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\centerline{\bf Plane curves and $p$-adic roots of unity}
\medskip
\centerline{\bf Jos\'e Felipe Voloch}
\bs

Let $\Cp$ be the completion of the algebraic closure of $\Qp$ with its 
usual norm extending that of $\Qp$.
In [TV], J. Tate and the author proved a result which implies the
following statement. If $f(x,y) \in \Cp [x,y]$ there exists a positive
constant $c$ such that, for any roots of unity $\z_1,\z_2$, either
$f(\z_1,\z_2)=0$ or $|f(\z_1,\z_2)| \ge c$ (A similar result holds for
polynomials with an arbitrary number of variables). In general, however, there
is little information about the value of $c$. In the case that $f$ is linear
and its coefficients are units in an unramified extension of $\Qp$, 
it was proved in [TV] that
the inequality $|f(\z_1,\z_2)| \le p^{-2}$ had at most $p$ solutions
$\z_1,\z_2$ roots of unity or zero. The purpose of this note is to obtain a similar
result for more general polynomials in two variables.
Recall that a binomial is a polynomial with (at most) two non-zero coefficients.
Our main result is then

\proclaim Theorem. Let $f(x,y)$ be a polynomial of degree $d$ in two variables 
whose coefficents are integers in an unramified extension of $\Qp$.
Assume that the reduction of $f$ modulo $p$ is irreducible of degree $d$ and
not a binomial. Assume also that $p > d^2 +2$. Then the number of solutions
of the inequality $|f(\z_1,\z_2)| < p^{-1}$, with $\z_1,\z_2$ roots of unity
in $\Qpb$ or zero, is at most $pd^2$.

{\it Proof:} We will first prove the theorem under the additional 
condition that we are dealing with roots of unity of order prime to $p$.
The inequality then translates into $f(\z_1,\z_2) \con 0 (\mod p^2)$.
The ring of integers of the completion of the maximal unramified extension of $\Qp$
can be viewed as the ring of Witt vectors over the algebraic closure of $\F_p$
and, since we are interested only in the situation modulo $p^2$, we can work
in the Witt vectors of length two over the algebraic closure of $\F_p$.
We are thus interested in the solutions of the equation 
$f((x,0),(y,0))=(0,0)$. This equation translates into the system
$f_0(x,y)=g(x,y)=0$, where $f_0$ is the reduction of $f$ modulo $p$ and
the polynomial $g$ is the reduction modulo $p$ of the polynomial 
$(f^{\s}(x^p,y^p)-f(x,y)^p)/p$ and $\s$ is the Frobenius automorphism of
the ring of Witt vectors. Clearly $g$ has degree at most $pd$ and, since
$f_0$ is assumed irreducible of degree $d$, the result we want follows from
B\'ezout's theorem unless $f_0$ divides $g$, which we proceed to show cannot
happen.

Let $X$ be the irreducible plane curve defined by $f_0(x,y)=0$. We will
derive a contradiction from the assumption that $g$ vanishes identically
on $X$. If $g=0$ on $X$ then, differentiating $g(x,y)=0$ we obtain
$g_x + g_ydy/dx =0$ and, from the definition of $g$ we have
$g_x = f^{\s}_x(x^p,y^p)x^{p-1}-f(x,y)^{p-1}f_x = f_{0x}^px^{p-1}$ on $X$.
Likewise $g_y= f_{0y}^py^{p-1}$ on $X$. Since $f_0$ is of degree less than $p$
and is not a binomial, we have that $f_{0x},f_{0y}$ are non-zero. So we obtain,
using that $dy/dx =-f_{0x}/f_{0y}$, the identity 
$f_{0x}^{p-1}x^{p-1}-f_{0y}^{p-1}y^{p-1}$, on $X$. This gives
$xf_{0x} = cyf_{0y}$ for some $c \in \F_p$. The lemma below ensures that this
cannot hold under the assumptions that $p > d^2$ and $f_0$ is not a binomial
and this will complete the proof in the case 
the roots of unity are of order prime to $p$.

If $\z_1,\z_2$ are arbitrary roots of unity satisfying 
the inequality $|f(\z_1,\z_2)| < p^{-1}$ we can write 
$\z_i = \lambda_i\eta_i, i=1,2$ where the $\lambda_i$ are of
order prime to $p$ and the $\eta_i$ are of $p$-power order and
are not both equal to one. We will show that this inequality has no
such solution. By a harmless change of coordinates we may assume
that $\lambda_i=1, i=1,2$. Further, perhaps after switching $x$ and
$y$ if necessary, we may assume that $\eta_2 = \eta_1^r$ for some
integer $r$. We write $\eta_1 = 1 + \pi$ and notice that the
inequality $|f(\z_1,\z_2)| < p^{-1}$ implies 
$f(1 + \pi,(1 + \pi)^r) \con 0 (\mod \pi^{p-1})$. On the other hand
if $\O$ is the ring of integer of the field $F(\eta_1)$, where $F$ is
a unramified extension of $\Qp$ containing the coefficients of $f$,
then $\O/\pi^{p-1}$ is isomorphic to $k[t]/t^{p-1}$, where $k$ is the
residue field of $F$. Therefore we obtain $f_0(1+t,(1+t)^r) \con 0 (\mod t^{p-1})$.
This implies, with notation as above, that $y/x^r -1$ has a zero of order at
least $p-1$ at some place of $X$ centered at $(1,1)$, 
so the differential $dy/y-rdx/x$ has a zero of order at least $p-2$ at that same
place. However, this differential has at most $3d$ poles counted with
multiplicity, so at most $3d + 2g-2$ zeros, where $g$ is the genus of $X$ unless
it is identically zero.
Now, $3d + 2g-2 \le 3d + d(d-3) =d^2 < p-2$, by hypothesis, so the
differential is identically zero, which using that $dy/dx =-f_{0x}/f_{0y}$
leads to a contradiction with the lemma below.

