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\centerline {\bf An analogue of the Weierstrass $\z$-function in characteristic
$p$}
\medskip
\centerline{\bf Jos\'e Felipe Voloch}
\bs
{\leftskip=2in \sevenrm 
To J.W.S. Cassels on the occasion of his 75th birthday.
\par}
\bs
{\bf 0. Introduction}
\medskip
Cassels, in [C], has noticed a remarkable analogy between an algebraic function
introduced by Deuring [D] in positive characteristic and the Weierstrass
$\z$-function in the classical theory of elliptic functions. The purpose of
this note is to take this analogy further. One of our main goals is to use this function to give a very
explicit description of the universal vectorial extension of elliptic
curves, which will give a characteristic $p$ analogue of results of Lang and
Katz.  Also, Mazur
and Tate [MT] have defined, for an elliptic curve defined over a local field
of residue characteristic $p$ with ordinary reduction, an analogue of the
Weierstrass $\s$-function which is defined on the formal group of the curve. In 
characteristic $p$ their construction can be performed for an ordinary curve
over any field and it turns out, as in the classical theory, that the 
logarithmic derivative of the Mazur-Tate $\s$-function is the characteristic $p$
$\z$-function. 
\medskip
{\bf Acknowledgements:} I would like to thank A. Broumas, A. Buium and
J. Tate for many helpful conversations and the NSF (grant DMS-9301157) 
and the Alfred P. Sloan Foundation for
financial support.
\medskip
\medskip
{\bf 1. Quick review of the classical case}
\medskip
Let $E$ be an elliptic curve over $\C$ and $\w$ a non-zero holomorphic
differential on $E$. By integrating $\w$ along closed paths on $E(\C)$ we
get the period lattice $\L$ and $E(\C)$ is isomorphic to $\C/\L$. If $E$ has
a Weierstrass equation $y^2 = x^3 + a_4x + a_6$, and $ \w = dx/2y$ then this
isomorphism is given by $z \mod \L \mapsto (\wp(z),(1/2)\wp'(z))$,
and $\w = dz$, where $\wp(z)$ is the Weierstrass $\wp$-function attached to
the lattice $\L$.
It is a periodic meromorphic function with
periods $\L$, holomorphic in $\C \setminus \L$ and having expansion 
$\wp(z) = z^{-2} + O(1)$ near $z = 0$. The Weierstrass $\z$-function is, by
definition, the unique odd meromorphic function satisfying $d\z/dz = -\wp$.
It is quasi-periodic, i.e., it satisfies $\z(z + \l) = \z(z) + \n(\l), \l \in
\L$, where $\n$ is linear in $\l$. The values $\n(\l), \l \in \L$ are called
the quasi-periods. Finally, it satisfies
$$\z(z + w) = \z(z) + \z(w) + {\wp'(z) - \wp'(w)\over2(\wp(z) - \wp(w))}.$$
(See [A], Ch. 7 and [L], Ch. 18).

The universal extension of $E$ by a vector group is a commutative algebraic
group $\Eu$ which sits in a non-split exact sequence $0 \to \Ga \to \Eu \to E
\to 0$. It can be constructed as the group of isomorphism classes of invertible
sheaves on $E$ with an integrable connection. Lang ([L], 18.1)
and Katz ([K], appendix C) describe
$\Eu$ in terms of $\z$ as follows: $\Eu(\C)$ is isomorphic to $\C^2/\L'$ where
$\L' = \{ (\l,\n(\l)) | \l \in \L\}$ and given $(a,v) \in \C^2$ it corresponds
to the integrable connection on $\O_E((P)-(0))$, where $P = (\wp(a),1/2\wp'(a))$,
given by the differential $(\z(z-a)-\z(z)+v)dz$.

\medskip
{\bf 2. The characteristic $p$ $\z$-function}
\medskip
Let $K$ be a field of characteristic $p$ and $E/K$ an
elliptic curve. Let $E^{(p)}$ be the target of Frobenius $F: E \to
E^{(p)}$. Let $V: E^{(p)} \to E$ be the dual isogeny, i.e., the Verschiebung.
Fix a holomorphic differential
$\w$ on $E$. Choose a Weierstrass equation 
$$y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6,$$
such that $ \w = dx/(2y +a_1x + a_3)$.
If $p \ne 2$, assume further that $a_1 = a_3 = 0$, put $f(x) = x^3 + a_2x^2 + a_4x + a_6$, and define polynomials
$U, T \in K[x]$, with $\deg U \le p-2$, and an element $A \in K$ by:
$$y^{p-1} = f(x)^{(p-1)/2} = U(x) + Ax^{p-1} + x^pT(x).$$
If $p =2$ let $A = a_1$. Then $A$ is the Hasse invariant
of $E$. Let $B$ be the coefficient of $x^{p-2}$ of $U(x)$ if $p$ is odd
and $B =1$ for $p=2$. 

