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Algebraic Solutions of Differential Equations
111
(7.4.3.1)
In order for the rankone differential equation (K, V~,) on K to
become trivial on a finite extension of K, it is necessary and sufficient
that there exist an integer n > 1 and a function g~ K* such that n co =
dg/g.
(If n co =
dg/g,
then f= g1/n is a nonzero algebraic solution of the equa
tion V~(f)=0. Conversely, iff is a nonzero algebraic solution, let n be
the degree of
K(f)/K.
Then n co =trace(d
f/f)=dg/g,
g= Norm(l/f).)
(7.4.3.2)
If (K, V~,) has pcurvature zero for almost all primes p (in the
sense of (5.4.3.2)) (i. e., if co is ~ locally logarithmic mod p" for almost all
primes p in the same sense (cf. (7.1.3.3)), then co has at worst first order
poles, and rational residues (i. e., there is an integer n > 1 such that n co
is a differential of the third kind on the complete nonsingular model of
K/k).
(For by ([24], Theorem 13.0), the differential equation (K, V~,) has
regular singular points, and rational exponents.)
(7.4.4)
Proposition.
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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