It remains only to prove:

\proclaim Lemma. Let $f(x,y)=0$ define an irreducible plane curve $X$
of degree $d$ over an algebraically closed field $k$ of characteristic $p$ 
satisfying $p > d^2$. If $xf_x = cyf_y$ on $X$ for some $c$ in $k$ then $f$ is
a binomial.

{\it Proof:} The hypothesis means an identity $xf_x - cyf_y = bf$ for some
$b$ in $k$. If $f(x,y) = \sum a_{ij}x^iy^j$ we get $a_{ij}(i-cj-b)=0$ for all
$i,j$. Suppose first that $b=0$. For any $i,j,i',j'$ with both $a_{ij},a_{i'j'}$
non-zero, we get $i-cj=i'-cj'=0$ which implies that
$ij'-i'j = (i-cj)j' -(i'-cj')j =0$ in $k$, which means that $p$ divides
$ij'-i'j$, but under our assuption that $p > d^2$, this implies that 
$ij'=i'j$ and this implies that the value of $i/j$ is constant for
all $i,j$ with $a_{ij} \ne 0$. So $f(x,y) = \sum_r a_{rm,rn}x^{rm}y^{rn}$
which can be written as a constant multiple of a product of terms of the
form $x^my^n-\alpha$ and, since $f$ is irreducible, we conclude that $f$ is
a binomial. 

Assume now that $b$ is not zero. First of all, if $f$ is a polynomial in
just one variable and is irreducible, then it is a binomial and we are done.
Therefore, we may assume that there exists $i_1,j_1$ with $a_{0j_1},a_{i_10}$
both non-zero and we get that $i_1=b$ and $cj_1=-b$, so $c$ is not zero and
$c=-i_1/j_1$. If $i,j$ are such that $a_{ij} \ne 0$ then 
$i+ji_1/j_1-i_1=0$ in $k$ so $ij_1+ji_1 \con i_1j_1 (\mod p)$. But
$i_1,j_1 \le d, i+j \le d$, therefore
$0 \le ij_1+ji_1, i_1j_1 \le d^2 < p$ so $ij_1+ji_1 = i_1j_1$.
Let $\d = (i_1,j_1), i_1=m\d , j_1 = n\d, (m,n)=1$. We get
$in+jm=mn\d$, so $m|i,n|j$ and writing $i=mu,j=mv$ we get $u+v=\d$.
Thus $f(x,y) = \sum_u a_{mu,n(\d -u)}x^{mu}y^{n(\d -u)}$
which can be written as a constant multiple of a product of terms of the
form $x^m-\alpha y^n$ and, since $f$ is irreducible, we conclude that $f$ is
a binomial.


{\it Remarks}(i) If $X$ is a projective curve of genus bigger than one
embedded in an abelian variety $A$, all defined over an unramified 
extension of $\Qp$, then Raynaud [R] proved that there are only finitely
many torsion points of $A$ of order prime to $p$ which are in $X$ modulo $p^2$
and Buium [B] gave an explicit bound for the number of those points.
Perhaps the techniques of Coleman [C] could be used to extend this result
to the full torsion and obtain an abelian analogue of the above result.

(ii) A special case of Lang's extension of the Manin-Mumford conjecture,
proved by Ihara, Serre and Tate (see [L], ch. 8, thm. 6.1) states that 
if $f(x,y)$ is an
irreducible polynomial, not a binomial, over a field of characteristic zero,
then there are only finitely many roots of unity $\z_1,\z_2$ with 
$f(\z_1,\z_2)=0$. This follows from the above theorem by choosing $p$ large
enough such that the field generated by the coefficents of $f$ embed in $\Qp$
and such that the hypotheses of the theorem hold.

{\bf Acknowledgements:}
The author would like to thank the TARP (grant \#ARP-006) and
the NSA (grant MDA904-97-1-0037) for financial support.

\bigskip
\centerline{\bf References.}
\bigskip
\noindent
[B] A. Buium, {\it Geometry of $p$-jets}, Duke Math. Journal {\bf 82}
(1996), 349--367.
\medskip
\noindent
[C] R. F. Coleman {\it Ramified torsion points on curves } Duke
Math. J. {\bf 54} (1987) 615--640.
\medskip
\noindent
[L] S. Lang,  Fundamentals of Diophantine Geometry.
Springer, New York 1983.
\medskip
\noindent
[R] M. Raynaud {\it Courbes sur une vari\'et\'e ab\'elienne et points
de torsion} Invent. Math. {\bf 71}(1983)207--233.
\medskip
\noindent
[TV] J. Tate and J. F. Voloch, {\it Linear forms in $p$-adic roots of unity},
Intern. Math.  Research Notices, 12 (1996) 589--601.
\medskip
\noindent
Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, USA
\smallskip
\noindent
e-mail: voloch@math.utexas.edu



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