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We prove two formulas in the style of the Gross-Zagier theorem, relating derivatives of L-functions to arithmetic intersection pairings on a unitary Shimura variety. We also prove a special case of Colmez's conjecture on the Faltings heights of abelian varieties with complex multiplication. These results are derived from the authors' earlier results on the modularity of generating series of divisors on unitary Shimura varieties., Comment: Final version. To appear in Asterisque

We form generating series of special divisors, valued in the Chow group and in the arithmetic Chow group, on the compactified integral model of a Shimura variety associated to a unitary group of signature (n-1,1), and prove their modularity. The main ingredient of the proof is the calculation of the vertical components appearing in the divisor of a Borcherds product on the integral model., Comment: Final version. To appear in Asterisque

We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big CM points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d,2). Our approach depends on the fact that the Green functions in question are constructed as regularized theta lifts of harmonic weak Mass forms, and it involves the Siegel-Weil formula and the central derivatives of incoherent Eisenstein series for totally real fields. In the case of a weakly holomorphic form, the formula is an explicit combination of quantities obtained from the Fourier coefficients of the central derivative of the incoherent Eisenstein series. In the case of a general harmonic weak Maass form, there is an additional term given by the central derivative of a Rankin-Selberg type convolution., Comment: 39 pages

This article describes results of joint work with Michael Rapoport and Tonghai Yang. First, we construct an modular form \phi(\tau) of weight 3/2 valued in the arithmetic Chow group of the arithmetic surface M attached toa Shimura curve over Q. The q-expansion of this function is an analogue of the Hirzebruch-Zagier generating function for the cohomology classes of curves on a Hilbert modular surface. This`arithmetic theta function' is used to define an `arithmetic theta lift' from modular forms of weight 3/2 to the arithmetic Chow group of M. For integers t_1 and t_2 with t_1t_2 not a square, the (t_1,t_2)-Fourier coefficient of the height pairing <\phi(\tau_1),\phi(\tau_2)> coincides with the (t_1,t_2)-Fourier coefficient of the restriction to the diagonal of the central derivative of a certain Eisenstein series of weight 3/2 and genus 2. Using this fact and results about the doubling integral for forms of weight 3/2, we prove that the arithmetic theta lift of a Hecke eigenform f is nonzero if and only if there is no local obstruction (theta dichotomy) and the standard Hecke L-function L(s,F) of the corresponding newform F of weight 2 has nonvanishing derivative, L'(1,F)\ne0, at the center of symmetry. This is an analogue of a result of Waldspurger according to which the classical Shimura lift of such a form is nonzero if and only if there is no local obstruction and L(1,F)\ne0. Detailed proofs will be given elsewhere., Comment: To appear in the proceedings of the Current Developments in Mathematics seminar held at Harvard University in November of 2002

This article sketches relations among algebraic cycles for the Shimura varieties defined by arithmetic quotients of symmetric domains for O(n,2), theta functions, values and derivatives of Eisenstein series and values and derivatives of certain L-functions. In the geometric case, results of joint work with John Millson imply that the generating functions for the classes in cohomology of certain algebraic cycles of codimension r are Siegel modular forms of genus r and weight n/2+1. A result of Borcherds shows that, for r=1, the same is true for the generating function for the classes of such divisors in the Chow group. By the Siegel-Weil formula, the generating function for the volumes of codimension r cycles coincides with a value of a Siegel-Eisenstein series of genus r. In particular, this gives an interpretation of the Fourier coefficients of these Eisenstein series as volumes of algebraic cycles. The second part of the paper discusses the possible analogues of these results in the arithmetic case, where the special values of derivatives of Eisenstein series arise. In this case, the Fourier coefficients of such derivatives are should be the heights (arithmetic volumes) of certain cycles on integral models of the O(n,2) type Shimura varieties. Relations of this sort would yield relations between central derivatives of certain L-functions and height pairings. The case of curves on a Siegel 3-fold and of the central derivative of a triple product L-function are discussed., Comment: to appear in the proceedings of the conference on Special Values of Rankin L series, held at MSRI in December of 2001

We describe connections between the Fourier coefficients of derivatives of Eisenstein series and invariants from the arithmetic geometry of the Shimura varieties $M$ associated to rational quadratic forms $(V,Q)$ of signature $(n,2)$. In the case $n=1$, we define generating series $\hat\phi_1(\tau)$ for 1-cycles (resp. $\hat\phi_2(\tau)$ for 0-cycles) on the arithmetic surface $\Cal M$ associated to a Shimura curve over $\Bbb Q$. These series are related to the second term in the Laurent expansion of an Eisenstein series of weight $\frac32$ and genus 1 (resp. genus 2) at the Siegel--Weil point, and these relations can be seen as examples of an `arithmetic' Siegel--Weil formula. Some partial results and conjectures for higher dimensional cases are also discussed.

