Publications (this might not be up to date, see
Recent Preprints too)
Book:
(with S. Kudla and
M. Rapoport) Modular forms and special cycles on Shimura
curves, Annals of Math. Studies series, vol 161, 2006.
Papers:
(with J.Bruinier) CM
values of automorphic Green functions on orthogonal groups
over
totally real fields, in Arithmetic Geometry and Automorphic Forms
(in
honor of S. Kudla's 60th birthday), pp55.
(with Ben Howard) Singular moduli
refined, to appear in a book in honor of Stephen Kudla's 60th birthday.
(with R. Masri), NONVANISHING OF HECKE
L-FUNCTIONS FOR CM FIELDS AND RANKS OF ABELIAN VARIETIES, to appear in
GAFA.
(with J. Bruinier) Faltings
heights of CM
cycles and derivatives of $L$-functions, Invent. Math.
177(2009), 631-681.
(with S. S. Kudla) Eisenstein series for SL(2),
Sci. China Math. 53 (2010), 2275–2316.
Arithmetic Intersection on a Hilbert Modular
Surface and the Faltings Height, preprint 2007, pp 47.
An arithmetic intersection
formula on Hilbert
modular surfaces, Amer. J. Math. 132(2010), 1275-1309.
Chowla-Selberg Formula and
Colmez's
Conjecture, Can. J. Math., 62(2010), 456-472.
(with B.D. Kim and R. Masri), NONVANISHING
OF HECKE L-FUNCTIONS AND THE BLOCH-KATO CONJECTURE, to appear in
Math. Ann., pp38.
(with K. Bringmann) On Jacobi Poincare
Series of small weights, IMRN(2007)
Minimal CM Liftings of
Supersingular Elliptic Curves, Pure and Appl. Math. Quarterly
(2007)
(with
Jan Bruinier) CM values of Hilbert modular
functions,
Invent. Math. 163(2006), 229-288.
(with Jan Bruinier) Twisted Borcherds
products
and their CM values, Amer. J. Math.,129(2007), 807-842.
(with N. Elkies and K. Ono) Reduction of CM
elliptic
curves and modular function congruences
, IMRN 44 (2005), 2695-2708.
(wtih S. Kudla and M. Rapoport) Derivatives
of Eisenstein Series and Faltings heights, Comp. Math.,
140 (2004), 887--951.
The second term of an Eisenstein series,
to
appear in 2nd ICCM proceeding, pp19.
CM number fields and modular forms,
Quarterly Jour. Pure Appl. Math. special issue
in memory of A. Borel
1(2005), 305-340..
A. Classical Modular forms
- Cusp forms of weight 1 associated to the Fermat curves, Duke
Math. J. 83(1996) 141-156
<>Using the theory of algebraic geometry and theta functions, we
discovered
a surprisingly simple formula for the dimension of the space of cusp
forms
of weight one associated to the Fermat curves. We also gave a basis for
the concerned space. We notice such a formula does not exist for
congruence
groups.
B. Theta liftings, Weil representations, Central Hecke L-values,
and elliptic
curves
- Theta liftings and Hecke
L-functions, J.
Reine Angew. Math. 485(1997)25-53.
<>Using the theory of theta lifting, we discovered a formula to
the
express
central L-value of certain Hecke characters of a CM number field as the
inner product of some theta lifting from $ U(1)$ to itself. It follows
immediately that the central L-value is nonnegative which is also
predicted
by the generalized Riemann hypothesis. The formula also recovers a
theorem
of Rogawski on nonvanishing of global theta lifting in a natural way.
Other
applications are given in papers 4-6. - Eigenfun
ctions of Weil
representation of unitary
groups of one variable, Trans. AMS, 350(1998)2393-2407.
<>We constructed {\it{explicit}} eigenfunctions of Weil
representation
of unitary groups of one variable by means of lattice model. In
addition
to its own interest, it is also needed in the next three papers. - (with
F. Rodriguez Villegas) Central
values of
Hecke L-functions of CM number fields, Duke Math. J.,
98(1999)541-564.
