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\centerline{\bf Elliptic Wieferich primes}
\medskip
\centerline{\bf Jos\'e Felipe Voloch}
\bs

Fermat's little theorem states that $a^{p-1} \con 1 (\mod p)$, if
$p$ is a prime number and $a$ is an integer coprime with $p$. The
primes for which $a^{p-1} \con 1 (\mod p^2)$ are sometimes called
base-$a$ Wieferich primes, or simply Wieferich primes if $a=2$.
This stems from a theorem of Wieferich that asserts that
the first case of Fermat's last theorem holds if $p$ is not a
Wieferich prime (similar results were obtained for a few other
bases). This is of historical interest only now, but Wieferich primes
appear in other contexts, most notably they are the primes for which
"the derivative of $a$ vanishes", when one pursues a number field-function
field analogy, such as in ([Bu],[I],[Sm]). Numerical evidence indicates
there are very few Wieferich primes for a given base, for instance
$p=1093,3511$ are the only (base-2) Wieferich primes less than $10^{12}$([CDP]).
A naive heuristic argument suggests that there should be about
$\log \log x$ Wieferich primes up to $x$, whereas the analogy with function
fields would suggest there are only finitely many Wieferich primes.
These two possibilities (and an unlikely third) are contrasted in [I].
However there are very few unconditional results on Wieferich primes.
For instance it is not known for any base $a$ if there are infinitely
many non-Wieferich primes to base $a$, although Silverman showed that
this follows from the $abc$-conjecture and Johnson [J] showed that
Mersenne primes are not (base-2) Wieferich primes. About the only 
unconditional result is due to Granville [G] and asserts that if
$a$ is prime (this condition may be removed, as we shall see) and
$a^{p-1} \con 1 (\mod p^2)$ for all sufficienly large primes $p$, then
$a^{p-1} \con 1 (\mod p^3)$ for infinitely many primes. Replacing
the multiplicative group by the group of an elliptic curve one
obtains an analogous notion of elliptic Wieferich primes, introduced
by Silverman [S], where it shown that the $abc$-conjecture implies
the existence of infinitely many non-elliptic Wieferich primes for
some special elliptic curves. 

The purpose of this note is to investigate further the elliptic Wieferich 
primes, we prove an analogue of Granville's result in the number field case
and prove some results in the function field case. We will also
discuss Silverman's result. In the process we will also indicate simple
proofs in the case of the multiplicative group.

Let $K$ be a global field, that is, a number field or a function field
in one variable over a finite field. If $v$ is a non-archimedian place
of norm $q_v$ of $K$ and $a \in K$, with $v(a)=0$ then $v(a^{q_v-1}-1) \ge 1$.
If $v(a^{q_v-1}-1) > 1$, we call $v$ a Wieferich place for base $a$.
If $a$ is a root of unity, all places will be Wieferich places, we exclude
this case in what follows.
Similarly, let  $E/K$ be an elliptic curve and $P \in E(K)$ a rational
point. If $v$ is a non-archimedian place of $K$ for which $E$ has good
reduction, let $N_v$ be the number of rational points,
over the residue field of $v$, of the reduction of $E$ at $v$. 
Then $N_vP$ belongs to the formal group $\hat E$ of $E$ at $v$, which admits
a canonical filtration ${\hat E} = E_1 \supset E_2 \cdots$ (see [S2]).
We call $v$ an elliptic Wieferich place for base $P \in E$ if 
$N_vP \in E_2$. Contrary to the case of the multiplicative
group, it can happen that $(q_v,N_v) >1$, which causes some complication.
We will call $v$ a strong elliptic Wieferich place for base $P \in E$ if
$n_vP \in E_2$, where $n_v$ is the order of $P$ modulo $v$.
We begin generalizing [G]. We say that an element $x$ of an infinite subset
of $K$ is almost an $n$-th power if there exists $y \in K$ such that
$xy^{-n}$ lies on a finite set.