In this note, we prove in full generality a conjecture of Jacquet concerning the nonvanishing of the triple product L-function at the central point. Let $\kay$ be a number field and let $\pi_i$, $i=1$, 2, 3 be cuspidal automorphic representations of $GL_2(\A)$ such that the product of their central characters is trivial. Then the central value $L(\frac12,\pi_1\otimes\pi_2\otimes\pi_3)$ of the triple product L--function is nonzero if and only if there exists a quaternion algebra $B$ over $\kay$ and automorphic forms $f_i^B\in \pi_i^B$, such that the integral of the product $f_1^B f_2^B f_3^B$ over the diagonal $Z(\Bbb A) B^\times(\kay) B^\times(\Bbb A)$ is nonzero, where $\pi_i^B$ is the representation of $B^\times(\A)$ corresponding to $\pi_i$. In a previous paper, we proved this conjecture in the special case where $\kay=\Q$ and the $\pi_i$'s correspond to a triple of holomorphic newforms. Recent improvement on the Ramanujan bound due to Kim and Shahidi, results about the local L-factors due to Ikeda and Ramakrishnan, results of Chen-bo Zhu and Sahi about invariant distributions and degenerate principal series in the complex case, and an extension of the Siegel--Weil formula to similitude groups allow us to carry over our method to the general case.

In his Inventiones papers in 1995 and 1998, Borcherds constructed holomorphic automorphic forms $\Psi(F)$ with product expansions on bounded domains $D$ associated to rational quadratic spaces $V$ of signature (n,2). The input $F$ for his construction is a vector valued modular form of weight $1-n/2$ for $SL_2(Z)$ which is allowed to have a pole at the cusp and whose non-positive Fourier coefficients are integers $c_\mu(-m)$, $m\ge0$. For example, the divisor of $\Psi(F)$ is the sum over $m>0$ and the coset parameter $\mu$ of $c_\mu(-m) Z_\mu(m)$ for certain rational quadratic divisors $Z_\mu(m)$ on the arithmetic quotient $X = \Gamma D$. In this paper, we give an explicit formula for the integral $\kappa(\Psi(F))$ of $-\log||\Psi(F)||^2$ over $X$, where $||.||^2$ is the Petersson norm. More precisely, this integral is given by a sum over $\mu$ and $m>0$ of quantities $c_\mu(-m) \kappa_\mu(m)$, where $\kappa_\mu(m)$ is the limit as $Im(\tau) -> \infty$ of the $m$th Fourier coefficient of the second term in the Laurent expansion at $s= n/2$ of a certain Eisenstein series $E(\tau,s)$ of weight $n/2 + 1$ attached to $V$. It is also shown, via the Siegel--Weil formula, that the value $E(\tau, n/2)$ of the Eisenstein series at this point is the generating function of the volumes of the divisors $Z_\mu(m)$ with respect to a suitable K\"ahler form. The possible role played by the quantity $\kappa(\Psi(F))$ in the Arakelov theory of the divisors $Z_\mu(m)$ on $X$ is explained in the last section.

In this paper we construct a modular form f of weight one attached to an imaginary quadratic field K. This form, which is non-holomorphic and not a cusp form, has several curious properties. Its negative Fourier coefficients are non-zero precisely for neqative integers -n such that n >0 is a norm from K, and these coefficients involve the exponential integral. The Mellin transform of f has a simple expression in terms of the Dedekind zeta function of K and the difference of the logarithmic derivatives of Riemann zeta function and of the Dirichlet L-series of K. Finally, the positive Fourier coefficients of f are connected with the theory of complex multiplication and arise in the work of Gross and Zagier on singular moduli.