Riccati Differential Equation for Hypergeometric Differential Equation

Riccati Differential Equation for Hypergeometric Differential Equation

TAKAHIRONAKAGAWA

ABSTRACT- In this paper, we study the solutions of the Riccati differential equation collesponding to ap-adic differential equation which is solvable on the generic disc. As an application, we consider the Grothendieck conjecture for Riccati differential equations. We see that the Riccati differential equation for some globally nilpotent differential equation with coefficients inQ(t) have, for almost all prime number, a solution inFp(t), butdo nothave any algebraic solutions.

1. Introduction

Letpbe a prime andK a complete non-archimedean valued field with mixed characteristics. In this paper, we consider the solutions of the Ric- cati differential equation for ap-adic differential equationLwhich is sol- vable on the generic discD(t;1 ) (see the Definition 2.5).

We attach the Riccati differential equationC(L)₁for each linear ordinary differential operatorLin section 3. LetEbe the completion ofK(t) under the Gauss norm. We see that a solution of the Riccati differential equationC(L)₁ forLonEis connected with a bounded solution ofLat the generic point.

LetbddKer_t(L) be the vector space consisting of the bounded solutions of L. If its dimension is one, then C(L)₁ has a unique solution on E. In particular if we can reduce Lmodulop, thenC(L)₁ has a solution on the rational function fieldK(t) over the residue field K ofK.

Let L be an n-th order linear ordinary differential operator with coefficients in Q(t). The Grothendieck conjecture (in its simplest form) asserts the following statements are equivalent (see [Put]):

(i) the equation L(y)ˆ0 has n linear independent(over Q^alg) solutions which are algebraic overQ(t).

(ii) For almostallp, the equationL(y)ˆ0 hasnlinear independent solutions over the fieldF_p(t^p) in the fieldF_p(t).

(*) Indirizzo dell'A.: Mathematical Institute CHIBA University, 263-8522 Japan.

E-mail: [email protected]

Van der Put studied the Grothendieck conjecture for non-linear differ- ential equations. He proved that the differential equationy⁰ˆ(1=t²)y‡1 has no algebraic solutions but there exists a solution inFp(t) for all primesp. He also proved that the conjecture is true for some Risch equations [Put], [Put2].

We will prove the following theorem.

THEOREM 1.1. Let L be a monic differential operator with coeffi- cients in Q(t). We assume that when we regard L as a p-adic differ- ential equation, L is solvable on the generic disc D(t;1) for almost all prime p, and the dimension of the vector space which consists of the bounded solutions of L is one. If L has no algebraic solutions, then C₁(L)modulo p has a solution onF_p(t)for almost all prime p and no algebraic solutions.

In particular some hypergeometric differential equations satisfy the condition of Theorem 1.1.

The author is grateful to Professor S. Matsuda for his helpful advice and thank referee for his suggestion.

2. Notation

2.1 ±Analytic elements

Let K be a complete non-archimedean valued field with mixed char- acteristics and K the residue field ofK. Let K(t) be the field of rational functions. For a polynomialf ˆP

a_itⁱwith coefficients inK, we set jfj₀ˆmaxjaij:

(2:1)

The field of rational functionsK(t) has the Gauss norm j j₀ defined by jfj₀ˆ jgj₀=jhj₀ if f ˆg=h and g;h2K[t]. Let E be a completion ofK(t) under the Gauss norm. An element ofEis called an analytic element. The derivationd=dtofK(t) is extended toE. We regard (E;d=dt) as a differ- ential field.

Fora2Eandr2R>0, let

D(a;r )ˆ fz2Ej jz aj₀5rg (2:2)

and

D(a;r)ˆ fz2Ej jz aj₀rg:

(2:3)

We define the ring of analytic functions on the discD(a;r ) t o be A(a;r)ˆ X¹

nˆ0

an(x a)ⁿ2E[[x a]]an2E;_n!1lim janjsⁿˆ0 for all 0s5r

( )

:

We define the subringE[[x a]]0of the ring of analytic functionsA(a;1) by E[[x a]]₀ˆ X¹

nˆ0

a_n(x a)ⁿ 2E[[x a]] sup

0n51ja_nj51

( )

:

