\ni{\bf Notation}

In $X(n,p)$ for $0\le v<1$, we have   subspaces $Z(N,p)(v)$ defined 
as follows: $ Z(N,p)=:Z(n,p)(0)=:W_\infty(N)- W_{0}(N)$.\
Suppose  $T_s: A_s\isom A(p^{-w_s},1)$ such that $|T_s(x)|\ra 1$ as $x\ra $.
Then if $1>v>0$, $Z(N,p)(v)$ be the set of $x\in W_\infty $, $x\in 
Z(N,p)$ or $x\in A_s$ for some
$s$ and $v(T_s(x))\le rw_s$.
We can also well define  $Z_1(N)(v)$ for $0\le v<1$,\vsk1

\ni{\bf Frobenius}

\proclaim Theorem.  Suppose $N>4$ and   $v<1/(p+1)$.
   There is a commutative diagram
of rigid morphisms;
$$\matrix{E(N,p)(v)&\lmr{\Phi}&E(N,p)(pv)\cr
\downarrow&&\downarrow\cr
X(N,p)(v)&\lmr{\phi}&X(N,p)(pv)\cr}$$
$$\phi(E,\iota,C)=(E/C,\beta_E\circ\iota,C')$$
where $\beta_E\colon E\ra E/K(E)$ and
$C'=K(E/C)$ (which exists).

\sk



\pr

  \proclaim Proposition.  There exists a section $t$ of $X(N,p)\ra X_1(N)$ over
$Z_1(N)(v)$ if $v<p/(p+1)$.  Moreover, in this case, $t(Z_1(N)(v))=Z(N,p)(v)$.


We will use
\proclaim Lemma.  If $f\colon X\ra Y$ is a morphism of reduced curves 
over $K$ and $U\subset X$ and $V\subset Y$ are affinoid subdomains 
such that  $f(U)\subseteq V$ and $\bar f\colon \bar U\ra \bar V$ is 
an isomorphism.  Then there exists a strict neighborhood $Z$ of $V$ 
in $Y$ and a section $Z\ra X$ of $F$.

and
\sk
\proclaim Lemma.  If $f\colon A(p\iv,1)\ra B(0,1)$ is a finite 
morphism of degree $p+1$ and  $\deg_{A[r]} f=1$ for $r$ near 1,
then there exist a section of $f$ on $A(p^{-{p\ov p+1} },1)$

\ni{\sl Proof  of proposition}\vsk1



Our $\phi$ will be $t\circ \pi_2$ where All we have to
  show is that $t(A,\alp)=(A,\alp,K(A))$

Let $V$ be the family of subgroups of order $p$ of $E(N,p)/X(N,p)$.

   Let $U$  be the family of kernels of reduction in $E_1(N)$ and if $r\in
p^\Q<1$, $U[r]$ the subfamily of affinoid disks of radius $r$.  If
$v<p/(p+1)$,  there exists  $r<r'<1$ such that
$$K_v=\b(E_1[N][p]\cap U[t]\b)_{X_1(N)(v)}$$
with $t=r$ or $r'$
is the family  of canonical subgroups over $X_1(N)(v)$.
In particular $(K_v)_\infty=\mu_p$.\sk

\proclaim Proposition.  $s^*V=K_v$.

\pr Claim:  $K_v|_{Z_1(0)}/Z_1(0)$ is finite.\vsk1

\sect{ A (little) higher level}

Let $X_1(Np)(v)$ be the inverse image of $X(N,p)(v)$ under the 
forgetful map $f$.
\proclaim Theorem.  Suppose $N>4$ and   $v<1/(p+1)$.
   There is a commutative diagram
of rigid morphisms;
$$\matrix{E_1(Np)(v)&\lmr{\Phi}&E_1(Np)(pv)\cr
\downarrow&&\downarrow\cr
X_1(Np)(v)&\lmr{\phi}&X_1(Np)(pv)\cr}$$
$$\phi(E,\iota,\alp)=(\beta_E(E),\beta_E\circ\iota,\alp')$$
where $\beta_E\colon E\ra \beta_E(E)=:E/K(E)$ and
$\alp'(\zeta)=\beta_E(a)$ where $a\in K_1(E)$ and $pa=\alp(\zeta)$.

\pr We have,
$\matrix{E_1(Np)(v)&\lmr{\Phi\circ F}&E(N,p)(pv)\cr
\downarrow&&\downarrow\cr
X_1(Np)(v)&\lmr{\phi\circ f}&X(N,p)(pv)\cr}$ so all we need is a 
rigid map $\beta\colon
X_1(Np)(pv)\ra K_{pv}$  compatible with tihe other maps,  of order $p$.