\proclaim Lemma. For $p$ odd we have $f'(x)T(x) + 
2f(x)T'(x) = -Ax + B.$

{\it Proof.} Since $f(x)^{(p-1)/2} = U(x) + Ax^{p-1} + x^pT(x)$, it follows
that $$((p-1)/2)f(x)^{(p-3)/2}f'(x) = U'(x) - Ax^{p-2} + x^pT'(x),$$ hence
$$x^p(f'(x)T(x) + 2f(x)T'(x)) = -f'(x)(U(x) + Ax^{p-1}) -2f(x)(U'(x) - 
Ax^{p-2}).$$ But the last polynomial has degree at most $p+1$, 
so comparing coefficients, the result follows.

We will consider the function field of $E$ as a subfield of the
function field of $E^{(p)}$ via $V$.

\proclaim Definition. The characteristic $p$ Weierstrass $\z$-function
is the rational function $z$ on $E^{(p)}$ satisfying
$z^p - Az = -yT(x), p \ne 2$ and $z^2 - Az = x + a_2, p = 2$
and, if $E$ is ordinary, such that $z + (y/x)$ vanishes at $0$.

It is easy to check that, in the ordinary case,
the equation defining $z$ describes a cyclic,
\'etale cover of $E$ of degree $p$, which has to be $E^{(p)}$ (see [D], pg.254
or [V], lemma 1.1).

Let $\w^{(p)}$ the differential 
on $E^{(p)}$ obtained from $\w$ on $E$ by transport of structure.

\proclaim Proposition 1 (Cassels [C] \S 5,6). If $E$ is ordinary,
the function $z$ satisfies
$z(P + P_0) = z(P) + \n(P_0)$ for all $P_0 \in \ker V$, where $\n$ is linear
in $P_0$ and satisfies $\n(P_0)^p - A\n(P_0) = 0$.
Furthermore, if $p \ne 2$, $z$ is odd and $dz/\w = -x + B/A$.

{\it Proof.} (Sketch) Since $z^p - Az$ is invariant under translation by
points of $\ker V$, the first part follows, except for the linearity, which
is straightforward. From the definition of $z$, $Adz = d(yT(x)) = (-Ax + B)\w$,
by the lemma.

The values $\n(P_0), P_0 \in \ker V$ are the analogues of the quasi-periods.
The derivation
$d/\w$ corresponds to $d/dz$ in the complex case, so $z$ satisfies a similar
differential equation as $\z$. We will discuss the discrepancy coming from the
constant $B/A$ in the next section.
The analogue of Proposition 1 in the supersingular case is the equation
$dz = -B^p\w^{(p)}$ which is easily verified.
Superficially, this result is analogous to the differential equation
satisfied by $z$ in the ordinary case, but is also an analogue of the
quasi-periodicity of $z$. Namely, over a ring with nilpotents, the map
$P \mapsto z(Q+P) - z(Q)$ induces an isomorphism $\n$ between $\ker V$ and
$\alpha_p$ such that the differential $dv$ on $\alpha_p$ (if $v$ is a
coordinate on $\alpha_p$) corresponds to $dz = -B^p\w^{(p)}$.


The following result gives an addition theorem for $z$.

\proclaim Proposition 2. $z(P + Q) = z(P) + z(Q) + {y(Q)-y(P) \over x(Q)-x(P)}
$.