PROPOSITION2.1 ([Christol, 2.5.1]). We define a ring homomorphism t:E!E[[x t]]₀ so that, for f 2E,

t:E!E[[x t]]₀ f 7!X¹

nˆ0

(d=dt)ⁿ(f)

n! (x t)ⁿ:

Thentis an isometric isomorphism from E to the subfield of E[[x t]]₀ consisting of elements f which satisfy:

t((d=dt)(f))ˆ(d=dx)(t(f)):

(2:4)

2.2 ±Differential modules

Let(H;D) be a differential field of characteristic zero and C_H the constant field of H. We say differential field (L;D⁰) is a differential ex- tensionof (H;D) ifLHandD⁰j_HˆD. AnH-differential module (M;r) is a free H-module M of finite type endowed with a CH-linear homo- morphismr(D) which satisfiesr(D)(am)ˆDa:m‡ar(D)(m) fora2H andm2M.

Let(M;r) be anH-differential module andLa differential extension of H. AnH-linear maps:M!Lis saidhorizontal ifs(rm)ˆD(s(m)) for m2M. We denote the CH-vector space by Homr(M;L):ˆ

fs2Hom_CH(M;K)js is horizontalg.

PROPOSITION 2.2. Let (M;r) be an H-differential module. For any differential extension L=H,

dim_CH(Hom_r(M;L))dim_H(M) (2:5)

REMARK2.3. By the existence of Picard-Vessiot field of HCHC^alg_H , there exists a differential extension L=H such that equality (2.5) holds (see [PS]).

PROPOSITION2.4. Let G be a matrix in M_n(E). There exists a unique matrix U of GL_n(E[[x t]])such that

U(t)ˆI_n; (d=dx)(U)ˆt(G)U:

(2:6)

LetfG_ig_i2Nbe the sequence of matrix in Mn(E)defined by G0ˆIn; G_i‡1ˆ(d=dt)(G_i)‡G_iG:

Then we have

UˆX¹

iˆ0

G_i(x t)ⁱ i! :

DEFINITION2.5. Let(M;r) be anE-differential module of rankn. Let e₁; ;e_n be a basis of M and Gˆ(g_ij) the matrix such that r(D)eiˆPⁿ

jˆ0gijej. We say (M;r) is solvableover D(t;r) if the matrix U which is defined by (2.6) is an elementof GL_n(A(t;r)).

For a differential field (H;D) which satisfiesK(t)HE, we call an H-differential module (M;r) is solvable overD(t;r) if t heE-differential module (MHE;r) is solvable overD(t;r).

3. Riccati differential equation

Let(H;D) be a differential field of characteristic zero andCHa constant field ofH. LetLˆDⁿ gnD^{n 1} g1be a differential operator with coefficients inH. Then (M;r)ˆH[D]=H[D]Lis anH-differential module of rankn. LetGbe the matrix of representation ofr(D) with respect to the basis 1; ;D^{n 1}of (M;r), i.e,

Gˆ

0 1 0 0

0 0 1 0

0... 1

g1 g2 gn

0 BB BB

@

1 CC CC A: (3:1)

We suppose thatLhas a decomposition

LˆL1R; L1;R2H[D] andR is monic.

We write RˆD^k c_kD^{k 1} c₁. Since the canonical morphism MˆH[D]=H[D]L!H[D]=H[D]R is surjective, the induced morphism (H[D]=H[D]R)^_!M^_is injective.

Let f1; ;fn (resp. l1; ;lk) be the dual basis of the basis 1;D; ;D^{n 1}ofM(resp. (H[D]=H[D]R)). If we regard (H[D]=H[D]R)^_as a submodule ofM^_then eachl_iis a linear combination off_i. In fact, by the definition of dual basis, we have

l1

l2

l..._k 0 BB B@

1 CC CAˆC

f₁ f₂ f..._n 0 BB B@

1 CC CA (3:2)

such thatCˆ fci;jgis theknmatrix defined by

D^{m 1}ˆc_k;mD^{k 1}‡c_{k 1;m}D^{k 2}‡ ‡c_1;m (modR):

(3:3)

In particularc_j;k‡1ˆc_jand ifjkthenc_i;jˆd_i;j.

