Potential Type Operators on Curves with Vorticity Points

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.

Torsion Points on Curves

EXAMPLES. (i) The multiplicative group A = Gm is the algebraic group whose points over a field are the nonzero elements of the field. Then for any field K, Tor Gm(K) are the roots of unity contained in K. (ii) A = Gm×Gm then Tor(A) = Tor(Gm)×Tor(Gm) = {(x, y) : x, y ∈ K are roots of unity}. (iii) Let A be an elliptic curve. Over the complex numbers we can uniformize A as A = C/L where L is a la...