The ring of differential Fourier expansions

For a fixed prime we prove structure theorems for the kernel and the image of the map that attaches to any differential modular function its differential Fourier expansion. The image of this map, which is the ring of differential Fourier expansions, plays the role of ring of functions on a "differential Igusa curve". Our constructions are then used to perform an analytic continuation between isogeny covariant differential modular forms on the differential Igusa curves belonging to different primes

Hecke operators on differential modular forms mod p

A description is given of all primitive differential series mod p of order 1 which are eigenvectors of all the Hecke operators and which are differential Fourier expansions of differential modular forms of arbitrary order and given weight; this set of differential series is shown to be in a natural one-to-one correspondence with the set of series mod p (of order 0) which are eigenvectors of all the Hecke operators and which are Fourier expansions of (classical) modular forms of appropriate weight

Differential overconvergence

We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation". This property can be viewed as a special kind of overconvergence property. One can also go in the opposite direction by using differential functions that arise in a ramified situation to construct "new" (unramified) differential functions

Solutions to arithmetic differential equations in algebraically closed fields

Arithmetic differential equations or δ-geometry exploits analogies between derivations and p-derivations δ arising from lifts of Frobenius to study problems in arithmetic geometry. Along the way, two main classes such functions, describable as series, arose prominently namely δ-characters of abelian schemes and (isogeny covariant) δ-modular forms. However, the theory of these δ-functions is not as straightforward in ramified settings. Overconvergence was introduced in [13] to account for these issues which essentially imposes growth conditions extensions of these series to a fixed level of ramification; necessary as such extensions have non-trivial fractional coefficients. In this article, we introduce a rescaling process which identifies a class of δ-functions we call totally overconvergent, which extend all the way to the algebraic closure of ring of integers of the maximally unramified extension of Q_p . Applications built on these functions allow one to remove boundedness assumptions on ramification. The bulk of the article is devoted to establishing that most δ-functions arising in practice, namely those in the applications described in [5], [7], [8], are totally overconvergent, which essentially extends results in [13] to unbounded ramification

Curvature in Noncommutative Geometry

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator. To carry explicit computations however one needs quite intriguing new ideas. We give an account of the most recent developments on the notion of curvature in noncommutative geometry in this paper.Comment: 76 pages, 8 figures, final version, one section on open problems added, and references expanded. Appears in "Advances in Noncommutative Geometry - on the occasion of Alain Connes' 70th birthday