Let F=Hbe a differential extension such that dim_CH(Homr(M;F))ˆ dimH(M) (see Remark 2.3). Lets1;. . .;snbe a basis of Homr(M;F). By the equality (3.2), we have

s1(l1) s2(l1) sn(l1) s1(l2) s2(l2) sn(l2)

... ...

s1(lk) s2(lk) sn(lk) 0

BB B@

1 CC CAˆC

s₁(f₁) s₂(f₁) s_n(f₁) s₁(f₂) s₂(f₂) s_n(f₂)

... ...

s₁(f_n) s₂(f_n) s_n(f_n) 0

BB B@

1 CC CA: (3:4)

OperatingDon both sides the equation (3.4), we have 0 c1

1 ...

1 c_k 0

BB BB

@

1 CC CC A

s1(l1) s2(l1) sn(l1) s₁(l₂) s₂(l₂) s_n(l₂)

... ...

s₁(l_k) s₂(l_k) s_n(l_k) 0

BB BB

@

1 CC CC A

ˆ D(C) C

0 g₁

1 g2

... ...

0 1 gn

0 BB BB

@

1 CC CC A 0

BB BB

@

1 CC CC A

s₁(f₁) s₂(f₁) s_n(f₁) s1(f2) s2(f2) sn(f2)

... ...

s1(fn) s2(fn) sn(fn) 0

BB BB

@

1 CC CC A

Since the matrix (si(fj))i;jis invertible, we have 0 c1

1 ...

1 c_k 0

BB B@

1 CC

CACˆD(C) C

0 g₁

1 g₂

... ...

0 1 g_n

0 BB B@

1 CC CA:

Therefore we have the equations

c1ck;n‡D(c1;n) g1 c1;k‡1gk‡1 c1;ngnˆ0;

c1;n‡c2ck;n‡D(c2;n) g2 c2;k‡1gk‡1 c2;ngnˆ0;

c_{k 1;n}‡c_kc_k;n‡D(c_k;n) g_k ...c_k;k‡1g_k‡1 c_k;ngn ˆ0:

(3:5)

We define the differential systemC(L)_kink-variabley₁;. . .;y_kby y₁y_k;n‡D(y_1;n) g₁ y_1;k‡1g_k‡1 y_1;ng_nˆ0;

y1;n‡y2yk;n‡D(y2;n) g2 y2;k‡1gk‡1 y2;ngn ˆ0;

y_{k 1;n}‡y_ky_k;n‡D(y_k;n) g_k ... y_k;k‡1g_k‡1 y_k;ngnˆ0 (3:6)

wherey_i;j (1ik; j1) is defined inductively by y1;1ˆ1;yj;1ˆ0 (2jk);

y_1;m‡1ˆD(y_1;m)‡y_k;m:y₁; y_j;m‡1ˆD(y_j;m)‡y_k;m:y_j‡y_{j 1;m} (2jk):

By definition, the vector (c₁; ;c_k) is solutions of differential systemC(L)_k. In particular,C(L)₁is a differential equation

y₁y_1;n‡D(y_1;n) g₁ y_1;2g₂ y_1;ng_nˆ0 (3:7)

wherey_1;jis defined inductively by

y1;1 ˆ1; y1;m‡1 ˆD(y1;m)‡y1;m:y1:

Conversely we assume differential system C(L)_k has a solution (c₁; ;c_k)2H^k. Let F=H be a differential extension which satisfies dimCH(fy2Fj(D^k ckD^{k 1} c1)yˆ0g)ˆk (see Remark 2.3). If y2Fsatisfies the equality (D^k ckD^{k 1} c1)yˆ0 then we have

G y Dy

...

D^{n 1}y 0 BB BB

@

1 CC CC AˆG:^tC

y Dy

...

D^{k 1}y 0 BB BB

@

1 CC CC A

ˆD(^tC) y Dy

...

D^{k 1}y 0 BB BB

@

1 CC CC A‡^tC:

0 1 0

0 1 0

...

c₁ c2 ck

0 BB BB

@

1 CC CC A

y Dy

...

D^{k 1}y 0 BB BB

@

1 CC CC A

ˆD(^tC) y Dy

...

D^{k 1}y 0 BB BB

@

1 CC CC A‡^tC:

Dy D²y ...

D^ky 0 BB BB

@ 1 CC CC Aˆ

Dy D²y

...