<>By computing the theta lifting in the formula of Paper 1, we
discovered
an interesting formula to express central Hecke L-values of CM number
fields
in terms of special values of classical theta functions at CM points.
The
theta functions involved is {\it{only}} dependent on the totally real
subfield
(not on the CM number field itself). In particular, we proved that the
central L-value $L(k+1, (\chi_{p, d})^{2k+1})=0$ if and only if all the
Heegner points of $X_0(4 d^2)$ with endomorphism ring $\Cal O_E$ are
roots
of a theta function only dependent on $d$ and $k$. Here $E=\Bbb
Q(\sqrt{-p})$,
$p\equiv 7 \mod 8$, $d \equiv 1 \mod 4$, every prime divisor of $d$ is
split in $\Bbb Q(\sqrt{-p})$, $(-1)^k =\hbox{sign} (d)$, and $(2k+1,
h_p)
=1$. Also $\chi_{p, d}$ is a Hecke character of $E$ associated to the
elliptic
curve $A(p)^d$ defined by Gross in his thesis (LNM 776). Applying this
result and a little calculus, we proved that for every integer $k \ge
0$,
there is an integer $M(k)$ such that for all the pairs $(p,d)$
satisfying
the conditions just mentioned and $\sqrt p > M(k) d^2$, the central
L-value
$L(k+1, (\chi_{p, d})^{2k+1})\ne 0$. We also computed $M(0)$ and $M(1)$
explicitly. - Nonvanishing of central Hecke
L-value
and rank of
certain elliptic curves, Compositios
Math.,
117(1999), 337-359.
<>In this paper, we tried to remove the condition that every
prime
divisor
of $d$ is split in $\Bbb Q(\sqrt{-p})$ from the result mentioned in the
above paper. We succeeded the goal by overcoming three techinal
problems.
One result in this paper is as follows: The elliptic curve $A(p)^d$
defined
in Gross's thesis (LNM 776) has rank $0$ if $p\equiv 7 \mod 8$, $d
\equiv
1 \mod 4$, and $\sqrt p > d^2 \log d$. Moreover When every prime
factor
of $d$ is inert in $\Bbb Q(\sqrt{-p})$, and is congruent to 1 modulo 4,
then $d^2$ can be replaced by $d$. $ (d >0)$. - Common z eros of theta functions and
central Hecke
L-values of CM number fields of degree 4,
Proc.
AMS., 126(1998)999-1004.
<>In this note, we gave two applications of the main formula in
paper
4 when the totally real field is a real quadratic number field. - (with
M. Stoll) On the L-function
of the curve
$y^2=x^5 +A$, to appear in J. London Math.
Soc.
<>In this note, we computed the special value of the L-series of
the
genus
two curve in the title at the center. We also studied its arithmetic
implications. - (with J. Jimenez-Urroz) Heegner
zeros of theta
functions, , to appear in Trans. AMS
<>Let $N \ge 1$ be an integer and let $f$ be a meromorphic
modular
form
of level $N$ with algebraic Fourier coefficients. The zeros and poles
of
$f$, viewed as points on the modular curve $X_0(N)$, are algebraic.
However,
if we let $\tau$ be a preimage of such a point, (if it is not a cusp),
in the upper half plane $\Bbb H$, then it is well-known that $\tau$ is
either quadratic (Heegner point) or transcendental. So it is very
interesting
to isolate and understand the Heegner zeros/poles of $f$. Although
Heegner
points play very important roles in many branches of number theory,
such
as the Gross-Zagier formula, Kolyvagin's Euler system, and the
Borcherds
product theory, to name a few, little is known about the Heegner zeros
of modular forms. In this note, we study Heegner zeroes for a family of
classical theta functions $$ \theta_d(z) $$ and raise a couple of
interesting
questions (to us).
