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.

We compute the index of certains groups of elliptic units. These groups are the analoguousof Sinnott's groups of circular units.

In this paper, we study the family of algebraic K3 surfaces generated by the smooth intersection of a (1, 1) form and a (2, 2) form in ${\open P}^2\times{\open P}^2$ defined over ${\open C}$ and with Picard number 3. We describe the group of automorphisms ${\cal A}={\rm Aut}(V/{\open C})$ on V. For an ample divisor D and an arbitrary curve C0 on V, we investigate the asymptotic behavior of the quantity $N_{{\cal A}(C_0)}(t)=\#\{C \in {\cal A}(C_0): C\cdot D<t\}$. We show that the limit $\lim_{t\to \infty } {\log N_{{\cal A}(C\,)}(t)\over \log t}=\alpha\fleqno{}$ exists, does not depend on the choice of curve C or ample divisor D, and that .6515<α<.6538.

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.

We construct a general semiregularity map for algebraic cycles as asked for by S. Bloch in 1972. The existence of such a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture. Aside from generalizing and extending considerably previously known results in this direction, we give new applications to deformations of modules that encompass, for example, results of Artamkin and Mukai. The formation of the semiregularity map here involves powers of the cotangent complex, Atiyah classes, and trace maps, and is defined not only for subspaces of manifolds but for perfect complexes on arbitrary complex spaces. It generalizes in particular Illusie's treatment of the Chern character to the analytic context and specializes to Bloch's earlier description of the semiregularity map for locally complete intersections as well as to the infinitesimal Abel–Jacobi map for submanifolds.

Let [ell ]>3 be a prime. Fix a regular character χ of F[ell ]2× of order ≤ [ell ] − 1, and an integer M prime to [ell ]. Let f∈S2(Γ0(M[ell ]2)) be a newform which is supercuspidal of type χ at [ell ]. For an indefinite quaternion algebra over Q of discriminant dividing the level of f, there is a local quaternionic Hecke algebra T of type χ associated to f. The algebra T acts on a quaternionic cohomological module M. We construct a Taylor–Wiles system for M, and prove that T is the universal object for a deformation problem (of type χ at [ell ] and semi-stable outside) of the Galois representation ρ¯f over F¯[ell ] associated to f; that T is complete intersection and that the module M is free of rank 2 over T. We deduce a relation between the quaternionic congruence ideal of type χ for f and the classical one.

We construct actions of the spheromorphism group of Neretin (containing Thompson's group V) on towers of moduli spaces of genus zero real stable curves. The latter consist of inductive limits of spaces which are the real parts of the Grothendieck–Knudsen compactification of the moduli spaces of punctured Riemann spheres. By a result of M. Davis, T. Januszkiewicz and R. Scott, these spaces are aspherical cubical complexes whose fundamental groups, the ‘pure quasi-braid groups’, can be viewed as analogues of the Artin pure braid groups. By lifting the actions of the Thompson and Neretin groups to the universal covers of the towers, we obtain extensions of both groups by an infinite pure quasi-braid group, and construct an ‘Euler class’ for the Neretin group. We justify this terminology by constructing a corresponding cocycle.

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.

Some curvature conditions about the geodesics emanating from a submanifold are obtained. These conditions are used to to study the topological and geometric properties of the ambient spaces which admit some minimal submanifolds.

A refinement of the rank 1 Abelian Stark conjecture has been formulated by B. Gross. This conjecture predicts some ${\frak p}$-adic analytic nature of a modification of the Stark unit. The conjecture makes perfect sense even when ${\frak p}$ is an Archimedean place. Here we consider the conjecture when ${\frak p}$ is a real place, and interpret it in terms of 2-adic properties of special values of L-functions. We prove the conjecture for CM extensions; here the original Stark conjecture is uninteresting, but the refined conjecture is nontrivial. In more generality, we show that, under mild hypotheses, if the subgroup of the Galois group generated by complex conjugations has less than full rank, then the refined conjecture implies that the Stark unit should be a square. This phenomenon has been discovered by Dummit and Hayes in a particular type of situation. We show that it should hold in much greater generality.

We give elementary proofs of the main theorems about the (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, the presentation, and a recent theorem of Fulton and Woodward about the minimal q-power which appears in a product of two Schubert classes.

The purpose of this paper is to investigate a conjecture about the universality of the circular distribution made by Robert Coleman. The algebraic property of the universal distribution is the main ingredient in studying Euler system of Kolyvagin and Rubin. We study the universality of the circular distribution by using the Iwasawa theory and the theory of the Euler systems. The conjecture is a characterization of Euler systems in the case of number field. The results here assert that Euler systems are essentially made out of cyclotomic units.