Dⁿy 0 BB BB

@ 1 CC CC A:

Thereforeysatisfies a solution of differential equation DⁿyˆgnD^{n 1}y‡gn 1D^{n 2}y‡ ‡g1y:

(3:8)

By proposition 2.2, we have a decomposition

Dⁿ gnD^{n 1} g12H[D](D^k ckD^{k 1} c1):

(3:9)

As a result, we have the following proposition.

PROPOSITION3.1. Let H be a differential field such that the character- istic of H is zero. Let LˆDⁿ gnD^{n 1} . . . g1 be a differential poly- nomial with coefficients in H.

There is a one to one correspondence between a k-th order differential operator R2H[D] which satisfies L2H[D]R and a solution of differ- ential system C(L)_k in H^k.

If there exists a differential polynomial RˆD^k ckD^{k 1} c1

such that L2H[D]R then (c1;c2;. . .;c_k)2H^k is a solution of C(L)_k. Conversely, if (c1;c2;. . .;c_k)2H^k is a solution of C(L)_k then the differ- ential polynomial RˆD^k c_kD^{k 1} c₁satisfies L2H[D]R.

By induction, we have the following lemma.

LEMMA 3.2. We assume HˆE. Let LˆDⁿ gnD^{n 1} . . . g1 be a differential polynomial with coefficients in E. Ifjgjj₀1for jˆ1; ;n, then, for any solution y2E of C1(L), we havejyj₀1.

EXAMPLE3.3. For a differential operatorLˆD² g2D g1of rank 2, C(L)₁ is a Riccati differential equation c²‡c⁰ g1 g2cˆ0. Young con- sidered in this case [Young].

4. Analytic element solution of Riccati differential equation

LetL be a differential operator with coefficients inEandDˆd=dt. LetC(L)_k be the differential system defined byL. We suppose thatLis solvable onD(t;1) (i.e.,E[D]=E[D]Lis solvable onD(t;1)). We denote the vector space which is generated by the solutions of (resp. the bounded solutions) ofLby

Ker_t(L)ˆ fy2 A(t;1)jt(L)yˆ0g;

(resp. Kerbddt(L)ˆ fy2E[[x t]]₀jt(L)yˆ0g):

In general, sinceLis solvable onD(t;1), we have dimK(Kerbddt(L))1 by [Robba, Theorem 3.5]. If dim_K(Kerbdd_t(L))n 1, then there exists a monic differential operatorR2E[D] such thatL2E[D]Rand Ker_t(R)ˆ Kerbdd_t(L) by [Robba, Theorem 2.6].

PROPOSITION 4.1. Let L2E[D] be a monic n-th order differential operator which is solvable on D(t;1).

(i) I f kˆdim_K(Kerbdd_t(L))n 1, then C(L)_khas a solution on E.

(ii) If1ˆdim_K(Kerbdd_t(L)), C(L)₁has a unique solution on E.

(iii) If there exists fundamental system of solutions y1;. . .ynof L at t such that yiis exactly of log-growth i 1(see the definition of log-growth in [C-T]) then, for each j, C(L)_j has a unique solution over E.

PROOF. There exists a monic differential operatorR2E[D] such that L2E[D]Rand Ker_t(R)ˆ fy2E[[x t]]jt…L†yˆ0 andy is log-growthig by [Robba, Theorem 2.6]. By Proposition 3.1, there exists a solution of C(L)_konEwherekˆdeg(R). The uniqueness of case (ii),(iii) is confirmed

by [Dwork1, Theorem 1]. p

By [D-R, Theorem 2.4.3 (iii)], we have the following proposition.

PROPOSITION4.2. Let L2E[D]be a monic n-th differential operator which is solvable on D(t;1). Assume kˆdim_K(Kerbdd_t(L))n 1. Let S be the setfa2K jL has no pole on D(a;1 )g. Then there exists a solution

(y1;. . .;yk)2E^k of C(L)k such that

fa2Sjdim(Kerbdd_t(L))>dim(Kerbdd_a(L))g fa2Sjyiis notanalytic on D(a;1 ) for someig

5. Hypergeometric differential equation and Riccati differential equation modp

Letpbe a prime number andDˆd=dx. Fora;b;c2Q\Zp, we setthe hypergeometric differential operator

L(a;b;c)ˆD²‡ c (1‡a‡b)x

x(1 x) D ab

x(1 x): (5:1)

We assume the following condition:

c a;c b;b;a all lie outsideZ:

(5:2)

c2Q f0; 1; 2;. . .g:

(5:3)

By [Dwork2, p. 11, Remark], we see the Riccati differential equation y²‡y⁰‡c (1‡a‡b)x

x(1 x) y ab

x(1 x)ˆ0 (5:4)

has no rational solutions.