C. Central derivatives of Eisenstein series and Automorphic
L-functions
- (with S. Miller) Nonvanishing of the
central
derivative
of canonical Hecke L-functions, with
applications
to the ranks and Shafarevich-Tate groups of Q-curves. Math.
Res.
letters 7(2000)263-277
- (with S. Kudla and M. Rapoport) On the
derivative
of an Eisenstein series of weight one, Intern. Math. Res.
Notices
7(1999) 347-385
<>The general philosophy is that if a `nice' function from number
theory
is forced to be zero at its center, its derivative at the center should
be very interesting and might be linked to arithmetic of a geometric
subject.
The most famous example is the Gross-Zagier formula. Recent work of
Kudla
on the central derivative of an Eisenstein series is another example.
There
is a systematic way to produce Eisenstein series on $Sp(n)$ which
vanishes
at the center from `incompatible' local quadratic space system of
dimension
$n+1$. Kudla has a general scheme to compute the central derivative of
such Eisenstein series, and conjecture that its restriction on
subgroups
of $Sp(n)$ (by doubling method for example) are closely related to the
global intersection numbers of cycles on Shimura varieties arising
naturally.
Moreover, this identity should be able to be proved term by term at
each
prime. He verified the case $n=2$ at good primes and infinity. As
usual,
bad primes cause a lot of technical difficulty. This joint paper is an
attempt to give a complete picture in a very simple example, i.e.,
Eisenstein
series on $Sp(1)$ coming from `incompatible' local quadratic space
system
of dimension $2$. We succeeded and the resulting central derivative of
this Eisenstein series turns out be a very interesting non-holmorphic
modular
form of weight 1. Its Mellon transform looks particularly neat.
Moreover,
we give a geometric interpretation of the Fourier coefficients of the
resulting
modular form. We also learned some tricks to attack in general
situations. - (with S. Kudla and M. Rapoport)
Derivatives of Eisenstein
series and Faltings heights, Compos. Math., 140 (2004),
887–951.
- Central derivatives of
Hecke
L-series,J.
number theory 85(2000) 130-157
<>This is both a continuation of project B and an example to
realize
the
general philosophy just mentioned. Let $\chi$ be a Hecke character
studied
in project B. When its global root number is one, it was studied in
project
B. When its global root number is $-1$, the central L-value is
automatically
zero, and it is very interesting and natural to study its central
derivative.
We have written the L-function as the integral of certain
{\bf{explicit}}
Eisenstein series using doubling method. We have also computed the
central
derivative of the Eisenstein series using Kudla's scheme, including bad
primes. We further computed its integral, and thus obtained a formula
for
the central derivative of the Hecke L-function. There are still two
problems
to work with. One is to make sense the formula, i.e., to give some kind
of geometric interpretation. The other is to somehow further simplify
the
formula and to derive some nonvanishing result similar to that of
project
B. This would in particular imply that Gross's elliptic curves $A(p)^d$
has rational rank one under certain easy to verify conditions. In this
note, we found an explicit formula for the central derivative L'(1,
\chi_p). - The derivative of Zagier's
Eisenstein series
and Faltings' height in Heegner points and Rankin L-series, MSRI
Publ.
49 (H. Darmon and S.W. Zhang eds.), 271-284.
- The second term of an Eisenstein series to
appear in Proceeding of the 2nd International Congress of Chinese
Mathematicians,
pp18
- Taylor expansion of an Eisenstein
series,
Trans.
Amer. Math. Soc., 355 (2003)2663-2674.
- (with S. Kudla and
M. Rapoport) Modular forms and special cycles on Shimura
curves, Annals of Math. Studies series, vol 161, 2006.
D. Local densities and local Whittaker functions
- An explicit formula for local densities
of
quadratic
forms , J. number theory 72(1998)309-356.
