Zero loci of differential modular forms

Abstract

The theory of p-adic modular forms initiated by Serre, Dwork, and Katz ( p-Adic Properties of Modular Schemes and Modular Forms, Lecture Notes in Mathematics, Vol. 350, Springer, Berlin, 1973) “lives” on the complement (in the p-adic completion of the appropriate modular curve) of the zero locus of the Eisenstein form E p−1 . On the other hand, most of the interesting phenomena in the theory of differential modular forms (J. Reine Angew. Math. (520) (2000) 95) take place on the complement of the zero locus of a fundamental differential modular form called f jet. We establish that the zero locus of the reduction modulo p for p not congruent to one modulo 12 of the Eisenstein form E p−1 is not contained in the zero locus of the reduction modulo p of the differential modular form f jet implying that the theory of differential modular forms is applicable in certain situations not covered by the theory of p-adic modular forms.