Residue Free Differentials and the Cartier Operator for Algebraic Function Fields of one Variable

1972 ◽  
Vol 24 (5) ◽  
pp. 905-914
Author(s):  
Tetsuo Kodama

Let K be a field of characteristic p > 0 and let A be a separably generated algebraic function field of one variable with K as its exact constant field. Throughout this paper we shall use the following notations to classify differentials of A/K:D(A) : the K-module of all differentials,G(A) : the K-module of all differentials of the first kind,R(A) : the K-module of all residue free differentials in the sense of Chevalley [2, p. 48],E*(A) : the K-module of all pseudo-exact differentials in the sense of Lamprecht [7, p. 363], (compare the definition with our Lemma 8).

1969 ◽  
Vol 12 (3) ◽  
pp. 339-341 ◽  
Author(s):  
Nobuo Nobusawa

Let A be an algebraic function field with a constant field k which is an algebraic number field. For each prime p of k, we consider a local completion kp and set Ap = Ak ꕕ kp. Then we have the question:Is it true that A/k is a rational function field (i.e., A is a purely transcendental extension of k) if Ap/kp is so for every p ? In this note we shall discuss the question in a slightly different and hence easier case.


1998 ◽  
Vol 09 (08) ◽  
pp. 1041-1066 ◽  
Author(s):  
ALEXANDRA SHLAPENTOKH

Let K be an algebraic function field over a finite field of constants of characteristic greater than 2. Let W be a set of non-archimedean primes of K, let [Formula: see text]. Then for any finite set S of primes of K there exists an infinite set W of primes of K containing S, with the property that OK,S has a Diophantine definition over OK,W.


1991 ◽  
Vol 72 (1) ◽  
pp. 67-79 ◽  
Author(s):  
Arnaldo Garcia ◽  
Henning Stichtenoth

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