Trivial points on towers of curves

Modular Towers of Noncongruence Curves

This proposal extends much work done during the last funding period. Two topics, however, which have separate goals, have brief summaries in x10. Together they form documentation of the results that came from the funding period of NSF GRANT #9622928. 1. An outline of the main problems of this proposal Let G be a nite group. Call a conjugacy class C a p 0-class if its elements have order prime t...

Cartier Points on Curves

Throughout this paper Y will denote a complete, nonsingular, and irreducible algebraic curve of genus g > 0 over the algebraically closed field k of characteristic p. Eventually we will assume that g ≥ 2. Denote by W the g-dimensional k-vector space of regular differentials of Y/k, which is just the vector space H(Y,ΩY/k). Also let K(Y ) denote the function field of Y , and let ΩK(Y )/k be the ...

Rational points on curves

Rational points on curves

2 Faltings’ theorem 15 2.1 Prelude: the Shafarevich problem . . . . . . . . . . . . . . . . 15 2.2 First reduction: the Kodaira–Parshin trick . . . . . . . . . . . 17 2.3 Second reduction: passing to the jacobian . . . . . . . . . . . 19 2.4 Third reduction: passing to isogeny classes . . . . . . . . . . . 19 2.5 Fourth reduction: from isogeny classes to `-adic representations 21 2.6 The isogen...

Cyclotomic Points on Curves

We show that a plane algebraic curve f = 0 over the complex numbers has on it either at most 22V (f) points whose coordinates are both roots of unity, or infinitely many such points. Here V (f) is the area of the Newton polytope of f. We present an algorithm for finding all these points.