We show that an elliptic modular form with integral Fourier coefficients in a number field K, for which all but finitely many coefficients are divisible by a prime ideal p of K, is a constant modulo p. A similar property also holds for Siegel modular forms. Moreover, we define the notion of mod p singular modular forms and discuss some relations between their weights and the corresponding prime p. We discuss some examples of mod p singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms. Finally, we apply our results to properties mod p of Klingen-Eisenstein series.

Sturm [13] 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 bound for Siegel modular forms of genus 2. We show the resulting bound is sharp. As an application, we study congruences involving Atkin's U (p)-operator for the Fourier coefficients of Siegel mdoular forms of genus 2.2000 Mathematics Subject Classification. 11F46,11F33.

No abstract

A congruence relation satisfied by Igusa's 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.

No abstract