Integrals of Borcherds forms

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

Special cycles and derivatives of Eisenstein series

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 200

Height pairings on Shimura curves and p-adic uniformization

We establish a relation between intersection numbers of special cycles on a Shimura curve and special values of derivatives of metaplectic Eisenstein series at a place of bad reduction where p-adic uniformization in the sense of Cherednik and Drinfeld holds. The result extends the one established by one of us (S. Kudla: Ann. of Math. 146 (1997)) for the archimedean place and for the non-archimedean places of good reduction. The bulk of the paper is concerned with the corresponding problem on the Drinfeld upper half plane (the formal scheme version).Comment: 82 pages, 1 figure, Postscrip

Arithmetic Hirzebruch Zagier cycles

We define special cycles on arithmetic models of twisted Hilbert-Blumenthal surfaces at primes of good reduction. These are arithmetic versions of these cycles. In particular, we characterize the non-degenerate intersections and partially determine the generating series formed from the intersection numbers of them relating it to the value at the center of symmetry of the derivative of a certain metaplectic Eisenstein series in 6 variables. These results are analogous to those obtained by us in the case of Siegel threefolds (alg-geom/9711025). We also study the case of degenerate intersections and show that in this case the intersection locus is a configuration of projective lines whose dual graph is described in terms of subcomplexes of the Bruhat-Tits building of PGL(2,F), where F is an unramified quadratic extension of Q_p.Comment: 106 page

Cycles on Siegel 3-folds and derivatives of Eisenstein series

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 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.Comment: AMSTe

On a conjecture of Jacquet

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

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 to a Shimura curve and the second term in the Laurent expansion at s=1/2 of an Eisenstein series of weight 3/2 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=1/2 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=1/2 coincides with the generating function for the degrees of the Heegner cycles on the generic fiber, and, in particular, does not vanish.Comment: 88 pages, AMS-Te

A peculiar modular form of weight one

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

Faltings heights of big CM cycles and derivatives of L-functions

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 page

Transmission of charge and spin in a topological-insulator-based magnetic structure

We discuss the effect of a magnetic thin-film ribbon at the surface of a topological insulator on the charge and spin transport due to surface electrons.\\ If the magnetization in the magnetic ribbon is perpendicular to the surface of a topological insulator, it leads to a gap in the energy spectrum of surface electrons. As a result, the ribbon is a barrier for electrons, which leads to electrical resistance.\\ We have calculated conductance of such a structure. The conductance reveal some oscillations with the length of the magnetized region due to the interference of transmitted and reflected waves. We have also calculated the Seebeck coefficient when electron flux is due to a temperature gradient.Comment: 7 pages, 7 figur