For a rational number a2Q\Zp, we set m_a2[0;p 1] such that m_aˆ a (modp). According t o Dwork, we say (a;b;c)2Qissplit(resp.

unsplit) if m_c5minfm_a;m_bg or m_c >maxfm_a;m_bg (resp. m_a5m_c5m_b or m_b5m_c5m_a). There exists an integer msuch that

F:ˆp^m X^{p 1}

nˆ0

(a;n)(b;n)

(c;n)n! xⁿ2Z_p[x] pZ_p[x];

(5:5)

where (a;n) is defined

(a;n)ˆ 1 (nˆ0)

a(a‡1) (a‡n 1) (n1):

If (a;b;c) is split , t henFis uniquely determined. SinceFis a solution of hypergeometric differential equation

D²‡c (1‡a‡b)x

x(1 x) D ab

x(1 x)

yˆ0 (modp);

(5:6)

by Lemma 5.2,F⁰=Fis a solution of the Riccati differential equation y²‡y⁰‡c (1‡a‡b)x

x(1 x) y ab

x(1 x)ˆ0 (modp):

(5:7)

Under the special condition, Dwork proved that if (a;b;c) is split, then L(a;b;c) has a unbounded solution and if (a;b;c) is unsplit, thenL(a;b;c) has two independent bounded solutions (see [Dwork2, Theorem 6.6, 9.4]).

We consider solutions of the Riccati differential equation (5.7).

LEMMA 5.1. We assume that(a;b;c)2(Q\Zp)³ is unsplit. Then the Riccati differential equation(5.7)has infinite solutions.

PROOF. In this case, the hypergeometric differential equation (5.6) has solution

u^minfmX^a;mbg

nˆ0

(a;n)(b;n)

(c;n)n! xⁿ‡vp^mmaxfmX^a;mbg

nˆmc‡1

(a;n)(b;n)

(c;n)n! xⁿ (u;v2F_p(x^p)) (5:8)

with m2Z which satisfies p^mmaxfmP^a;mbg

nˆmc‡1

(a;n)(b;n)

(c;n)n! xⁿ2Z_p[x] pZ_p[x]. By Lemma 5.2, the Riccati differential equation (5.7) has infinite solutions. p LEMMA 5.2. Let RˆDⁿ g_nD^{n 1} g_{n 1}D^{n 2} . . . g₁ be a differ- ential operator with coefficients F_p(x). Then R has a decomposition RˆQ(D c)where Q2F_p(x)[D]and c2F_p(x)if and only if c2F_p(x) is a solution of the Riccati differential equation C(R)₁.

PROOF. Assume that Rˆ(D^{n 1}‡bn 1D^{n 2}‡ ‡b1)(D c) where b1; ;bn 1;c2Fp(x). Comparing the both sides, we have the equalities

gnˆc bn 1

gn 1ˆ n 1 1

c⁽¹⁾‡cbn 1 bn 2

...

giˆ n 1 n i

c^{(n i)}‡ n 2 n i 1

c^{(n i 1)}bn 1‡ ‡c⁰ i

1 bi‡1‡cbi bi 1

...

g2ˆ n 1 n 2

c^{(n 2)}‡ n 2 n 3

c^{(n 3)}bn 1‡ ‡c⁰ 2

1 b3‡cb2 b1

g1ˆc^{(n 1)}‡c^{(n 2)}bn 1‡ ‡c⁽¹⁾b2‡cb1:

We setcm;Cmby

cm‡1ˆc⁰_m‡ccm; c1ˆ1 C_mˆX^m

iˆ1

m 1

i 1

c_{m i‡1}c^{(i 1)} c_m‡1:

We have the equality

g1 c2g2 cngnˆcn‡1‡Cn‡X^{n 1}

iˆ1

C_ib_i:

To prove thatcis a solution ofC(R)₁, we have only to proveC_m ˆ0 for allm.

We use induction onm. Formˆ1,C₁ˆc₁c⁽⁰⁾ c₂ˆ0. Therefore the case mˆ1 is true.

