions at cusps
In this article we study the fields generated by the Fourier coefficients of modular forms at arbitrary cusps. We prove that these fields are contained in certain cyclotomic extensions of the field generated by the Fourier coefficients at…
The object of this work is to present the status of art of an open problem: to provide an analogue for Shimura curves of the Ihara's lemma \cite{Ihara73} which holds for modular curves. We will describe our direct result towards the…
In this paper, we extend previous results to prove that generalized modular forms with rational Fourier expansions whose divisors are supported only at the cusps and certain other points in the upper half plane are actually classical…
We prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide a table of…
Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). Then there is a notion of modular forms for $G$, anchored on the…
In this paper, we prove the existence of an efficient algorithm for the computation of $q$-expansions of modular forms of weight $k$ and level $\Gamma$, where $\Gamma \subseteq SL_{2}({\mathbb{Z}})$ is an arbitrary congruence subgroup. We…
Bruinier and Raum, building on work of Ibukiyama-Poor-Yuen, have studied a notion of ``formal Siegel modular forms". These objects are formal sums that have the symmetry properties of the Fourier expansion of a holomorphic Siegel modular…
We present and discuss an algorithm and its implementation that is capable of directly determining Fourier expansions of any vector-valued modular form of weight at least $2$ associated with representations whose kernel is a congruence…
The aim of this article is to study the fields generated by the Fourier coefficients of Hilbert newforms at arbitrary cusps. Precisely, given a cuspidal Hilbert newform $f$ and a matrix $\sigma$ in (a suitable conjugate of) the Hilbert…
We prove a formula of Petersson's type for Fourier coefficients of Siegel cusp forms of degree 2 with respect to congruence subgroups, and as a corollary, show upper bound estimates of individual Fourier coefficient. The method in this…
The goal of this paper is to improve existing bounds for Fourier coefficients of higher genus Siegel modular forms of small weight.
In this note, we study the arithmetic nature of values of modular functions, meromorphic modular forms and meromorphic quasi-modular forms with respect to arbitrary congruence subgroups, that have algebraic Fourier coefficients. This…
In this paper, we explore a two-way connection between quasimodular forms of depth $1$ and a class of second-order modular differential equations with regular singularities on the upper half-plane and the cusps. Here we consider the cases…
In [6], Kohnen proves that if $\Gamma=\Gamma_0(N)$ where $N$ is a square-free integer, then any modular function of weight $0$ for $\Gamma$ having a divisor supported at the cusps is an $\eta$-product. Under the condition of having rational…
The goal in this preprint is to give an efficient algorithm to compute Puiseux expansions at cusps of X0(N). It is based on a relation with a hypergeometric function that holds for any N.
Quaternionic modular forms on $\mathsf{G}_2$ carry a surprisingly rich arithmetic structure. For example, they have a theory of Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. For quaternionic…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
We modify the proof of the basic lemma of a paper of Saks and Zygmund on additive functions of rectangles.
In this paper, we investigate the algebraic nature of the value of a higher Green function on an orthogonal Shimura variety at a single CM point. This is motivated by a conjecture of Gross and Zagier in the setting of higher Green functions…
We prove that the space of cuspidal quaternionic modular forms on the groups of type $F_4$ and $E_n$ have a purely algebraic characterization. This characterization involves Fourier coefficients and Fourier-Jacobi expansions of the cuspidal…