%plain TeX \normalbaselineskip=1.6\normalbaselineskip\normalbaselines \magnification=1200 \def\max{\mathop{\rm max}} \def\min{\mathop{\rm min}} \def\rk{\mathop{\rm rk}} \def\uple#1{(#1_1,\ldots,#1_n)} \def\puple#1{(#1_1:\ldots:#1_n)} \def\pmb#1{\setbox0=\hbox{#1}% \kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0 \kern-.025em\raise.0433em\box0 } \def\w{\pmb{$\omega$}} \def\Gm{{\bf G}_m} \def\Ga{{\bf G}_a} \def\ra{\rightarrow} \def\Z{{\bf Z}} \def\topn{\buildrel{p^n}\over\to} \def\Q{{\bf Q}} \def\C{{\bf C}} \def\F{{\bf F}} \def\zmp{\Z/p\Z} \def\O{{\cal O}} \def\isom{\cong} \def\Ext{\mathop{\hbox{Ext}}} \def\Lie{\mathop{\hbox{Lie}}} \def\mod{\mathop{\rm mod}\nolimits} \def\Qp{{\bf Q}_p} \def\Qpb{\bar{\bf Q}_p} \def\Cp{{\bf C}_p} \def\P{{\bf P}} \def\Hom{\mathop{\rm Hom}\nolimits} \def \H{{\cal H}} \def \X{{\cal X}} \def\Spec{\mathop{\rm Spec}\nolimits} \def \bs{\bigskip} \def\Gal{\mathop{\rm Gal}\nolimits} \def\End{\mathop{\rm End}\nolimits} \def\limproj{\mathop{\oalign{\hfil$\rm lim$\hfil\cr $\longleftarrow$\cr}}} \def \L{\Lambda} \def\bk{\bar{k}} \def \ra{\rightarrow} \def \op{\frac{1}{p}} \def \st{\stackrel} \def \da{\downarrow} \def \R{{\bf Rings}} \def \d{\delta} \def \s{\sigma} \def \a{\alpha} \def \b{\beta} \def \x{\chi} \def \t{\tau} \def \z{\zeta} \def \con{\equiv } \def \e{\epsilon} \def \bC{\bar{C}} \def \bX{\bar{X}} \def \k{\kappa} \def \bF{{\bar{\bf F}}_p} \centerline{\bf A note on the arithmetic of differential equations} \medskip \centerline{\bf Jos\'e Felipe Voloch} \bs In this note we give a method for computing the differential Galois group of some linear second-order ordinary differential equations using arithmetic information, namely the $p$-curvatures. \bs {\bf 1. Introduction} \bs Let $K$ be a number field and consider a finite extension $F/K(x)$, where $x$ is an indeterminate, with derivation $D=d/dx$. To a linear differential equation $Ly=\sum_{i=0}^n c_iD^i y =0, c_i \in F$, one associates its differential Galois group $G$, which is a linear algebraic subgroup of $GL_n$ defined over $F$ and isomorphic over $\bar F$ to a group defined over $K$. A conjecture of Katz [K3], which generalizes a conjecture of Grothendieck, predicts that the Lie algebra of $G$ is the smallest $F$-Lie subalgebra of of the Lie algebra of $GL_n$, whose reduction modulo primes $p$ contains the $p$-curvature of $L$, for all sufficiently large $p$. The $p$-curvature of $L$ is the $n\times n$ matrix $A_p (\mod p)$ where $(D^py,D^{p+1}y,\ldots,D^{p+n-1}y)^t=A_p(y,Dy,\ldots,D^{n-1}y)^t$. Katz has shown ([K3]) that the $p$-curvatures of $L$ all belong to the Lie algebra of $G$. We will show, in some cases where we have an a priori restriction on $G$, that the fact that its Lie algebra contains the $p$-curvatures is enough to determine $G$. This gives an affirmative answer, in these cases, to the question posed at the end of the introduction of [K4]. From now on, assume that $n=2$, that is $L$ is a second-order linear differential operator. We will also assume that $c_1/c_2$ is the logarithmic derivative of an element of $F$. As is well-known, this is equivalent to the Galois group $G$ of $L$ be a subgroup of $SL_2$. Let us write $$A_p =\pmatrix{a_p&b_p\cr a_{p+1}&b_{p+1}\cr}$$ From the fact that $c_1/c_2$ is a logarithmic derivative, it follows that the $p$-curvature of the determinant of $Ly=0$ is always zero, which means that the trace of the $p$-curvature of $Ly=0$ is zero, that is $b_{p+1}=-a_p$. For an example that will be useful in the sequel, take the equation $D^2y = ay$, where $a$ has expansion $\a x^k + \cdots$, with $k > 0$ at a place $P$ of $F/K$ above $x=\infty$, so $a$ has a pole there. Then $D^ny = a_ny + b_nDy$, where $a_2=a,b_2=0$ and, for $n >2$, $a_{n+1}= Da_n + ab_n, b_{n+1}= Db_n + a_n$. It follows by induction that the $a_n,b_n$ have the following expansion at $P$: $$\displaylines{a_{2n} = \a^n x^{kn}+ \cdots, b_{2n} = kn(n-1)\a^{n-1} x^{k(n-1)-1}+\cdots,\cr a_{2n+1} = kn^2\a^n x^{kn-1}+ \cdots, b_{2n+1}=\a^n x^{kn}+ \cdots.