We assume that the assertion is true formˆk 1.

c⁰_kˆ X^{k 1}

iˆ1

k 2 i 1

c_{k i}c^{(i 1)}

!₀

ˆX^{k 1}

iˆ1

k 2 i 1

(c⁰_{k i}c^{(i 1)}‡c_{k i}c⁽ⁱ⁾)

ˆX^{k 1}

iˆ1

k 2 i 1

((c_{k i‡1} cc_{k i})c^{(i 1)}‡c_{k i}c⁽ⁱ⁾)

ˆX^{k 1}

iˆ1

k 2 i 1

(ck i‡1c^{(i 1)}‡ck ic⁽ⁱ⁾) cck

ˆX^{k 1}

iˆ2

k 2 i 1

‡ k 2 i 2

c_{k i‡1}c^{(i 1)}‡c^{(k 1)}

ˆX^{k 1}

iˆ2

k 1 i 1

c_{k i‡1}c^{(i 1)}‡c^{(k 1)}ˆC_k cc_k‡c_k‡1

This completes the induction.

Conversely let c2F_p(x) be a solution of C(R)₁. We define

b₁;. . .;b_{n 1}2F_p(x) as above. Then we have the decomposition

Rˆ(D^{n 1}‡b_{n 1}D^{n 2}‡ ‡b₁)(D c). p

LEMMA5.3. We assume that(a;b;c)2 Q\Zp₃

is split. The Riccati differential equation(5.7)has a unique solution if and only if

x^mc(1 x) ^{1‡ma‡mb mc} F²

!_{(p 1)}

6ˆ0 (modp) (5:9)

PROOF. Substituting yˆz‡F⁰=F for y in the Riccati differential equation (5.7), we have

z²‡z⁰‡ 2 F⁰

F‡c (a‡b‡1)x x(1 x)

zˆ0 (modp):

(5:10)

Substitutingzˆhx^mc(1 x) ^{1‡ma‡mb mc}=F²forzin the Riccati differential equation (5.7), we have

x^mc(1 x) ^{1‡ma‡mb mc}h²

F²‡h⁰ˆ0 (modp):

(5:11)

We supposeh6ˆ0. By this equality, 1

h

0ˆx^mc(1 x) ^{1‡ma‡mb mc}

F² (modp):

(5:12)

The differential equation (5.12) has a solution if and only if x^mc(1 x) ^{1‡ma‡mb mc}

F² 2F_p(x^p)xF_p(x^p). . .x^{p 2}F_p(x^p)

(5:13) i.e.

x^mc(1 x) ^{1‡ma‡mb mc} F²

!_{(p 1)}

ˆ0 (modp):

(5:14)

p PROPOSITION5.4. We assume thatm_cˆp 1and(a;b;c)is split. Then the Riccati differential equation(5.7)has a unique solution F⁰=F.

PROOF. Let v_x be a valuation ofF_p(x) with respect to a prime ideal xF[x] of F[x]. Since p 1ˆm_c>maxfm_a;m_bg by assumption, vx(F)ˆ0.

Therefore we have vx 1 h 0

ˆvx x^mc(1 x) ^{1‡ma‡mb mc} F²

!

ˆp 1:

(5:15)

Since there exists noh2F(x) which satisfies (5.15), we havehˆ0. p

6. Algebraic solution of Riccati differential equation

Let L be a monic differential operator with coefficients in Q(t) and Dˆd=dt. For almostall primep, we can reduce L modulo p. If we can reduceLmodulop, then we denoteLmodulopbyL_p.

THEOREM 6.1. (i) If L has an algebraic solution, then C(L)₁ has an algebraic solution.

(ii) We assume that for almost all prime p, L is solvable on the generic disc D(t;1) as a p-adic differential equation. If C(L)₁ has a ra- tional solution then L has an algebraic solution.

(iii) We assume that for almost all prime p, L is solvable on the generic disc D(t;1) and dim(bddKer_t(L))ˆ1 as a p-adic differential equation. Then for almost all prime p, C(L)₁has a solution inFp(t). And if C(L)₁ has an algebraic solution, then L has an algebraic solution.

PROOF. IfLhas an algebraic solution then there exists an algebraic function h such that L2Q(t)^alg[D](D h). Hence the Proposition 3.1 implies (i).