\proclaim Theorem 1. Let $K$ be a number field. 
\item (i) Let $a \in K^*$, not a root of unity. If 
$v(a^{q_v-1}-1) \ge 2$ for all but finitely many places of $K$ then
$v(a^{q_v-1}-1) \ge 3$ for infinitely many places of $K$.
\item (ii) Let  $E/K$ be an elliptic curve and $P \in E(K)$ a
rational point of infinite order. If $n_vP \in E_2$
for all but finitely many of the places of $K$ satisfying
$(n_v,q_v)=1$, then $n_vP \in E_3$ 
for infinitely many places of $K$ satisfying $(n_v,q_v)=1$.

{\it Proof:} (i) Assume that $v(a^{q_v-1}-1) = 2$ for all but finitely many
places of $K$. If $r$ is a prime number and $v$ a place with $v(a^r-1) > 0$,
but $v(a-1)=0$, it follows that $v(a^r-1) = 2$, unless $v$ is on a fixed
finite set. Hence, there exists $b \in K$ such that, for infinitely many 
$r \con 1 (\mod 5)$, say, we have $a^r-1 =by^2$, for some $y \in K$.
Writing $x=a^{(r-1)/5}$ we thus obtain infinitely many points on the
curve $ax^5-1=by^2$, contradicting Faltings' theorem!

(ii) Assume that $n_vP \in E_2 \setminus E_3$ for all but finitely many
places of $K$ satisfying $(n_v,q_v)=1$. 
We may assume, enlarging $K$ if necessary, that there
exists $Q,R \in E(K), 2Q=P, 2R=0$.
Let $x$ be a function on $E$ with divisor $2(R-0)$. Then, since the
divisor of $x$ is divisible by $2$, $x(T)$ is almost a square for $T \in E(K)$.
If $r$ is a prime number which splits completely in $K$
 and $v$ a place with $v(x(rP)) < 0$ then
$rP \in E_1$, then $r=n_v$ and
$rP \in E_2 \setminus E_3$, unless $v$ is on a fixed finite set, provided
that $(q_v,n_v)=1$. If this doesn't happen, then $r=n_v|N_v$, but also
$2|N_v$, since $R$ is rational. However $q_v=r$ since $(q_v,r) >1$ and
$r$ splits in $K$, but then $2r \le N_v \le r + 1 + 2\sqrt r$, which is
a contradiction for $r$ large. We conclude then that $v(x(rP)) = -4$ for
such $r$. If $v$ is a place with $v(x(rP)) > 0$ then arguing with $2rP$
instead of $rP$ as above we conclude that $v(x(rP)) = 4$.
Therefore, for infinitely many $r$, $x(rP)$ is almost a $4$-th power.
We have thus produced infinitely many $K$-rational points on the
cover of $E$ defined by $by^4=x$, for some $b \in K^*$,
again contradicting Faltings' theorem!

One can get by with Siegel's theorem by being more careful.
One can also make similar arguments in the function field
case but there much more is true. For instance, it is easy to see
that there are only finitely many Wieferich places to base $a \in K^*$,
$a$ not a $p$-th power, ($p$ being the characteristic of $K$).
Indeed if $v(a^{q_v-1}-1) > 1$, then, applying
a non-trivial derivation $D$ of $K$ and using that $q_v = 0 \in K$,
we get $v(a^{q_v-2}Da) > 0$, so $v(aDa) > 0$, which can only be satisfied
by finitely many $v$, unless $Da=0$, as desired. Note that the heuristic
argument that gives infinitely many Wieferich places in the number field
case also seem to apply to function fields.

As for the elliptic case, there can be infinitely many $v$'s with 
$p$ dividing $n_v$ which forces these places to be Wieferich. However
we have:

\proclaim Theorem 2. Let $K$ be a function field of characteristic 
$p$, $E/K$ an elliptic curve, $P \in E(K)$, such that $P \notin pE(K_s)$,
where $K_s$ is the separable closure of $K$. Then there are only finitely
many strong elliptic Wieferich places to base $P$ with $p$ not dividing $n_v$.

