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 ‘δ-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.

Let $X/\overline{\open F}_p$ be an Artin–Schreier curve defined by the affine equation yp − y = $\tilde{f}$(x) where $\tilde{f}$(x) ∈ $\overline{\open F}_p$[x] is monic of degree d. In this paper we develop a method for estimating the first slope of the Newton polygon of X. Denote this first slope by NP1($X/\overline{\open F}_p$). We use our method to prove that if p>d ≥ 2 then NP1($X/\overline{\open F}_p$) ≥ [lceil ](p−1)/d[rceil ]/(p − 1). If p > 2d ≥ 4, we give a sufficient condition for the equality to hold.

In his Inventiones papers in 1995 and 1998, Borcherds constructed holomorphic automorphic forms Ψ(f) with product expansions on bounded domains D associated to rational quadratic spaces V of signature (n2), starting from vector valued modular forms f of weight 1 − n2 for SL2($\open Z$) which are allowed to have poles at the cusp and whose nonpositive Fourier coefficients are integers cμ(−m), m ≥ 0. In this paper, we use the Siegel–Weil formula to give an explicit formula for the integral κ(Ψ(f)) of − log||Ψ(f)||2 over X = Γ\D, where || ||2 is the Petersson norm. This integral is given by a sum for m0 of quantities cμ(−m)κμ(m), where κμ(m) is the limit as Im(τ) → ∞ of the mth Fourier coefficient of the second term in the Laurent expansion at s = n2 of a certain Eisenstein series E(τs) of weight (n2) + 1 attached to V. The possible role played by the quantity κ(Ψ(f)) in the Arakelov theory of the divisors Zμ(m) on X is explained in the last section.