<>Local densities of quadratic forms was first introduced by
Siegel
to study the problem to represent a number (or a positive definite
quadratic
form) by a (another) positive definite integral form. Local densities
are
very closely related to local Whittaker functions. In fact, they are
two
languages to express the same thing, the local factors in the Fourier
coefficients
of an Eisenstein series. It is very important to have explicit formulas
for them in studying arithmetics of Eisenstein series or quadratic
forms.
However, very few is known except for the `unramified' case (even in
the
unramified cases, not much is known). The usual method is to use some
kind
of reduction formula and functional equation to reduce it simple cases
one can handle. This kind of method works very poorly in `ramified
case',
where the reduction formula does not work until in later stage. In this
two preprints, we instead compute the local Whittaker functions
directly,
and have succeeded in obtaining an explicit formula in a quite general
situation. In the process, we have also completely determined a1. Gauss
integral over $GL_2(\Bbb Z_p)$. In examples related to quaternions, we
obtained very nice formulas, which will be used by Kudla and Rapoport
to
prove an important local identity at a bad prime relating the central
derivative
of Whittaker functions with local intersection number of cycles in
certain
Shimura curves. - 2-adic density ,
J.
number theory,
108(2004), 287-345.
E. Hecke characters and CM abelian varieties
- On CM abelian varieties over an imaginary
quadratic
field whose CMs are defined over the field.
Math. Ann. 329(2004), 87-117.
- On the existence of algebraic
Hecke
characters,C.R.
Acad. Sci. Paris Ser. I Math. 332(2001)1041-1046
<>In this short note, we proved that a weak
version of the
Brumer-Stark
conjecture is equivalent to the existence of certain algebraic Hecke
characters.
As a consequence, we showed that the CM case of the Brumer-Stark
conjecture
implies the general case as expected.
F. Borcherds Products and their CM values.
1.(with
Jan Bruinier) CM values of Hilbert modular
functions,
Invent. Math. 163(2006), 229-288.
2. (with Jan Bruinier) Twisted
Borcherds
products
and their CM values, to appear in Amer. J. Math., pp30.
G.
Supersingular elliptic curves and Quaternions
1. Minimal CM Liftings of
Supersingular Elliptic Curves, preprint.
2. (with N.
Elkies and K. Ono) Reduction of CM
elliptic
curves and modular function congruences
, IMRN 44 (2005), 2695-2708.
<>
H. Algebra
The following are my graduate work (toward my MS degree) in ring
and
modular theory, and holomorphic algebra. They are in good quality and
indicate
my solid algebra background. - Cocritical modules and
finiteness relative to a module, Comm. Algebra (1989),
217-235.
- (with Huaiding Tang, Gianlong Chen) Flat modules and ML
modules
relative
to an hereditary torsion theory, J. Math (Chinese) 12 (1992),
213-220.
- On the algebraic structure of group rings, J. University
of
Science
and Technology of China, (1990) 1-8.
- On noncommutative semicoherent rings, Chinese Ann. Math.
Ser.A 10
(1989), 148-152.
- (with Huaiding Tang, Gianlong Chen) Fp*-injective, fp-flat
modules and
coherent rings, J.Anhui Normal Univ. 1(1988), 1-7.
- (with Huaiding Tang, Gianlong Chen) Fp*-injective dimension,
pf-flat
dimension and FPQ ring, J. Anhui Normal Univ., 2(1987), 11-16.
- (with Gianlong Chen) Weakly globally homological dimension
and
direct(inverse)
limits, J. Anhui Normal Univ., 4(1986), 79-85.
- (with Gianlong Chen) Some results on coherent rings, J.
Anhui Normal
Univ., 4(1986), 73-78.
F. Other Preprints
- Two dimensional Artin representations of conductor dividing
256,
preprint,
1995.
- Special values of Hecke L-functions and symmetric power of
CM
elliptic
curves: odd case, preprint, 1995.
- Special values of Hecke L-functions and symmetric power of
CM
elliptic
curves: even case, in preparation.