Some papers and preprints of ChingLi Chai
The more recent ones are preprints. Click to download the
.dvi, .pdf or .ps files.
Work on these articles have been supported
by the National Science Foundation since 1990, including the following
grants: DMS1200271, DMS0901163, DSM0400482, DMS0100441,
DMS9800609, DSM9502186,
a Simons Fellowship 561646, and a Simons collaboration grant
701067.
 Tatelinear structures
on formal schemes. A survey article on Tatelinear structures
and their rigidity.
An older version
 Tatelinear formal varieties
and orbital rigidity. A draft of chapter 11 of the book Hecke
orbits with Frans Oort. The main result here generlizes the rigidity
statements for pdivisible formal groups and their biextensions to
all Tatelinear formal varieties, including the formal completions
at closed points of central leaves.
 Stratifying Lie strata of
Hilbert modular varieties (with ChiaFu Yu and Frans Oort). This is
a survey of the Liecongruity stratification of Hilbert modular
varieties in characteristic p>0. A consequence of the main result is that
every nonsupersingular distinguished central leaf in a
Hilbert modular variety in characteristic p is irreducible.
 Sustained pdivisible
groups and a foliation on moduli spaces of abelian varieties. Expanded
version of a plenary talk at the ICCM Annual Meeting in Taipei,
December 2018. A precise definition of "Tatelinear formal subscheme"
of the sustained deformation space attached to a pdivisible group
in characteristic p is formulated, as the expected answer of the
local rigidity question for the space of sustained deformations of
a pdivisible group.
 Sustained pdivisible groups: a
foliation retraced (with Frans Oort).
Chapter 7 of Problems in Arithmetic
Algebraic Geometry edited by Frans Oort, on the progress of a collection
of problems "Some questions in algebraic geometry" posed in 1995,
Higher Education Press and International Press, December 2018, pp. 209233.
 The Hecke orbit conjecture:
a survey and outlook
(with Frans Oort).
Chapter 8 of Problems in Arithmetic
Algebraic Geometry edited by Frans Oort, on the progress of a collection
of problems "Some questions in algebraic geometry" posed in 1995,
Higher Education Press and International Press, December 2018, pp. 235262.
 Moduli of abelian
varieties (with Frans Oort).
Chapter 5 of Problems in Arithmetic
Algebraic Geometry edited by Frans Oort, on the progress of a collection
of problems "Some questions in algebraic geometry" posed in 1995,
Higher Education Press and International Press, December 2018, pp. 95177.
 Sustained pdivisible
groups (with Frans Oort), preliminary version.
A summary of the notion of sustained pdivisible groups. Part of a chapter
of a book project Hecke Orbit with Frans Oort.
a chapter of Arithmetic Algebraic Geometry, a collection
of articles on questions posted in the 1995 collection of questions
Some questions in algebraic geometry, edited by Frans Oort.
 Rigidity for biextensions of
formal groups (with Frans Oort), draft version for a chapter of the
Hecke Orbits book project.
Updated version May 2022.
Statement of the main result: Let E be a biextension of two pdivisible
formal groups X, Y by a pdivisible formal group Z, and the slopes
of the three pdivisible groups are mutually disjoint. Suppose that
have a padic Lie group G operating strongly nontrivially on E,
and we have a formal subvariety W in E which is stable under the action
by G. Then W is a subbiextension of E.
In the proof we introduced a procedure which takes complete equicharacteristic
p local domains as inputs; the outputs are completions of suitable subrings
of the perfections of the input rings. These rings seem to be new.
 Life and work
of Alexander Grothendieck (with Frans Oort), ICCM Notices 5 (2017), 2250.
 A refinement
of the Artin conductor and the base change conductor (with Christian
Kappen), Algebraic Geometry 2 (2015), 446475.
 Mean field equations, hyperelliptic
curves and modular forms: I (with ChangShou Lin and ChinLung Wang),
version 03/15/2015.
Cambridge Journal of Mathematics 3 (2015), 127274.
The motivation of this paper is to study a special case of mean field
equation with a critical parameter: solving the prescribed curvature
equation on an elliptic curve with a singular source placed at the origin.
When the parameter of the singular source is critical, this nonlinear
partical differential equation is integrable in a strong sense:
solving the equation is equivalent to finding meromorphic functions
on the complex plane such that translation by the lattice point
changes the function by a linear fractional transformation coming from
a unitary 2by2 matrix; in addition this soughtafter function has
a fixed multiplicity at the lattice point corresponding to the
value of the critical parameter. The configuration of the zeros and
poles of such function are described by a hyperelliptic curve if
the monodromy condition is relaxed from 2by2 unitary matrices to
2by2 nonsingular matrices. This problem turns out to be very closely
related to the classical theory of Lame functions and Lame differential
equations whose index (one of its parameters) are integers.
