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\line{\hfil \sixrm{Version of May 21,1992}}





\line{\hfil{\bf Companion Forms and  Kodaira-Spencer Theory}\hfil}
\line{\hfil Robert F. Coleman and Jos\'e Felipe Voloch
\footnote{*}{Supported by a CNPq grant}\hfil}

\beginsection 0.  Introduction.

 Let $p$ be a rational prime and $N$ a positive integer relatively prime to $p$. 
Suppose $f=\sum a_nq^n$ is a normalized eigenform on $X_1(N)$  modulo $p$ of weight $k$.
 Then there is a
representation $\rho_f$ of the absolute
Galois group of the rationals to $GL_2(E)$ attached to $f$, where $E$ is a finite field of
characteristic $p$
 (see [D]).
   When $a_p\ne 0$ and $k>1$, Deligne has shown that the restriction $\rho_{f,p}$ of
the representation to a decomposition group at $p$ stabilizes a line.  

Suppose 
 $f$ has  nebentypus $\eps$.
Then, if $2\le k\le p$,
and $a_p\ne 0$, Serre conjectured [S2] that $\rho_f$ is tamely ramified above $p$
 if and only if there exists 
 an eigenform $g=\sum b_nq^n$ modulo $p$ of weight $k'=: p+1-k$ on $X_1(N)$ such that
   $na_n = n^{k}b_n$.   
If $g$ exists it is called a companion form of $f$. 
Gross [G] proved this conjecture in most cases. More precisely, he proved it under the additional 
assumption that the semi-simplification of $\rho_{f,p}$ 
is the sum of two distinct one dimensional representations which is  true \ff $k<p$ or 
$a_p^2\ne\eps(p)$.  This will be called the exceptional case.
The main result of this paper is:

\proclaim Theorem 0.1.  Suppose $f$ is an ordinary cuspidal eigenform on $X_1(N)$ of weight $k$ where 
$2<k\le p$.  Then the representation $\rho_f$ is tamely ramified above $p$ 
if and only if $f$ has a companion form.

\ni   We also use Kodaira-Spencer theory to shed some 
light on the essence of a 
companion form. In contrast to [G], the results proven here do not depend on any unproven
 compatibilities (cf the introduction to [G]).
(The case in which k=p=2 and the 
semi-simplification is the sum of two copies of a 
one dimensional representation remains open.)  


 This  is closely connected to Serre's conjecture [S1] 
that asserts that every odd irreducible representation of the absolute Galois
group of the rationals to $GL_2(E)$ is that attached to  a modular
form modulo $p$ , as above, whose weight, level and
nebentypus character are described  in terms of the representation
 .
 Indeed, if $f$ is as above and $\rho_f$ is tamely ramified above $p$,
then Serre's conjecture on Galois representations (modified for weight $p$
as in the introduction of [E])
predicts that 
 $\rho_f\otimes \chi=\rho_g\otimes\chi^{k}$,  where $g$ is an eigenform of weight
 $k'$ and $\chi$ is the cyclotomic 
character.  This is implied by the above theorem and [G], for $p>2$, since if a
 companion form $f'$ exists
 one can take $g=f'$.
 This paper,  together with [E], 
 settles the question of the 
weight in Serre's conjecture (when $p>2$).  (See [R] for a more detailed 
discussion of the current status of Serre's conjecture.)
The implication, 
if a companion form exists
then $\rho_f$ is tamely ramified above $p$, proven when
$N=1$ in Serre's letter to Fontaine  [S2], does not seem to be a consequence of this 
conjecture.


While we use many of the results and ideas of [G] 
the point of divergence between our proof and Gross's
 is           
[G, Proposition 13.14 1)].  That is, we interpret the existence of a 
companion form to  $f$ as the vanishing of a certain class $h$ in the de Rham 
cohomology of the Igusa curve.  We generalize [G, Proposition 13.14 4)] 
in Proposition 6.8
and answer the question at the end of [G, \S 13].  That is, we give a 
formula for the cup product of ${h}$ with the de Rham class of a global one form 
on $I$.    We then use this formula to establish Serre's
 conjecture.

