Tonghai Yang

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

\frac32

An arithmetic intersection formula on Hilbert modular surfaces

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.

Heights of Kudla–Rapoport divisors and derivatives of \(L\)-functions

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)

In this paper, we obtain an explicit arithmetic intersection formula on a Hilbert modular surface between the diagonal embedding of the modular curve and a CM cycle associated to a non-biquadratic CM quartic field. This confirms a special case of the authors conjecture with Bruinier, and is a generalization of the beautiful factorization formula of Gross and Zagier on singular moduli. As an application, we proved the first nontrivial non-abelian Chowla-Selberg formula, a special case of Colmez conjecture.

Nonvanishing of central Hecke L-values and rank of certain elliptic curves

We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature

Twisted borcherds products on hilbert modular surfaces and their cm values

(n-1,1)

The Main Results

(n-1,1). We construct an arithmetic theta lift from harmonic Maass forms of weight