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 kf,"(a~/s(log O) a")
is of formation compatible with ar
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 coefficients of p (D) lie in m. Because B is a
f initely gene
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/sc(log Oc) V),
x~A
becomes trivial on a finite &ale cove
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 "passage to the limit" shows that if the property
(5.4
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 stalk over the generic point of Sc of the coherent (9sm
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 finite rank with an integrable Tconnection. Recall that
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 fields is
K,
, k . KPt . . . .
K+_
\k.K p
o 61
k ~ xp~x k
B
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
i
W~!
b ecause all Wi_<p 1. For any two exponent V, W
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 connection V on M / f a nd only if i ts coefficients
s
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). Then (6.0.6.0) gives
(6.0.6.2)
K v ~ (KP)vo0 " " 9 (KV)
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 A~ where the section 2 ( 1  2 ) e
F (A~, (9A~) is inv
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], Theorem 1.3.0), (6.2.5)
implies that E(a, b, c) h as
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, ., p  1. I n order that the hypergeometric equation wi
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
(5.1.I.1)
(R"fc,(~ic/sc(log D), V
becomes trivial on a
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 at E 1. Let us recall why this is so. By Deligne's mixe
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,(O'x/s(log D), V)
is isomorphic to (gs,)b"(x), d) as a c
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 f (4.1.1.2) obtained by taking germs of local horizont
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 exercise in the definition o f the GaussManin
c onnection (cf
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.
Q.E.D.
4.2. Families of Mixed Hodge Structures
(4.2.0
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 symmetric and positive definite.
T he positivedefinitenes
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. The
c onsiderations of (4.2.0.1113) apply mutatis mutand
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~*( Hz, W, F) of ( Hz, W, F) on s
is a constant
f amily o
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)
s ponding rational local system
(4.3.0.2)
R" n," (Q) =
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 mixed Hodge structures are
isogenous t o various of the E
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 < [n/21
is an i sogeny. T hus it remains to polarize Prim"
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 particular
t hat the local system (~oan)*(R"n."(C) is
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
the largest constant sublocal system of R" ft. n (Z).
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 idempotent
in Q [G]. Every indecomposable central idempotent
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)
t@ ( 2 )
."."."." ' .
It's proper values are thus its diagonal terms. If a = b, then all but one of
these diagonal terms is nonzero, and hence zero is a prop
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 vanishes, this matrix obviously has as i mage t he subspace