Michael Rapoport

Derivatives of Eisenstein series and Faltings heights

We prove a relation between a generating series for the heights of Heegner cycles on the arithmetic surface associated with a Shimura curve and the second term in the Laurent expansion at s = ½ of an Eisenstein series of weight

On the derivative of an eisenstein series of weight one

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Cycles on Siegel threefolds and derivatives of Eisenstein series

for SL(2). On the geometric side, a typical coefficient of the generating series involves the Faltings heights of abelian surfaces isogenous to a product of CM elliptic curves, an archimedean contribution, and contributions from vertical components in the fibers of bad reduction. On the analytic side, these terms arise via the derivatives of local Whittaker functions. It should be noted that s = ½ is not the central point for the functional equation of the Eisenstein series in question. Moreover, the first term of the Laurent expansion at s = ½ coincides with the generating function for the degrees of the Heegner cycles on the generic fiber and, in particular, does not vanish.

Local models in the ramified case. III unitary groups

In [17], a certain family of Siegel Eisenstein series of genus g and weight (g + 1)/2 was introduced. They have an odd functional equation and hence have a natural zero at their center of symmetry (s = 0). It was suggested that the derivatives at s = 0 of such series, which we will refer to as incoherent Eisenstein series, should have some connection with arithmetical algebraic geometry. Some evidence was provided in the case of genus 2 and weight 3/2. In that case, certain of the Fourier coefficients of the central derivative were shown to involve (parts of) the height pairing of Heegner points on Shimura curves. Additional evidence occurred earlier in the work of Gross and Keating [12], where, implicitly, derivatives of Siegel Eisenstein series on Sp3 of weight 2 arise. Higher dimensional cases are studied in [19] (Sp4, weight 5/2) and [21] (Sp3, weight 2). In the present paper,we consider the simplest possible example of an incoherent Eisenstein series and its central derivative. More precisely, let q > 3 be a prime congruent to 3 modulo 4. There are two types of Eisenstein series of weight 1 associated to the imaginary quadratic field k = Q(√−q). The first is a coherent Eisenstein series. For τ = u+ iv in the upper half-plane and s ∈ C with Re(s) > 1, this series has the form

Modular Forms and Special Cycles on Shimura Curves. (AM-161)

Abstract We consider the Siegel modular variety of genus 2 and a p-integral model of it for a good prime p>2, which parametrizes principally polarized abelian varieties of dimension two with a level structure. We consider algebraic cycles on this model which are characterized by the existence of certain special endomorphisms, and their intersections. We characterize that part of the intersection which consists of isolated points in characteristic p only. Furthermore, we relate the (naive) intersection multiplicities of the cycles at isolated points to special values of derivatives of certain Eisenstein series on the metaplectic group in 8 variables.

A finiteness theorem in the Bruhat-Tits building: an application of Landvogt's embedding theorem

We continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primes p at which the group defining the Shimura variety ramifies. We describe ‘good’ p -adic integral models of these Shimura varieties and study their etale local structure. In the present paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties that we developed in a previous paper.

On the arithmetic fundamental lemma in the minuscule case

is published by Princeton University Press and copyrighted,

The supersingular locus of the Shimura variety for \hbox {GU}(1, n-1) over a ramified prime

Abstract Let G be a connected reductive group over Qp and let B(G, Qp) be the extended Bruhat-Tits building of G over Qp. Let L be the completion of the maximal unramified extension of Qp and let B(G, L) be the building of G over L. By the theorem of Bruhat and Tits, B(G, Qp) may be identified with the fixed point set of the Frobenius automorphism σ acting on B(G, L). A special case of our main result states that for any c>0 there exists C > 0 with the property that any point x σ B(G, L) with distance d(x, σ(x))

Stratifications in the reduction of Shimura varieties

The arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of

On the arithmetic transfer conjecture for exotic smooth formal moduli spaces

p