Ching-Li Chai

Complex Multiplication and Lifting Problems

Introduction Algebraic theory of complex multiplication CM lifting over a discrete valuation ring CM lifting of

Hecke orbits on Siegel modular varieties

p

A note on the existence of absolutely simple Jacobians

-divisible groups CM lifting of abelian varieties up to isogeny Some arithmetic results for abelian varieties CM lifting via

METHODS FOR

p

p

-adic Hodge theory Notes on quotes Glossary of notations Bibliography Index

-ADIC MONODROMY

We sketch a proof of the Hecke orbit conjecture for the Siegel modular variety \(\mathcal{A}_{g,n}\) over \(\overline {\mathbb{F}_p }\), where p is a prime number, fixed throughout this article. We also explain several techniques developed for the Hecke orbit conjecture, including a generalization of the Serre-Tate coordinates.

A Note on Manin's Theorem of the Kernel

Abstract Suppose given a prime number p, and a positive integer g. We show there exists a curve of genus g over a finite field of characteristic p such that its Jacobian variety is absolutely simple.

An algebraic construction of an abelian variety with a given Weil number.

We explain three methods for showing that the p-adic monodromy of a modular family of abelian varieties is ‘as large as possible’, and illustrate them in the case of the ordinary locus of the moduli space of g-dimensional principally polarized abelian varieties over a field of characteristic p. The first method originated from Ribet’s proof of the irreducibility of the Igusa tower for Hilbert modular varieties. The second and third methods both exploit Hecke correspondences near a hypersymmetric point, but in slightly different ways. The third method was inspired by work of Hida, plus a group theoretic argument for the maximality of -adic monodromy with = p.

Moduli of Abelian Varieties

The purpose of this note is to show that Manins original statement of the theorem of the kernel is an easy consequence of Delignes theory of differential equations with regular singularities and Delignes Hodge theory. Since this note is primarily of historical interest, some discussion about the history may be beneficial to the reader, and I stand to be corrected for any inaccuracy. More than a quarter of a century ago Manin in [M] proved Mordells conjecture for function fields over C. His major tool was what is now commonly called the Gauss-Manin connection, introduced by Manin. A major step is a statement called the theorem of the kernel, an equivalent form of which will be recalled later. Recently, Robert Coleman found a mistake in elementary linear algebra in the proof of the theorem of kernel in [M], which unfortunately invalidated the whole proof. Coleman however proved a weaker version of the theorem of kernel, and deduced Mordells conjecture over function fields following Manins original ideas, and he used analogue of Siegels theorem on integral points over function fields over C. In this note, it will be shown that Manin was right after all, and it is possible to make a local correction using Delignes theorems. It may be of interest to note that our proof uses global monodromy of the Gauss-Manin connection systematically instead of the local monodromy as in [M]. I learned of Manins beautiful ideas from lectures of Professor Coleman on Jan. 7-8, 1989 at the Tucson conference on the arithmetic of curves. This note is a response to the stimulating question he raised at Tucson, and I thank him very much. I would also like to thank Carlos Simpson for pleasant discussions and a pertinent remark. I strongly recommend Manins original paper [M] for readers interested in Manins circle of ideas about Mordels conjecture for function fields (over C). For a modern treatment of Manins proof using algebraic DeRham cohomology, the reader is referred to [C].

Toroidal Compactification of A g

A classical theorem of Honda and Tate asserts that for every Weil q-number , there exists an abelian variety over the nite eld Fq, unique up to Fq-isogeny, whose q- standard proof (of the existence part in the Honda-Weil theorem) uses the the fact that for a given CM eld L and a given CM type for L, there exists a CM abelian variety with CM type (L; ) over a eld of characteristic 0. The usual proof of the last statement uses complex uniformization of (the set of C-points of) abelian varieties over C. In this short note we provide an algebraic proof of the existence of a CM abelian variety over an integral domain of characteristic 0 with a given CM type, resulting in an algebraic proof of the existence part of the Honda-Tate theorem which does not use complex uniformization. Dedicated to the memory of Taira Honda.