In [1], the authors prove a conjecture of Calegari and Stein regarding mod p congruences between cusp forms of weight four for Γ0(p) and the derivatives of cusp forms of weight two for the same congruence subgroup. In this paper, we investigate whether or not the result remains valid for cusp forms of level Np.

We prove a conjecture of Calegari and Stein regarding mod p congruences between modular forms of weight four and the derivatives of modular forms of weight two.

Abstract. The purpose of this article is to develop the theory of differential modular forms introduced by A. Buium. The main points are the construction of many isogeny covariant differential modular forms and some auxiliary (nonisogeny covariant) forms and an extension of the 'classical theory' of Serre differential operators on modular forms to a theory of 'd-Serre differential operators' on differential modular forms. As an application, we shall give a geometric realization of the space of elliptic curves up to isogeny.

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Now we turn to the main objects of this paper: Abelian varieties and their isogenies. For any ring S, let A g (S) denote the collection of all principally polarized Abelian schemes over S of relative dimension g. On A g (S) we consider the equivalence relation defined by "isogeny": A 1 , A 2 ∈ A g (S) are called equivalent (write A 1 ∼ A 2 ) if there is an isogeny (i.e., a finite flat homomorphism) A 1 → A 2 . (We do not ask that our isogenies be compatible with the polarizations!) Let A g (S)/ ∼ denote the quotient of A g (S) by this equivalence relation. We would like to construct a geometric object that is "reasonably close" to representing the functor(1.1)As noted above, this cannot be done in "usual" algebraic geometry even if we allow the use of algebraic spaces, algebraic stacks, and so forth. (Indeed, if S is the complex field C, then any isogeny class in A g (C) is dense in the complex topology.) In this paper, we use an "extension" of usual algebraic geometry in order to make the functor (1.1) fit into a geometric picture. There are two such "extensions" that prove to be useful. One is the Kolchin geometry [5,21], in which a derivation δ is adjoined to the usual algebraic geometry. This viewpoint in studying A g (S)/ ∼ was pursued in [7]; however this "extension" of algebraic geometry, although relevant for Diophantine problems over function fields [4,7], is, by its very nature, not suited for purely arithmetic questions, in particular for p-adic questions. The second "extension" of algebraic geometry that we have in mind, initiated in [6,8], is obtained by adjoining a "p-derivation" δ to the usual algebraic geometry; p-derivations (whose definition is recalled below) are arithmetic analogues of usual derivations that, in particular, act via the Fermat quotient operator δx = (x − x p )/p on the integers x ∈ Z. This latter geometry is sensitive to arithmetic problems and an investigation of A g (S)/ ∼, from the viewpoint of this geometry was undertaken in [1, 10], in case when g = 1. Our purpose here is to extend the main results in [1, 10] to the case of dimension g > 1. However, note that our strategy will not be to generalize the methods in [1, 10] from the case g = 1 to the case g > 1; surprisingly, what we do is to reduce the proofs of our main results from the case when g > 1 to the case when g = 1. So the present paper does not supersede [1, 10] but, rather, it relies on them in an essential way.In the rest of this introduction, we informally discuss our theory in a simple situation. We fix a prime number p and let our rings S belong to the category S of p-adic completions of smooth finitely generated Z p -algebras. Then we have at our disposal the "p-jet" functors