Work on these articles have been supported by the National Science Foundation since 1990, including the following grants: DMS-1200271, DMS09-01163, DSM04-00482, DMS01-00441, DMS98-00609, DSM95-02186,

- A refinement
of the Artin conductor and the base change conductor (with Christian
Kappen), accepted for publication by
*Algebraic Geometry.*

*Mean field equations, hyperelliptic curves and modular forms: I (with Chang-Shou Lin and Chin-Lung Wang), version 03/15/2015, 122 pp. Accepted for publication by**Cambridge Journal of Mathematics*.

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 non-linear 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 2-by-2 matrix; in addition this sought-after 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 2-by-2 unitary matrices to 2-by-2 non-singular 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 p-function. 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, for a volume celebrating the 150th year after Riemann,*

*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 Honda-Tate theorem, without using complex uniformization of abelian varieties. Accepted for publication by**Algebraic Geometry*.

*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 978-1-4704-1014-8.*

There are three main results in this book on CM lifting problems for abelian varieties and p-divisible 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 p-divisible 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 p-divisible groups with a given p-adic CM type over the reflex field (appendix A),

(7) alternative proofs of the existence of CM lifting using p-adic Hodge theory (appendix B).

*Abelian varieties isogenous to a Jacobian (with Frans Oort), Annals of Math. 176 (2012), 589-635. Under either the GRH or the Andre-Oort 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, 387-389.*

*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 non-flat Pic^{\tau}. In Seminaire Bourbaki 1961/62, no. 236, Grothendieck remarked that Mumford had an example of a non-flat 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 non-supersingular Newton polygon strata and leaves are irreducible, and their p-adic monodromy are maximal. Annals of Math. 173 (2011), 1359-1396.*

*Moduli of abelian varieties and p-divisible 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.), 441-536.*

*Methods for p-adic monodromy (.pdf file) We explain three methods for showing that the p-adic monodromy attached to a modular family of abelian varieties is "as large as possible". J. Inst. Math. 7 (2008), 247-268. published version*

*Hecke orbits and irreducibility of leaves(with Frans Oort) Notes for a talk at the Workshop on abelian varieties, Amsterdam, May 29-May 31, 2006.*

*Hecke orbits as Shimura varieties in positive characteristic. Proceeding of ICM2006 Madrid, 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 Honda-Tate theorem. This notation was motivated by the Hecke orbit problem. Pure Appl. Math. Quaterly 2 (Coates special issue), 2006, 1-27.*

*Canonical coordinates on leaves of p-divisible groups: The two-slope case, .pdf file The formal completion at any point of a central leaf in a modular variety of PEL-type is built-up from p-divisible formal groups by a family of successive fibrations. In this article we treat the essential case where the Barsotti-Tate group has exactly two slopes. Then the formal completion at a point of a leaf is the maximal p-divislble subgroup of the "extension part" of the local deformation space. A "triple Cartier module", defined to be the set of all p-typical 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. Progress in Math. 235, Birkhauser, 2004, pp. 71-107.*

*Families of ordinary abelian varieties: canonical coordinates, p-adic monodromy, Tate-linear 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 Mumford-Tate 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 Hecke-invariant subvarieties, .pdf file of published version, or Monodromy of Hecke-invariant subvarieties, .dvi file. We use a group-theoretic argument to show that for any l-adic monodromy group attached to the Zariski closure of the Hecke orbit of a non-supersingular 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, 291-303.*

*A rigidity result for p-divisible 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), 193-202.*

*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 (p-adic) 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, 715-736, 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, .dvi file or Congruences of Neron Models for Tori and the Artin Conductors, .ps file*

(Joint work with J.-K. Yu and E. de Shalit, Annals of Math. 154, 2001, 347-382) 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, 115-120.)

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 Wen-Ching Winnie Li. The main tools used are the machinery of l-adic 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, 967-990.)

A combinatorial study of Newton points for*F*-isocrystals 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**p*-corank one, .dvi file or Density of Hecke Orbits for Abelian Varieties of*p*-corank one, .ps file Any such symplectic isogeny class in the moduli space of*g*-dimensional principally polarized abelian varieties in characteristic*p*is dense.

*Local Monodromy for Deformations of One-Dimensional Formal Groups, .dvi file or Local Monodromy for Deformations of One-Dimensional Formal Groups, .ps file*

(J. reine angew. Math. 524, 2000, 227-238.)

Over an equal-characteristic*p*formal power series ring, if a one-dimensional formal group has a closed fiber with height*h*and generic fiber with height*1*, then the*(h-1)*-dimensional*p*-adic 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 Lubin-Tate Moduli Space, .dvi file or The Group Action on the Closed Fiber of the Lubin-Tate Moduli Space, .ps file*

Motivated and written before the main result of the previous paper was proved, this whole paper is a long-and-dirty calculation of the*initial terms*of the action of the stabilizer subgroup in the Lubin-Tate case. (This group is also known as the Morava stabilizer subgroup in stable homotopy theory.) This is one of the few cases where high-order deformations are calculated, and may be of interest only to those who like to make further progress in understanding this action.