60
N.M. Katz:
T he corresponding spectral sequence in locally free (gs.nmodules of
f inite rank
tyn,k+n_
nknIr~an
F lan/~'an)
Wt'I (an)
(4.1.1.2)

(~
D R ~xlil n , . . n L " i ~ / ~
!
i,<.<i,
R
76
N.M. Katz:
z ero. This implies that for every maximal ideal m of B such that B /m B
h as characteristic p~2;, the inverse image on B /m B o f the mapping p(D)
v anishes, i.e., all the matrix coeffi
Algebraic Solutions of Differential Equations
77
Then after the change of base S pec(C)~ T (cf. the diagram (5.1.1.0)
each of the coherent sheaves with integrable connection on Sc
(P(x) R"fc.(t2~Cc/s
78
N.M. Katz:
a fter the base change Spec(Fq)~ T
V QFq
I
S pec(Fq)
>~cJ#
1
. ,T
t he pcurvature of the free (_9~ Fqmodule with integrable connection
(M, 17)1~/) F0
is zero.
(5.4.3.2) A standard
Algebraic Solutionsof DifferentialEquations
79
o ver S and cross normally relative to S, such that the fibre of X (respect ively of Di) over the generic point of Sc is XK (resp. Di, K).
(5.4.5.1) The
80
N.M. Katz:
6. Applications to the Hypergeometric Equation
6.0. Relations with Ordinary Differential Equations
(6.0.0) Let T a scheme, S a smooth Tscheme, and (M, V) a locally free
(gsmodule of fi
Algebraic Solutionsof DifferentialEquations
81
a nd "decorate" it with the relative Frobenius F : S~ Stp)
S
F
~, S(p)
a
~S F
~, S(p)
(6.0.2.5)
T
f ~bs ~ T
T he corresponding diagram of function field
82
N.M. Katz:
t hat whenever h orizontal s ections m w , o ne for each exponentsystem W
a s above verify
~ XW mw=O
(6.0.3.1)
in M
W
t hen all m w= O.
IWl=~ W~eZ,
; t his has a meaning
and put D W=[I
A lgebraic Solutions of Differential Equations
83
L et eo . , ~,_ 1 denote the d ual b asis of the dual module AS/. A section
(6.0.4.1)
~ fi ei,
i=o
f/ local sections of (9s
is horizontal for the dual
84
N . M . Katz:
ordv (i~i fip ti) =mini(ordv(ff ti)
(6.0.6.1)
= min i (i + p ordvo (f~P).
L et Kv (resp. (KV)~o) d enote the completion of K (resp. K P) with
r espect to the valuation v (resp. Vo). T
Algebraic Solutions of Differential Equations
85
6.1. The Hypergeometric EquationDefinition
(6.1.0) For any scheme T, we will denote by 2 the standard coordinate
o n A~, and by S r the open subset of
86
N.M. Katz:
P roof By (5.4.1), (6.2.1).r
and by (6.0.4.3), (6.2.2)r
Bec ause 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]
Algebraic Solutions of Differential Equations
87
6.4. Solutions of the Hypergeometric Equation in Characteristic p
(6.4.0) Proposition (compare [ 23]). L et a, b, c be integers contained in
cfw_0, 1,
Algebraic Solutions of Differential Equations
75
(5.1.1) Then after the change of base Spec(C)* T,
Dc
.* D
Xc~
X
(5.1.1.0)
1
Sc~S
S pec(C)
, T
the coherent sheaf with integrable connection on S c
74
N.M. Katz:
l ocally free module of finite rank. (Such a 0/always exists.) Then, over ~,
t he Hodge ~ De Rham spectral sequence
(5.0.1.0)
E f'~=Rqf, (s
(log D) ~ R"+qf, (f2~/s (log O)
is degenerate
A l g e b r a i c Solutions of Differential E q u a t i o n s
73
(4.4.2.3) There exists a finite ~tale covering q~: S'~ S such that, for every
irreducible Z on the Qconjugary class A, q~*(P(x) R"f,(
Algebraic Solutions of DifferentialEquations
61
(4.1.1.9) Lemma. The morphism (4.1.1.8) o f spectral sequences induces an
isomorphism between (4.1.1.6) and the subspectral sequence in local systems
o
62
N.M. Katz:
is an isomorphism between its source and the subsheaf Rq f,(f2~c/~) v of
g erms of horizontal sections (for the GaussManin connection) of
R q/, (f2~./~).
P roof T he proof is an exercis
Algebraic Solutionsof DifferentialEquations
63
C onsequently, the spectral sequence (4.1.2.1) is degenerate at E 2, a nd
t he canonical mappings
(4.1.2.11)
Rqf, (O~/c) ~ E ~ q , E ~
a re isomorphisms
64
N.M. Katz:
A p olarization of a pure Hodge structure H of weight n is a homom orphism of Hodge structures
(4.2.0.8)
(,)H
s uch that the real bilinear form on H R
(4.2.0.9)
(2 g i)" (x, C y)
is sym
A lgebraic Solutions of Differential Equations
65
w hichpoint by point is a polarization (4.2.0.8). A family of pure Hodge
s tructure is called polarizable i f it admits at least one polarization. Th
66
N.M. Katz:
s ense that it comes ffom a filtration by sublocal systems of the complexified
local system H c. Then there exists a finite Otale covering ~z: Sf'~ 5t~ such
t hat the inverse image z~*
A lgebraic Solutions of Differential E q u a t i o n s
67
For each integer n>0, the sheaf R"n,"(Z) on San is a local
system o f Zmodules of finite type. As we have seen (4.1.1), the corre
(4.3.0.1)
68
N.M. Katz:
W hen tensored with Q, it degenerates at E3 and defines (a renumbering
of) the filtration W on R"n,n(Q). Thus the assosciated graded families
o f Hodge structures grw of our family of mi
Algebraic Solutions of Differential Equations
69
A ccording to the "primitive decomposition" ([43], pp. 7779), for n < N
t he mapping
(4.3.1.7)
t~
Prim.2if, n(z)(_i)
~L, , R " f , " ( Z )
0 < i < [
70
N.M. Katz:
B y (4.2.2.3), (4.3.1), and "Riemann's existence theorem" (cf. [36] and
t he appendix) we have (4.3.3.1) ~ (4.3.3.2).
T o see that (4.3.3.2) ~ (4.3.3.3), notice that (4.3.3,2) implies in
A lgebraic Solutions of Differential Equations
71
m ixed Hodge structures on S an
( n n( O an, Z)san ~ R" 7~,n (Z)
(4.3.5.0)
has as image a constant subfamily of mixed Hodge structures. Its image is
72
N.M. Katz:
F or each Qconjugacy class A of irreducible Crepresentations of G,
t he element
(4.4.0.2)
P (A) der'" ~ P(Z)
Z~A
o fC [G] lies in fact in Q [G], and is an indecomposable central idempo
88
N.M. Katz:
T hen the matrix of f (  a ,  b ,  c ) on N is
/ P(O)
Q(O)
P(1) Q(I!.
(6.4.0.9)
[email protected] ( 2 )
."."."." ' .
It's proper values are thus its diagonal terms. If a = b, then all but one of
the
Algebraic Solutionsof DifferentialEquations
89
w hich has the shape
0 Q(a)
*
\
Q (a+ 1)
(6.4.0.13)
", Q(b 1)
o
/
t he ,'s indicating nonzero diagonal terms. If n one o f the offdiagonal
t erms vani