Robert F. Coleman

Torsion points on curves and p-adic Abelian integrals

THEOREM A. Let f: C --* J be an Albanese morphism defined over a number field K, of a s-mooth curve of genus g into its Jacobian. Suppose J has potential complex multiplication. Let S denote the set of primes p of K satisfying i) p does not divide 2 or 3. ii) K/Q is unramified at P. iii) C has good ordinary reduction over K. Then the set T of torsion points of A defined over an algebraic closure of K which lie on the image of C is defined over an extension of K unramified above S. Moreover #To pg

Hodge-Tate periods and p-adic abelian integrals.

The object of this paper is to present two results. The first is a new description of one of the maps involved in the Hodge-Tate decomposit ion of an Abelian variety with good reduction. This result complements recent work of Fontaine IF] and combined with it yields a new proof of this decomposition. The second result is a limit formula involving the integrals described in [C] for what may properly be called Hodge-Tate periods. We will now describe in more detail the contents of this paper. The notations used below are standard and are also explained in Sect. I. Let p be a fixed rational prime. Let II~p be the field of p-adic numbers and ~p the completion of a fixed algebraic closure of II~p. Let A be an Abelian variety defined over a complete discretely valued subfield K of Cp. Let /i denote the dual Abelian variety, and T(A) the p-Tate module of A. Set V(A) = T(A) | 1Igp as a G=Autr The theorem of Tate and Raynaud which describes the decomposit ion of V(A) is equivalent to the following assertion: There exist canonical G-equivariant ~2flinear maps,

A p-Adic Inner Product on Elliptic Modular Forms

This chapter describes a p -Adic inner product on elliptic modular forms. It presents a cohomology class in the first crystalline cohomology group. It is not true however that the map is always an injection. It is found that if F is an eigenform with complex multiplication such that its eigenvalue has valuation 1, it can be shown that the image is zero. This will follow from the theorem as the q invariant of the p -divisible group of an ordinary CM elliptic curve is a root of unity. It is not known whether this map is an injection when restricted to new forms or equivalently whether the inner product is nondegenerate when restricted to new forms. If W is assumed to be a basic wide open and y be a minimal underlying affinoid of W , W is a rigid space isomorphic to a complete curve C with good reduction X minus an affinoid, which after a finite etale base extension becomes a finite nonzero disjoint union of closed disks lieing in distinct residue classes of C .