As in [G],  our proof is based on the \hb{$p$-adic}
geometry of $X_1(pN)$ combined with one of Katz's formulas [K2] for the Serre-Tate local moduli 
for the deformations of an ordinary Abelian variety in  characteristic $p$ in terms
of the relative de Rham cohomology of the universal deformation.
  What we need is contained 
in Theorem 1.1 which is a formula for the logarithmic derivative $dq/
q$ of the Serre-Tate $q$-pairing
between the Tate module of the reduction of a family 
of ordinary
 $p$-divisible groups $G$  over a complete local p-adic ring $R$
and that of its dual $\td G$
in terms of the Kodaira-Spencer pairing between the family of 
relative invariant differentials on $G$ and $\td G$.  (In this case, the values
of $dq/q$ lie in $\Om R{\Z_p}$.)  (This is an important distinction between our 
proof and Gross's.  In [G], Gross used Katz's formula for $\log q$ which does not contain
the information about $q$ necessary to handle the exceptional case.)   
In \S 2 and \S3 we obtain a general formula for the Kodaira-Spencer 
pairing attached to
 a semi-stable curve over a one-parameter infinitesimal deformation of a point.  
In \S4, using [DR] and [G], we apply these results to modular forms 
regarding $X_1(pN)$ as a family over $Spec(\Z_p[\zeta_p])$ with base $Spec(\Z_p)$.
  
Specifically, let $G$ be the ordinary factor
of the Tate module of the jacobian of $X_1(pN)$ cut out by the natural  action of 
$(\Z/p\Z)^*$.  (This will be made more precise in \S 4.)  Let $R$
be the ring of integers in the completion of the maximal unramified extension 
of $\Z_p[\zeta_p]$.  Let $\alpha$ be an element of
 $T_pG\otimes R$
whose reduction (in two senses, see \S 4) corresponds via the Cartier-Serre isomorphism to
the the global one-form
 on the Igusa curve
$I_1(N)$ with $q$ expansion
$f(q)dq/q$ .
We obtain a formula, Theorem 4.4, for the
leading term of $dq_G(\alpha,\beta)/q_G(\alpha,\beta)$ where $q_G$ is the 
Serre-Tate pairing attached to $G$ and $\beta$ is an element of the Tate-module of the
dual of $G$.
  This formula is expressed in terms
of the cup product  of     a global one-form attached to $\beta$ with the class $h$ 
mentioned above.    In \S5, we show that 
$h$ lies in the unit root subspace for the action of Frobenius.  
This means that it is determined by
its cup product with global one-forms and in particular the vanishing of 
the above leading terms of $dq/q$ is equivalent to the vanishing of $h$.  
Finally, in \S6, we complete the proof of Theorem 0.1 by observing that the vanishing of 
these leading terms
is equivalent to the tameness of the ramification of 
$\rho_{f,p}$.

The above theorem has the following interesting corollary which was established
 in [DS] for forms $f$
 which can be lifted to characteristic zero.  However, Mestre {[M]} has found examples of forms of 
weight one
which cannot be lifted. 

\proclaim Corollary  0.2.  Suppose $p>2$.  If $g$ is a
cuspidal eigenform on $X_1(N) \mod p$
of weight one then $\rho_g$ is unramified above $p$.

Conversely, Edixhoven has used our results to show that if $rho$
is an unramified "modular" representation, then $\rho=\rho_g$ for some form $g$
of weight  one.  In the case  $ a_p^2\ne  4\eps(p)$   this is  also a
consequence of  [G, Cor. 13.11].  

In [G] it is observed that, in the cases handled,
 $\rho_{f,p}$ is tamely ramified \ff it is split.  This this not true in the exceptional case and 
finding a criterion for the splitting of $\rho_{f,p}$ remains an interesting open problem.

We would like to thank A. Ogus and K. Ribet for several helpful conversations and 
S. Edixhoven and B.
Gross for useful remarks.
The second author would like to thank U. C. Berkeley for its hospitality 
during the academic year 1991/92, when this work was done.


