Shoyu Nagaoka

On some p-adic properties of Siegel–Eisenstein series

Abstract We introduce a formula for the p -adic Siegel–Eisenstein series which demonstrates a connection with the genus theta series and the twisted Eisenstein series with level p . We then prove a generalization of Serres formula in the elliptic modular case.

On -adic Hermitian Eisenstein series

In this paper we generalize the notion of p-adic modular form to the Hermitian modular case and prove a formula that shows a coincidence between certain p-adic Hermitian Eisenstein series and the genus theta series associated with Hermitian matrix with determinant p.

Note on Igusa's cusp form of weight 35

A congruence relation satisfied by Igusas cusp form of weight 35 is presented. As a tool to confirm the congruence relation, a Sturm-type theorem for the case of odd-weight Siegel modular forms of degree 2 is included.

On p-Adic Properties of Siegel Modular Forms

We show that Siegel modular forms of level \(\Gamma _{0}(p^{m})\) are p-adic modular forms. Moreover we show that derivatives of such Siegel modular forms are p-adic. Parts of our results are also valid for vector-valued modular forms. In our approach to p-adic Siegel modular forms we follow Serre [18] closely; his proofs however do not generalize to the Siegel case or need some modifications.

Notes on theta series for Niemeier lattices

Some explicit expressions are given for the theta series of Niemeier lattices. As an application, we present some of their congruence relations.

On the mod p kernel of the theta operator

Siegel modular forms in the space of the mod p kernel of the theta operator are constructed by the Eisenstein series in some odd-degree cases. Additionally, a similar result in the case of Hermitian modular forms is given.

On the restriction of the Hermitian Eisenstein series and its applications

We introduce a simple construction of a Siegel cusp form obtained by taking the difference between the Siegel Eisenstein series and the restricted Hermitian Eisenstein series. In addition, we present applications of the Siegel cusp form.

Congruence properties of Hermitian modular forms

We study the existence of a modular form satisfying a certain congruence relation. The existence of such modular forms plays an important role in the determination of the structure of a ring of modular forms modulo p. We give a criterion for the existence of such a modular form in the case of Hermitian modular forms.

On the kernel of the theta operator mod p

We construct many examples of level one Siegel modular forms in the kernel of theta operators mod p by using theta series attached to positive definite quadratic forms.

On Level p Siegel Cusp Forms of Degree Two

We give a simple formula for the Fourier coefficients of some degree-two Siegel cusp form with level p.