}$$ \noindent it follows that, for a prime $p=2n+1$, $$A_p =\pmatrix{kn^2\a^n x^{kn-1}+ \cdots&\a^n x^{kn}+ \cdots\cr \a^{n+1}x^{k(n+1)}+ \cdots&kn(n+1)\a^n x^{kn-1}+ \cdots \cr}.\eqno(*)$$ \bs Another result that will be useful in the sequel is the following lemma. \proclaim Lemma. Let $p$ be a prime sufficiently large and consider the equation $Ly=0$ modulo $p$ and assume it has a solution $y \ne 0$ with $u=Dy/y$ separable algebraic over $\F_p(x)$ and that $A_p$ has trace zero. Then $u$ satisfies $b_pu^2+2a_pu-a_{p+1}=0$. {\bf Proof:} Since $u$ is separable algebraic over $\F_p(x)$, we have $D^pu=0$. Since $D^p$ is a derivation we get $D^{p+1}y=D^p(uy)=uD^py$. On the other hand, by definition of $A_p$, $D^py=a_{p}y+b_{p}Dy=y(a_{p}+b_{p}u),D^{p+1}y=a_{p+1}y+b_{p+1}Dy= y(a_{p+1}+b_{p+1}u)$. Thus, $a_{p+1}+b_{p+1}u=u(a_{p}+b_{p}u)$ and using that $b_{p+1}=-a_p$, we get the equation stated in the lemma. \bs {\bf 2. Proper subgroups of $SL_2$} \bs The list of proper algebraic subgroups of $SL_2$, up to conjugation, is well-known and we will go through the list pointing out facts relevant to our purposes. Our arguments in this section are based on the work of van der Put [P]. \medskip {\bf 2.1 $G$ finite} \medskip In this case all the solutions to the equation $Ly=0$ are algebraic. This leads to infinitely many $Dy/y$ which are also algebraic and the equation of the lemma therefore has infinitely many solutions, which proves that the $p$-curvature is zero. \medskip {\bf 2.2 $G \isom \Gm$} \medskip In this case $G$ is conjugate to the group of diagonal matrices with determinant $1$. The action of $G$ on the space of solutions of $Ly=0$ has two invariant lines, generated by $y_1,y_2$, say. Since the lines are invariant, the logarithmic derivatives $u_i=Dy_i/y_i$ are invariant under $G$, which means that the $u_i$ are in $F$ but are not logarithmic derivatives, for otherwise we would be in the case $G$ finite. We then obtain from the Lemma the following two relations among the entries of the $p$-curvature $b_pu_i^2+2a_pu_i - a_{p+1}=0, i=1,2$. Recall that we also have $b_{p+1}=-a_p$, since the $p$-curvature has trace zero. These relations are easily seen to be independent. \medskip {\bf 2.3 $G$ extension of $\Z/2$ by $\Gm$} \medskip In this case $G$ is conjugate to the group of diagonal and antidiagonal matrices with determinant $1$. This case is similar to the previous case, except that now the $u_i$ are in a quadratic extension $E/F$ and are conjugate over $F$, since the $\Z/2$ must permute the lines invariant under the subgroup of $G$ isomorphic to $\Gm$. \medskip {\bf 2.4 $G$ extension of $\Gm$ by $\Ga$} \medskip In this case $G$ is conjugate to the group of triangular matrices with determinant $1$. In this case there is an unique invariant line under $G$ in the solution space of $Ly=0$, generated by $y_1$, say. As before, the logarithmic derivative $u_1 = Dy_1/y_1$ is in $F$ and we get a relation $b_pu_1^2+2a_pu_1 - a_{p+1}=0$ as well as $b_{p+1}=-a_p$. \medskip {\bf 2.5 $G \isom \Ga$} \medskip In this case $G$ is conjugate to the group of triangular matrices with both diagonal entries equal to $1$. In this case $G$ acts trivially on the invariant line so there is a solution of $Ly=0$, say $y_1$, in $F$. As before, $u_1 = Dy_1/y_1$ is also in $F$ and this gives a relation $b_pu_1^2+2a_pu_1 - a_{p+1}=0$. But we do not get all relations among the entries of $A_p$ this way. In order to get all relations we use that $y_1$ is in $F$ to get $D^py_1= D^{p+1}y_1=0$. This gives two relations $a_py_1+b_pDy_1=a_{p+1}y_1+b_{p+1}Dy_1=0$ and, again as before, we have $b_{p+1}=-a_p$. As an application of the above, we expand an argument of van der Put [P] for the Airy equation (and correct a small error there) in order to reprove a theorem of Katz. \proclaim Theorem 1. The equation $D^2y=ay$, where $a$ is a polynomial of odd degree, has Galois group $SL_2$. {\bf Proof:} Suppose $G$ is not $SL_2$. Notice that, from (*) above, we get in particular that the $p$-curvature is not zero so $G$ is not finite. In all other cases, $Ly=0$ has a solution for which $u=Dy/y$ is algebraic. So from the lemma we get the relation $b_pu^2+2a_pu-a_{p+1}=0$, which is a quadratic equation for $u$. Again from (*), we get the top terms of the $a_p,b_p,a_{p+1}$ and this gives that the discriminant of the quadratic for $u$, $4(a_p^2+b_pa_{p+1})$ is a polynomial of degree $kp$, which is odd for $p$ odd. Therefore the quadratic equation cannot have a rational function as root, since the discriminant is not a square. So the Galois group can only be conjugate to (2.4). To rule out (2.4), we proceed as in [P]. From [P] 4.1 (our $b_p$ is $f$ there), $b_p$ satisfies a third order equation with polynomial coefficients $D^3b_p -4aDb_p -2Dab_p=0$. If the Galois group is assumed to be (2.4), $u$ satisfies a quadratic equation over $K(x)$ and since it also satisfies $b_pu^2+2a_pu-a_{p+1}=0$, we get that $2a_p/b_p = -Db_p/b_p$ is independent of $p$. However, $b_p$ has degree $(p-1)/2$ (from (*))and, because of the differential equation it satisfies, $b_p$ cannot have any triple zeros ([P] erroneously asserts the zeros are simple but gives counterexamples a paragraph earlier!). Thus $-Db_p/b_p$ has at least $[(p-1)/4]$ poles and thus cannot be independent of $p$. This contradiction completes the proof. \bs {\bf 3. Globally nilpotent equations} \bs As before we consider a second-order equation $Ly=0$ with Galois group $G$ contained in $SL_2$. We will assume that $Ly=0$ is globally nilpotent, that is, its $p$-curvatures are nilpotent for all sufficiently large $p$. Katz [K3] has shown that factors of the Gauss-Manin connection on the cohomology of families of algebraic varieties are globally nilpotent. In particular, the Gauss hypergeometric equation $x(x-1)D^2y+((a+b+1)x-c)Dy+aby=0$ is globally nilpotent, a result which is also proved directly in [M]. Dwork [D] conjectured that the only second-order equations over $K(x)$ which are globally nilpotent are obtained from Gauss hypergeometric equation by a change of variable or have a rational solution. In this section we will compute the Galois group of some second order globally nilpotent equations using the $p$-curvatures. We note that, for the Gauss hypergeometric equation, our result follows from [BH]. Katz [K3] has shown that globally nilpotent equations have regular singular points and rational exponents. The $p$-curvature of the determinant of $Ly=0$ is the trace of the $p$-curvature of $Ly=0$ which is zero, by nilpotence. So the determinant of of $Ly=0$ has all its $p$-curvatures equal to zero, hence finite Galois group if the one-dimensional case of the Grothendieck conjecture holds. Thus, passing to a finite extension of $F$ we may assume that $G$ is a subgroup of $SL_2$. As shown by Honda [H], the nilpotence of $A_p$ implies that $Ly=0$ has a non-zero solution $y_1$ in the reduction of $F$ modulo $p$. Then, as before, $u_1=Dy_1/y_1$ satisfies the quadratic equation $b_pu^2+2a_pu-a_{p+1}=0$, as does any other algebraic $Dy/y$ with $Ly=0$, by the Lemma. However, the discriminant of the quadratic equation is $4(a_p^2+b_pa_{p+1})= -4\det A_p =0$, by the nilpotence of $A_p$. So this quadratic equation has only one solution if it is not identically zero, i.e. if $A_p \ne 0$. Hence the only possibilities, if $A_p \ne 0$, for the Galois group are $SL_2$,(2.4) and (2.5). In both cases (2.4) and (2.5), there a unique $u \in F$, of the form $Dy/y, Ly=0$. and so, from the above $u_1=u$. Suppose that $Ly=0$ has $m$ singularities and denote by $\rho_1',\rho_1'',\ldots,\rho_m',\rho_m''$ their respective exponents. We say that $Ly=0$ satisfies the {\it exponent restriction} if for all choices of $\rho_i$ from $\rho_i',\rho_i'', i=1,\ldots,m$, we have that $\sum \rho_i$ is not a nonpositive integer. Suppose that $Ly=0$ satisfies this exponent restriction. Now, the valuation of $y_1$ is congruent modulo $p$ to $0$ or $1$ at the regular points. Let $k$ be the number of regular points where the valuation of $y_1$ is congruent to $1$ modulo $p$. The valuation of $y_1$ is congruent modulo $p$ to either $\rho_i'$ or $\rho_i''$ at the $i$-th singular point and therefore, by the residue theorem applied to $dy_1/y_1$, $\sum \rho_i + k$ is divisible by $p$. If $u_1dx=dy_1/y_1$ has a bounded number of poles, then $k$ is bounded and therefore $\sum \rho_i$ is congruent modulo $p$ to a bounded nonpositive integer (viz. $-k$) for the choice of $\rho_i$ corresponding to the valuations of $y_1$. From the exponent restriction and the fact that the $\rho_i$'s are rational numbers, it follows that this cannot happen for infinitely many primes. Therefore $u_1$ cannot have a bounded number of poles and therefore cannot be congruent to some $u \in K(x)$ independently of $p$. From the above discussion this implies that the Galois group is $SL_2$. This argument proves the following theorem. \proclaim Theorem 2. Let $Ly=0$ be a globally nilpotent, second order differential equation whose $p$-curvatures do not vanish for all sufficiently large $p$ and which satisfies the above exponent restriction. Then the differential Galois group of $Ly=0$ is $SL_2$. {\bf Acknowledgements:} The author would like to thank N. Katz for helpful comments and the NSA for financial support. \bigskip \centerline{\bf References.} \bigskip \noindent [BH] F. Beukers and G. Heckman, {\it Monodromy for the hypergeometric function ${}_nF_{n-1}$}, Invent. math. 95 (1989) 325-354. \medskip \noindent [D] B. Dwork, {\it Differential operators with nilpotent $p$-curvature}, Amer. J. Math. 112 (1990), 749--786. \medskip \noindent [H] T. Honda, {\it Algebraic differential equations}, Symposia Math. XXIV (1979) 169-204. \medskip \noindent [K1] N. Katz, {\it Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin}, Publ. Math. IHES 39, (1970), 175--232. \medskip \noindent [K2] N. Katz, {\it Algebraic solutions of differential equations ($p$-curvature and the Hodge filtration)}, Invent. Math. 18 (1972), 1--118. \medskip \noindent [K3] N. Katz, {\it A conjecture in the arithmetic of differential equations}, Bull. Soc. Math. France 110 (1982) 203-239. Corrections, ibid. 347-348. \medskip \noindent [K4] N. Katz, {\it On the calculation of some differential galois groups}, Invent. math. 87 (1987) 13-61. \medskip \noindent [P] M. van der Put, {\it Reduction modulo $p$ of differential equations}, Indag. Math. 7 (1996) 367--387. \medskip \noindent Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, USA \smallskip \noindent e-mail: voloch@math.utexas.edu \end