Suppose that C(L)₁ has a rational solution y. By Proposition 3.1, we haveL2Q(t)[D](D y). Letpbe a prime such that when we regardLas a p-adic differential equation,Lis solvable on the generic disc D(t;1) and that we can reduceLandD ymodulop. By [DGS, 3. Proposition 5.1],L_p is nilpotent. And by [DGS, 3. Theorem 2.1], (D y)_pis also nilpotent and therefore has a solution ofFp(t). Since for all almostprimep, (D y)_phas a solution ofFp(t), by Honda's theorem([Put2, Proposition 2.2]),D yhas an algebraic solution. This implies (ii).

We will prove (iii). By Lemma 3.2 and Proposition 4.1(ii), for all almost prime p, C(L)₁ has a solution on F_p(t). We assume that C(L)₁ has an al- gebraic solution y which is not a rational function. Let G be the Galois group which corresponds to a finite Galois extension ofQ(t) containsy. For alls2G,s(y) is a solution ofC(L)₁. Sinceyis nota rational function,C(L)₁ has at least two algebraic solutions. By Proposition 3.1, this implies dim(bddKer_t(L))>1. This is a contradiction. p COROLLARY 6.2. Let L be a monic differential operator with coeffi- cients inQ(t). We assume that for almost all primep, when we regardLas ap-adic differential equation,Lis solvable on the generic discD(t;1)and dim(bddKer_t(L))ˆ1. I f L has no algebraic solutions, then C(L)₁ has a solution onFp(t)for almost all primepand no algebraic solutions.

REFERENCES

[Andre] Y. ANDREÂ,Dwork's conjecture on the logarithmic growth of solu- tions of p-adic differential equations, Compos. Math.,144(2008), pp. 484-494.

[Baldassarri] F. BALDASSARRI,Continuity of the radius of convergence of differ- ential equations on p-adic analytic curves, Invent. Math.,182, (3) (2010), pp. 513-584.

[C-T] B. CHIARELLOTTO- N. TSUZUKI,Logarithmic growth and Frobenius filtrations for solutions of p-adic differential equations, J. Inst.

Math. Jussieu,8(2009), pp. 465-505.

[Christol] G. CHRISTOL, Modules diffeÂrentiels et eÂquations diffeÂrentielles p- adiques, Queen's Papers in Pure and Appl. Math. 66, Queen's University, Kingston, ON, 1983.

[Dwork1] B. M. DWORK,On p-adic differential equations II, Annals of Math., 98, (2) (1973), pp. 366-376.

[Dwork2] B. M. DWORK,Lectures on p-adic differential equations, Grundleh- ren Math. Wiss. 253, Springer- Verlag, New York-Berlin, 1982.

[DGS] B. DWORK - G. GEROTTO - F. SULLIVAN, An Introduction to G- Functions, Annals of Math. Studies 133, Princeton University Press, Princeton, 1994.

[D-R] B. M. DWORK- P. ROBBA,On Ordinary Linear p-Adic Differential Equations, T.A.M.S.,231(1977), pp. 1-46.

[Kedlaya] K. KEDLAYA,p-adic differential equations, Cambridge University Press, Cambridge (2010).

[Put] M. VAN DER PUT, Reduction modulo p of differential equations.

Indag. Math. N. S.,7, (3) (1996), pp. 367-387.

[Put2] M.VAN DERPUT,Grothendieck's conjecture for the Risch equation y⁰ˆay‡b. Indag. Math. (N.S.),12(2001), pp. 113-124.

[Put3] M.VAN DERPUT, Bounded p-adic differential equations Circum- spice, Nijmegen, 2001.

[PS] M.VAN DER PUT- M. SINGER,Galois theory of linear differential equations. Grundlehren der Mathematischen Wissenschaften 328.

Springer 2003.

[R-C] P. ROBBA- G. CHRISTOL,EÂquation diffeÂrentielles p-adiques. Appli- cation aux sommes exponentielles. ActualiteÂs Math., Hermann, Paris, 1994.

[Robba] P. ROBBA,On the index of p-adic differential operators I, Ann. of Math. (2),101(1975), pp. 280-316.

[Young] P. YOUNG, Radii of p-adic convergence of generic solutions of homogeneous linear differential equations. Thesis (Ph.D.)-Oklaho- ma State University (1988), pp. 71.

Manoscritto pervenuto in redazione il 20 maggio 2011.