For each positive integer n one gets a hyperelliptic curve of genus n
varying holomorphically with the moduli of the elliptic in question.
This hyperelliptic curve goes back to Hermite and Halphen, and
also appeared in KdV theory as a spectral curve when the potential
is of the form n(n+1) times the Weistrass pfunction.
This paper provides a connected account of the many facets of this
family of integrable system in the simpest case.

The period matrices
and theta functions of Riemann
An article
which gives a short exposition four concepts due to Riemann:
(A) Riemann bilinear relations, (B) Riemann matrices, (C) Riemann
theta function and (D) Riemann's theta formula,
plus some historical information about these concepts,
in The Legacy of Bernhard Riemann after One Hundred and Fifty Years,
eds. L. Ji, F. Oort and S.T. Yau, Advanced Lectures in Math. 35,
International Press and Higher Education Press, 2016, pp. 79106.
 An algebraic construction
of an abelian variety with a given Weil number
(with Frans Oort) We give an algebraic proof of the existence of
a CM abelian variety with a given CM type, and deduce from it
the existence part of the HondaTate theorem, without using
complex uniformization of abelian varieties.
Algebraic Geometry 2 (2015), 654663.
 Complex Multiplication and
Lifting Problems
(with Brian Conrad and Frans Oort),
Mathematical Surveys and Monographs, volume 195, American Mathematical
Society, 2014, 387 + ix pp, ISBN 9781470410148.
There are three main results in this book on CM lifting problems
for abelian varieties and pdivisible groups.
(1) a necessary and sufficient condition for the existence of a CM lifting
for an abelian variety over a finite field
with a given CM structure to a characteristic zero
normal domain (chapter 2),
(2) an obstruction for the existence of CM liftings for a pdivisible
group, extending Oort's abelian variety example (chapter 3),
(3) existence of a CM lifting for an an abelian variety with
a given CM structure over a finite field to a characteristic 0 domain
(chapter 4).
Other materials in the book include:
(4) a review of basic CM theory (chapter 1),
(5) a "modern style" proof of the
main theorem of complex multiplication and a converse (appendix A),
(6) existence of algebraic Hecke characters with a given algebraic
part over the field of moduli of the algebraic part which has good
reduction over a given finite set of finite places, existence
of CM pdivisible groups with a given padic CM type over the reflex field
(appendix A),
(7) alternative proofs of the existence of CM lifting using padic Hodge theory
(appendix B).
 Abelian varieties isogenous
to a Jacobian (with Frans Oort),
Annals of Math. 176 (2012), 589635. Under either the GRH or
the AndreOort conjecture, we show that for every g at least 4,
there exists an abelian variety over the field of algebraic numbers
which is NOT isogenous to a Jacobian. Furthermore, there
are at most a finite number curves of genus g whose Jacobian
has complex multiplication by a CM field of degree 2g whose
Galois group has maximal order, 2^g g!.
We also establish the generalization to Shimura varieties.
 Correction to "A note on Manin's
theorem of the kernel", Amer. J. Math. 113, 1991, 387389.
 Local monodromy of Hilbert
modular varieties. We prove a constancy result of the local monodromy
of a Hilbert modular variety on the zero locus of Hasse invariants,
and prove that they are maximal.
 Mumford's example of nonflat
Pic^{\tau}. In Seminaire Bourbaki 1961/62, no. 236, Grothendieck
remarked that Mumford had an example of a nonflat Pic^{\tau}, and said
that it is a deformation of an Igusa surface over an artin ring.
We work out such an example.
 Monodromy and irreducibility
of leaves (.pdf file) (with Frans Oort) We prove that for the
moduli space of principally polarized abelian varieties,
the nonsupersingular Newton polygon strata and leaves are irreducible,
and their padic monodromy are maximal. Annals of Math. 173 (2011),
13591396.
 Moduli of abelian varieties
and pdivisible groups: Density of Hecke orbits, and a conjecture
of Grothendieck (.pdf file) Expository article based on notes for a
Conference on Arithmetic Geometry, Goettingen, July 17  August 11,
2006. In "Arithmetic Algebraic Geometry", Clay Mathematics Proceedings 8,
2009 (Darmon, Ellwood, Hassett, Tschinkel eds.), 441536.
