Hilbert modular forms of weight 1/2 and theta functions

Serre and Stark found a basis for the space of modular forms of weight 1/2 in terms of theta series. In this paper, we generalize their result—under certain mild restrictions on the level and character—to the case of weight 1/2 Hilbert modular forms over a totally real field of narrow class number 1. The methods broadly follow those of Serre-Stark; however we are forced to overcome technical difficulties which arise when we move out of Q.     

A short remark on sign changes of Fourier coefficients of half-integral weight modular forms

We give an upper bound on the first sign change of the Fourier coefficients of a cusp form of half-integral weight.     

A remark on non-integral p-adic slopes for modular forms

We give a sufficient condition, namely “Buzzard irregularity”, for there to exist a cuspidal eigenform which does not have integral p-adic slope.     

Differential operators on Hilbert modular forms

We investigate differential operators and their compatibility with subgroups of SL₂n(R). In particular, we construct Rankin-Cohen brackets for Hilbert modular forms, and more generally, multilinear differential operators on the space of Hilbert modular forms. As an application, we explicitly determine the Rankin-Cohen bracket of a Hilbert-Eisenstein series and an arbitrary Hilbert modular form. We use this result to compute the Petersson inner product of such a bracket and a Hilbert modular cusp form.     

A simple proof of Zagier duality for Hilbert modular forms

In this paper, we give a simple proof of an identity between the Fourier coefficients of the weakly holomorphic modular forms of weight 0 arising from Borcherds products of Hilbert modular forms and those of the weakly holomorphic modular forms of weight satisfying a certain property.

     

On the Fourier coefficients of certain Hilbert modular forms

We prove that given any \(\epsilon >0\), a non-zero adelic Hilbert cusp form \({\mathbf {f}}\) of weight \(k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n\) and square-free level \(\mathfrak {n}\) with Fourier coefficients \(C_{{\mathbf {f}}}(\mathfrak {m})\), there exists a square-free integral ideal \(\mathfrak {m}\) with \(N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }\) such that \(C_{{\mathbf {f}}}(\mathfrak {m})\ne 0\). The implied constant depends on \(\epsilon , F\).

     

A note on the Fourier coefficients of half-integral weight modular forms

In this note, we show that the algebraicity of the Fourier coefficients of half-integral weight modular forms can be determined by checking the algebraicity of the first few of them. We also give a necessary and sufficient condition for a half-integral weight modular form to be in Kohnen’s +-subspace by considering only finitely many terms.     

Petersson norms and liftings of Hilbert modular forms

We prove the algebraicity of the ratio of the Petersson norm of a holomorphic Hilbert modular form over a totally real number field and the norm of its Saito-Kurokawa lift. We prove a similar result for the Ikeda lift of an elliptic modular form. In order to obtain these we combine some results on local symplectic groups to generalize a special value of the standard L-function attached to a Siegel-Hilbert cuspform.     

On galois representations associated to Hilbert modular forms

Supported by a Sloan Foundation Doctoral Dissertation Fellowship     

Coefficients of half-integral weight modular forms

In this paper, we study the distribution of the coefficients a(n) of half-integral weight modular forms modulo odd integers M. As a consequence, we obtain improvements of indivisibility results for the central critical values of quadratic twists of L-functions associated with integral weight newforms established in Ono and Skinner (Fourier coefficients of half-integral weight modular forms modulo ?, Ann. of Math. 147 (1998) 453-470). Moreover, we find a simple criterion for proving cases of Newman's conjecture for the partition function.