{\it Proof.} Assume first that $E$ is ordinary.
Let $Q$ be a fixed arbitrary point of $E^{(p)}$ and consider
$z(P + Q) - z(P) - z(Q)$ as a function of $P$. It is clearly invariant under
translations by points of $\ker V$ and is therefore a function on $E$. As
$z$ has simple poles on the points of $\ker V$ and no others, it follows
that $z(P + Q) - z(P) - z(Q)$, as a function on $E$, has simple poles at
$0$ and $-V(Q)$ and no others, thus
$$z(P + Q) - z(P) - z(Q)  = c(Q){y(Q)-y(P) \over (x(Q)-x(P))}+d(Q)$$
where $c(Q)$ and $d(Q)$ are some constants depending on $Q$. 
But interchanging the roles
of $P$ and $Q$, it follows that $c(Q) = c, d(Q) = d$ are absolute constants.
(I learned this trick from A. Broumas in a analogous context). One can then
easily check that $c = 1,d=0$ by making $P,Q \to 0$ and looking at the formal
group or by taking $d/\w$ and using the addition formula for $x$.
An alternate proof can be given using the addition formula for the $\z$
function and proposition 3 of the next section.
The supersingular case follows from the generic case by specialization.

\medskip
{\bf 3. The Mazur-Tate $\s$-function}
\medskip
Let $K$ be an algebraically closed field of characteristic $p$ and $E/K$ an
ordinary elliptic curve. Let $E^{(p^n)}$ be the image of
the $n$-th iterate of Frobenius $F^n: E \to
E^{(p^n)}$. Let $V_n: E^{(p^n)} \to E$ be the dual isogeny, so that $V_n$
are separable, by hypothesis. We will consider the function field of $E$ as
a subfield of the function field of $E^{(p^n)}$ via $V_n$. We will also 
identify the formal groups of $E$ and $E^{(p^n)}$ via $V_n$.
Assume $p \ne 2$ and choose a function $s_n$ on $E^{(p^n)}$ with divisor
$\sum_{P \in \ker V_n}(P) - p^n(0)$. Choose a local parameter $t$ at $0$ so that
$\w = (1 + O(t))dt$ near $0$. Mazur and Tate define the $\s$-function to be
the power series in $t$ given by $\s = \lim_{n \to \infty}t^{p^n}s_n$ after 
normalizing $s_n = t^{p^n-1} + O(t^{p^n})$. We refer to [MT] for the many
interesting properties of this function. Let, just as in the classical theory,
$\z = \s^{-1}d\s/\w$.

\proclaim Proposition 3. With the above definitions $\z = z$.

{\it Proof.} Let $\z_n = s_n^{-1}ds_n/\w$, then $\z = \lim\z_n$. Note that
$\z_n$ is odd and has simple poles at the points of $\ker V_n$ and no others
and that the residues of $\z_n\w$ at the poles is $1$. Clearly, these properties
uniquely determine $\z_n$ as a function on $E^{(p^n)}$. But $\z_1$, as a 
function on $E^{(p^n)}$ has these properties also, so $\z_1 = \z_n,\forall n$.
Finally, $z$ was shown to have these same properties (for $n=1$) in [V],
lemma 1.1. However, in [V], there is an inaccuracy, in that the differential
$dx/y$ was used when it should have been $dx/2y$ and leads to a factor of $-2$
being missing throughout the paper. This was pointed out to me by A. Broumas.
Also, what is denoted by $z$ in [V] is what is denoted here by $-z$.
Anyhow, this completes the proof. 

This result leads to an alternative definition of the 
quasi-periods, as follows. Given $P \in \ker V$, let $f$ be the function on
$E^{(p)}$ with divisor $p((P)-(0))$. Then $\n(P) = -df/f\w$. To see this, let 
$\t$ be the operator $\t h(Q) = h(Q+P)$ on functions $h$ on $E^{(p)}$. Then:
$$\n(P)\w = (\t z - z)\w = \t z\w -z\w = \t (ds_1/s_1)-ds_1/s_1 = 
d(\t s_1/s_1)/(\t s_1/s_1).$$
Now, it is clear that $(\t s_1/s_1)$ has divisor $p(0 - (-P))$ and, therefore
$f(\t s_1/s_1) = h^p$ for some function $h$ on $E^{(p)}$ and the result
follows.

Note that the function $x$ in the Weierstrass equation for $E$ is determined
by $\w$ up to the addition of a constant. Mazur and Tate then noted that a
choice for $x$ can be made by letting $x = -d\z/\w$ and called this a 
canonical eks-function (sic). It now follows from propositions 1 and 3 that
this canonical eks-function is characterized, in characteristic $p > 2$, by
$B = 0$.