 Methods for padic monodromy
(.pdf file)
We explain three methods for showing that the padic
monodromy attached to a modular family of abelian varieties is
"as large as possible". J. Inst. Math. Jussieu 7 (2008), 247268.
published version
 Hecke orbits and
irreducibility of leaves(with Frans Oort)
Notes for a talk at the Workshop on abelian varieties, Amsterdam,
May 29May 31, 2006.
 Hecke orbits as Shimura
varieties in positive characteristic.
Proceeding of ICM 2006 Madrid, vol II, pp. 295312, a survey areticle on the Hecke orbit
problem. Some errors in the published version are corrected here.
 Hypersymmetric
abelian varieties, .pdf file of published version
(Joint with Frans Oort) We explain the notation of hypersymmetric
abelian varieties. Included are the existence of simple hypersymmetric
abelian varieties over a finite field with a given Newton polygon
satisfying a suitable condition (called "balanced"), as well as
a partial converse to the HondaTate theorem.
This notation was motivated by the Hecke orbit problem.
Pure Appl. Math. Quaterly 2 (Coates special issue), 2006, 127.
 Canonical coordinates on
leaves of pdivisible groups: The twoslope case, .pdf file
The formal completion at any point of a central leaf in a modular
variety of PELtype is builtup from pdivisible formal groups
by a family of successive fibrations. In this article we treat
the essential case where the BarsottiTate group has exactly two
slopes. Then the formal completion at a point of a leaf is
the maximal pdivislble subgroup of the "extension part" of
the local deformation space. A "triple Cartier module", defined
to be the set of all ptypical curves of the Cartier ring functor,
plays an important role.
 Hecke orbits on Siegel
modular varieties, .pdf file
This is a survey article on the Hecke orbit conjecture for
Siegel modular varieties. We sketch a proof of the conjecture and
describe the theories developed for the conjecture.
In "Geometric Methods in Algebra and Number Theory",
eds. F. Bogomolov and Y. Tschinkel,
Progress in Math. 235, Birkhauser, 2004, pp. 71107.
 Families of ordinary abelian
varieties: canonical coordinates, padic monodromy, Tatelinear
subvarieties and Hecke orbits, .pdf file
This paper was motivated by the ordinary case of the Hecke orbit
problem. Methods were developed along several related threads.
Several conjectures were formulated, including an analog of the
MumfordTate conjecture for ordinary abelian varieties.
These conjectures hold in the case of Hilbert modular varieties.
 Hypersymmetric abelian
varieties, .pdf file
This is an older version, introducing the notion of
"hypersymmetric abelian varieties":
their ring of endomorphism up to isogeny is as large as
the slope condition allowed. They play a useful role in the
proof of the Hecke orbit conjecture for Siegel modular varieties.
The new version
is joint with Frans Oort.
 Monodromy of Heckeinvariant
subvarieties, .pdf file of published version, or
Monodromy of Heckeinvariant
subvarieties, .dvi file.
We use a grouptheoretic argument to show that for any
ladic monodromy group attached to the Zariski closure of
the Hecke orbit of a nonsupersingular point in the moduli
space of principally polarized abelian varieties of characteristic
p is equal to the full symplectic group.
Pure Appl. Math. Quarterly 1 (Borel special issue), 2005, 291303.
 A rigidity result
for
pdivisible formal groups, .pdf file
We prove a generalization of the following statement.
Let Z be an irreducible closed formal subvariety of
a formal torus T, and suppose that Z is stable under
multiplication by 1+p^n for some integer n>1.
Then Z is a formal subtorus of T. Asian J. Math. 12 (2008), 193202.
 Elementary divisors of the
base change conductor for tori, .pdf file, or
Elementary divisors of the
base change conductor for tori, .dvi file.
This note contains some estimates for the "elementary divisors"
of the base change conductor for tori over local fields.
One can view these "elementary divisors" as numerical invariants
of local Galois representations, with values in a general linear
group with coefficients in (padic) integers.
The estimates in this note are definitely not sharp, except perhaps
the first and the last one among the elementary divisors.
 A bisection of the Artin conductor,
.pdf file,
or
A bisection of the Artin conductor,
.dvi file. In this paper we give a formula for the base change
conductor of an abelian variety over a local field with potentially
ordinary reduction.
 Neron modesl for semiabelian
varieties: congruence and change of base field, .ps file,
or
Neron modesl for semiabelian
varieties: congruence and change of base field, .pdf file
(Asian Journal of Math. vol. 4, No. 4, 715736, December 2000.)