{\it Proof:} We will assume $p >2$ for simplicity but the proof can be
modified to handle $p=2$. Suppose first that $E$ is defined over $K^p$
and choose a Weierstrass equation $y^2=f(x)$ with coefficients in $K^p$.
The map $\mu: E(K) \to K, (x,y)\mapsto Dx/y$ is a homomorphism with
kernel $E(K^p)=E(K)\cap pE(K_s)$, for any non-trivial derivation $D$ of $K$.
This is proved, e.g., in [V]. Then $\mu(P) \ne 0$, so there are only
finitely many $v$ with $v(\mu(P)) > 0$. If 
$n_vP=(x,y) \in E_m\setminus E_{m+1}$, then $v(x)=-2m,v(y)=-3m$ and
$v(Dx/y) \ge m-1$, if $v$ is outside a finite set. However,
$Dx/y=\mu(n_vP)=n_v\mu(P)$, and we are done.
If $E$ is not defined over $K^p$, we use the map $\mu:E(K) \to K$,
with kernel $pE(K)$ constructed in [V] and a similar argument gives the
result.

If $E$ is defined over a finite field $k$, we can refine further the
notion of elliptic Wieferich place, by saying that $v$ is a very strong
elliptic Wieferich place to base $P \in E(K)$ if $P - P_v \in E_2$, where
$P_v$ is the unique point in $E({\bar k})$ with $P - P_v \in E_1$.
Now it is easy to see there are only finitely many very strong
elliptic Wieferich places to base $P \in E(K)$ under the assumption
of theorem 2. Indeed, if $C/k$ is a curve with function field $K$ then
$P$ corresponds to a map $f:C\to E$ and $v$ is a very strong
elliptic Wieferich place if and only if $df$ vanishes at $v$, so if 
$df \ne 0$, there are only finitely many such places, as desired.

In the hypotheses of theorem 2, it follows from a result of Scanlon [Sc]
that there exists $m$ depending on $E,P$ but not on $v$ with 
$P \notin E_m$. It can be shown, along the lines of Silverman, that
in the number field case there exists $m$ depending on $E,P$ such that
$P \notin E_m$ for infinitely many $v$, assuming
the $abc$ conjecture. Take
$x:E \to \P^1$ ramified only above the $2$-torsion $E[2]$ of $E$. By an argument
similar to the proof of theorem 1, it is  enough to show that $P$ does
not get $v$-adically close to $E[2]$. Now by Belyi's theorem, there exists
$f:\P^1\to \P^1$ ramified only above $\{0,1,\infty\}$ with 
$f(x(E[2])) \subset \{0,1,\infty\}$. Applying $abc$ to $f(x(rP))$ for primes
$r$ yields the result. The simple form of Belyi's map $f$ when $j(E)=0, 1728$
allowed Silverman to get $m=2$ in those cases.

{\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. J., to appear.
\medskip
\noindent
[CDP] R. Crandall, K. Dilcher, C. Pomerance,{\it A search for Wieferich 
and Wilson primes}. Math. Comp. {\bf 66} (1997) 433--449.
\medskip
\noindent
[G] A.Granville, {\it Refining the conditions on the Fermat quotient},
Math. Proc. Cambridge Philos. Soc. {\bf 98} (1985) 5--8.
\medskip
\noindent
[I] Y. Ihara, {\it On Fermat quotient and "differentiation of numbers"}
RIMS Kokyuroku {\bf 810} (1992) 324-341, (In Japanese). English translation
by S. Hahn, Univ. of Georgia preprint.
\medskip
\noindent
[S] J. Silverman,{\it Wieferich's criterion and the $abc$-conjecture},
J. Number Theory {\bf 30} (1988), 226--237. 
\medskip
\noindent
[S2] J. Silverman, {\it The Arithmetic of Elliptic Curves}, Springer 1987.
\medskip
\noindent 
[Sc] T. Scanlon, {\it The abc theorem for commutative algebraic groups 
in characteristic p}, Intern. Math. Research Notices, (1997), {\bf 17} 881-898. 
\medskip
\noindent
[Sm] A. L. Smirnov, {\it Hurwitz inequalities for number fields}, St. Petersburg
Math. J. {\bf 4} (1993) 357-375.
\medskip
\noindent
[V] J. F. Voloch, {\it Explicit p-descent for elliptic curves in characteristic
p}, Compositio Math. {\bf 74} (1990) 247-258.
\medskip
\noindent
Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, USA
\smallskip
\noindent
e-mail: voloch@math.utexas.edu



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