If $K$ is a local field and $E$ has split multiplicative reduction then
$E$ is a Tate curve, so it admits a parametrization $E = \Gm/q^{\Z}$,
where $q$ is in the maximal ideal of the valuation ring of $K$ and, if
$u$ is the parameter on $\Gm$ and we choose $\w = du/u$, then $\s$ has the
following product expansion (See [MT] or [L], Ch. 18): 
$$\s(u) = (u^{1/2}-u^{-1/2})\prod_{n=1}^{\infty}{(1-q^nu)(1-q^nu^{-1})\over(1-q^n)^2}.$$
By direct substitution one gets the functional equation
$\s(qu) = -(q^{1/2}u)^{-1}\s(u)$. Logarithmic differentiation of these
formulas lead to a series expansion for $\z$, namely, 
$$\z(u) = {(u + u^{-1} + 2) \over 2(u - u^{-1})} + \sum_{n=1}^{\infty}
\left({q^n \over (u - q^n)} - {q^nu \over (1 - q^nu)}\right),$$ and to the
equation $\z(qu) = \z(u) - 1$. Now, $E^{(p)}$ is the Tate curve with
multiplicative period $q^p$, and the Verschiebung is induced by the identity
on $\Gm$. Therefore $q$ represents a point in $E^{(p)}$  
which generates $\ker V$ and the last formula then reads $\n(q) = -1$,
giving the quasi-periods of the Tate curve in characteristic $p$. 
\medskip
{\bf 4. The universal vectorial extension}
\medskip
The universal extension of $E$ by a vector group is a commutative algebraic
group $\Eu$ which sits in a non-split exact sequence $0 \to \Ga \to \Eu \to E
\to 0$. It can be constructed as the group of isomorphism classes of invertible
sheaves of degree zero on $E$ with an integrable connection. In characteristic
$p > 0$ it is known (Rosenlicht [R], pg. 704) that $\Eu$ splits up to isogeny 
and that the maximal abelian subvariety of $\Eu$ is $E^{(p)}$. The next result 
makes these facts explicit.

\proclaim Theorem. $\Eu$ is isomorphic to the quotient of $E^{(p)}\times\Ga$ by 
the subgroup-scheme given by the graph of the homomorphism $\n: \ker V \to \Ga$.
The isomorphism is obtained by
associating to each point $(P,v) \in E^{(p)}\times\Ga$ the connection on
$\O_E((V(P))-(0))$ given by the differential
$(f_P + v)\w$ where $f_P(Q) = z(Q-P)-z(Q)$ is a function on $E$
that has simple poles at $V(P)$ and $0$ and the residues of $f_P\w$ at these
points are $1,-1$ respectively.

{\it Proof.} Clearly $f_P$ is invariant by the action of $\ker V$ so defines
a function on $E$ and the statement about the poles and residues follow from
the corresponding properties of $z$. Thus we have a map from $E^{(p)}\times\Ga$
to $\Eu$, which is clearly surjective. Since $f_{P+P'}(Q) = f_{P'}(Q+P) +f_P(Q)$
it follows that this map is a homomorphism. Finally its kernel corresponds
to $P \in \ker V$, so that $f_P = -\n(P)$, and $v = -f_P = \n(P)$, as desired. 
\medskip
{\bf References}
\medskip
\noindent
[A] L.V. Ahlfors, {\it Complex Analysis}, McGraw-Hill, New York, 1966.
\medskip
\noindent
[C] J.W.S. Cassels, {\it Some elliptic function identities}, Acta Arithmetica
{\bf XVIII} (1971) 37-52.
\medskip
\noindent
[D] M. Deuring, {\it Die Typen der Multiplicatorenringe elliptischer
Functionenk\"orper}, Hamburger Abhandlungen {\bf 14} (1941) 197-272.
\medskip
\noindent
[K] N. M. Katz, {\it The Eisenstein measure and $p$-adic interpolation},
Amer. J. of Math. {\bf 99} (1977) 238-311. 
\medskip
\noindent
[L] S. Lang, {\it Elliptic Functions} Addison Wesley, Reading, Mass, 1973.
\medskip
\noindent
[MT] B. Mazur and J. Tate, {\it The $p$-adic sigma function}, Duke J. of
Math. {\bf 62} (1991) 663-688.
\medskip
\noindent
[R] M.Rosenlicht, {\it Extensions of vector groups by abelian varieties},
Am. J. of Math., {\bf 80} (1958), 685-714.
\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

\bye


\end