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86
N.M. Katz:
Proof
By (5.4.1), (6.2.1).r
and by (6.0.4.3), (6.2.2)r
Be
cause the hypergeometric equation has regular singular points, (6.2.3)~
(6.2.4). By (5.4.4), we have (6.2.2)~ (6.2.5). By ([24], Theorem 1.3.0), (6.2.5)
implies that
E(a, b, c)
has regular singular points on
Sc,
and that its
local monodromy
around each singular point 0, 1, oo is of finite order. In
particular, (6.2.5) implies that
E(a, b, c)
on Sc has
rational
exponents at
each of 0, 1, ~, or, what is the same, that the hypergeometric equa
tion (6.1.2) with parameters a, b, c has rational exponents at 0, 1, ~. As
the exponents are 0 and 1 c at 0, 0 and c a b at 1, and a and b at ~,
(6.2.5) implies that a, b, c~Q. The rest of the implication (6.2.5)~ (6.2.6)
results from the fact that
E(a, b, c)
"comes from" Tan open subset of
Spec(Z), so that we may test for "pcurvature zero for almost all p" by
seeing if the reduction modp on
E(a, b, c)
on Svp has pcurvature zero
for almost all p (and perform this latter test prime by prime availing
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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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