869 research outputs found
Integrals of Borcherds forms
In his Inventiones papers in 1995 and 1998, Borcherds constructed holomorphic
automorphic forms with product expansions on bounded domains
associated to rational quadratic spaces of signature (n,2). The input
for his construction is a vector valued modular form of weight for
which is allowed to have a pole at the cusp and whose non-positive
Fourier coefficients are integers , . For example, the
divisor of is the sum over and the coset parameter of
for certain rational quadratic divisors on the
arithmetic quotient . In this paper, we give an explicit formula
for the integral of over , where
is the Petersson norm. More precisely, this integral is given by a
sum over and of quantities , where
is the limit as of the th Fourier
coefficient of the second term in the Laurent expansion at of a
certain Eisenstein series of weight attached to . It
is also shown, via the Siegel--Weil formula, that the value of
the Eisenstein series at this point is the generating function of the volumes
of the divisors with respect to a suitable K\"ahler form. The
possible role played by the quantity in the Arakelov theory
of the divisors on 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 , , 2, 3 be cuspidal
automorphic representations of GL_2(\A) such that the product of their
central characters is trivial. Then the central value
of the triple product L--function is
nonzero if and only if there exists a quaternion algebra over \kay and
automorphic forms , such that the integral of the product
over the diagonal Z(\Bbb A) B^\times(\kay) B^\times(\Bbb
A) is nonzero, where is the representation of B^\times(\A)
corresponding to .
In a previous paper, we proved this conjecture in the special case where
\kay=\Q and the '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
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