We study an invariant, called the "base change conductor",
for semiabelian varieties over local fields.
The case for tori is dealt with in the joint paper with J.K. Yu and
E. de Shalit.
 Congruences of Neron Models for
Tori and the Artin Conductors .pdf file, or
.dvi file, or
.ps file
(Joint work with J.K. Yu and E. de Shalit,
Annals of Math. 154, 2001, 347382)
We prove that the Neron
models of two tori are congruent if their Galois representation on
the character groups are sufficiently congruent. The point is that the
two tori may be defined over local fields with different
characteristics. The generic characteristic zero is easier to analyse,
and the case of positive characteristic is "reduced" to that.
 A Note on the Existence of
Absolutely Simple Jacobians, .dvi file or
A Note on the Existence of
Absolutely Simple Jacobians, .ps file
(Joint work with Frans Oort,
Jour. Pure Appl. Alg. vol. 155, 2001, 115120.)
We show that the subset
of curves of genus g over finite fields whose Jacobians are absolutely
simple has positive density in the moduli space of curves.
 Character Sums, Automorphic
Forms, Equidistribution, and Ramanujan Graphs, part1, .dvi file,
Character Sums, Automorphic
Forms, Equidistribution, and Ramanujan Graphs, part1, .pdf file;
and
Character Sums, Automorphic
Forms, Equidistribution, and Ramanujan Graphs, part2, .dvi file,
Character Sums, Automorphic
Forms, Equidistribution, and Ramanujan Graphs, part2, .pdf file.
Joint work with WenChing Winnie Li. The main tools used
are the machinery of ladic cohomology and the converse theorem for
automorphic representations.
 Geometry of Shimura Varieties in
Positive Characteristics, .dvi file or
Geometry of Shimura Varieties in
Positive Characteristics, .ps file
Slides of a talk at ICCM 1998, Beijing.
 Newton Polygons as Lattice
Points, .dvi file or
Newton Polygons as Lattice
Points, .ps file
(Amer. J. Math. 122, 2000, 967990.)
A combinatorial study of Newton points for Fisocrystals with
additional structures, cast against the background of roots and weights.
In connection with Shimura varieties, there is a formula expressing the
predicted dimension of the Newton strata of the reduction of a Shimura variety.
 Density of Hecke Orbits
for Abelian Varieties of pcorank one, .dvi file
or Density of Hecke Orbits
for Abelian Varieties of pcorank one, .ps file
Any such symplectic isogeny class in the moduli space of
gdimensional principally polarized abelian varieties in characteristic
p is dense.
 Local Monodromy for Deformations of
OneDimensional Formal Groups, .dvi file or
Local Monodromy for Deformations of
OneDimensional Formal Groups, .ps file
(J. reine angew. Math. 524, 2000, 227238.)
Over an equalcharacteristic p formal power series ring, if a
onedimensional formal group has a closed fiber with height h
and generic fiber with height 1, then the (h1)dimensional
padic Galois representation is irreducible.
 Density of Members
with Extra Hodge Cycles in a Family of Hodge Structures, .dvi file
or Density of Members
with Extra Hodge Cycles in a Family of Hodge Structures, .ps file
of the galley
You will find an easily computable invariant c such that the set of
all such members is dense if the codimension of the image of the period
map is at most c and the period space is a hermitian symmetric domain.
 The Naturality in
Kirwan's Decomposition, .dvi file or
The Naturality in
Kirwan's Decomposition, .ps file
This is a short joint paper with Amnon Neeman. Please contact us
if you can either prove or disprove the conjectures here.
 Every Ordinary Symplectic
Isogeny Class in Characteristic p is Dense in the Moduli,
.pdf file of published version,
Every Ordinary Symplectic
Isogeny Class in Characteristic p is Dense in the Moduli, .dvi
file or
Every Ordinary Symplectic
Isogeny Class in Characteristic p is Dense in the Moduli, .ps
file
The main result is already stated in the title. Please let me know
if you find an application of the main theorem here.
 The Group Action on the Closed
Fiber of the LubinTate Moduli Space, .dvi file or
The Group Action on the Closed
Fiber of the LubinTate Moduli Space, .ps file
Motivated and written before the main result of the previous paper was proved,
this whole paper is a longanddirty calculation of the
initial terms of the action of the stabilizer subgroup in the LubinTate case.
(This group is also known as the Morava stabilizer subgroup in
stable homotopy theory.) This is one of the few cases where
highorder deformations are calculated, and may be of interest only to those
who like to make further progress in understanding this action.
Back to ChingLi Chai's Home Page