\beginsection 1.  Katz's formula for ${\bf dq/q}$.

 Suppose $T$ is a scheme, $S$ 
is a scheme over $T$ and $A$ is an abelian scheme over $S$. 
  Let $\Omega_A$ denote the sheaf 
on $S$ of invariant relative one-forms on $A/S$ and $\td A$  the dual of $A$.  
Then one has the Kodaira-Spencer  pairing:
$$\kappa\colon\Omega_{A} \otimes \Omega_{\td A} \ra \Omega^1_{S/T}.$$
This is constructed,  in several equivalent
ways                                                                                                                                                          
 in {[I]}, [FC, III \S 9] and [K, \S 1].  In particular,  suppose
$S=Spec(R)$ where $R$ is a complete local ring with residue field $F$
of
characteristic $p$ and $T=Spec(W(F))$ where $W(F)$ nis the ring of Witt vectors of $F$.
  Let $f$ denote the map $\td A\ra S$ and
$\Omega_{R/W(F)}=\Omega^1_{S/T}$.  Then the sequence of sheave on $\td A$
$$0\ra
f^*\Omega_{R/W(F)}\ra \Omega^1_{\td A/W(F)}\ra \Omega^1_{\td A/R}\ra 0$$ is
exact.  Let $Kod$ denote the composition $$\Omega_{\td A}\isom f_*\Omega^1_{\td
A/R}\ra R^1f_*f^*\Omega^1_{R/W(F)}\isom R^1f_* \O {\td A}\otimes
\Omega_{R/W(F)},$$ 
where the second map is the boundary map. 
 Then, $\kappa(\omega,\td \omega) = \omega.Kod(\td\omega)$.  

 One can  also define a pairing between the group of 
invariant one-forms of a $p$-divisible group over $R$
 and that of its dual into $\Omega_{R/W(F)}$ using the construction
 in [I, Corollaire 4.8 (iii)]. 
One gets the pairing discussed above when 
the $p$-divisible
group is the one attached to an abelian variety, identifying 
the corresponding modules of invariant differentials.  This pairing is 
clearly functorial for morphisms of $p$-divisible groups over $R$.


For an object $X$ over $R$, we let $\bar X$ denote its special fiber. 
We say a $p$-divisible group $G$ is  ordinary  if the dual of the connected 
subgroup of $\bar G$ is \'etale.

Suppose the residue field $F$ of $R$ is algebraically closed.  
If $m$ is the maximal ideal of $R$, a construction of Serre and Tate gives a pairing
 $q\colon T_p\bar G\times T_p\td\bar G\ra 1+{ m}$ for an 
ordinary $p$-divisible group $G$
which in turn gives
local parameters on the local moduli 
space of $p$-divisible groups over an Artin local ring of residue characteristic
p (see [K2] and [G,\S 14]).  As is already clear from [G],  
understanding the Serre-Tate parameters is fundamental for a proof of 
Serre's conjecture on
companion forms.  In [K2], Katz gives  formulas for the Serre-Tate parameters 
in terms of the  Kodaira-Spencer pairing.  
We will need the following theorem which is a corollary  of Katz's results.


Suppose $G$ and $\td G$ are dual 
$p$-divisible groups over $R$.  If $\alpha\in T_pG$, we let  $\omega_{\alpha}
 = \alpha^*(dt/t)\in \Omega_{\td G}$, viewing $\alpha$ as a homomorphism 
from $\td G$ to $\Gm$ the
formal multiplicative group and where $dt/t$ is the canonical invariant form on 
$\Gm$.  For $a\in R^*$, let $d\log(a)=d_{R/W(F)}a/a \in \Omega_{R/W(F)}$.


\proclaim Theorem 1.1. Suppose $R$ 
is as above, $F$ is algebraically closed  and $G$ is 
an ordinary $p$-divisible
group over $R$.
If $\alpha \in T_p\bar G$ and $\td\alpha \in T_p\td\bar G$, we have:
  $$d\log q(\alpha,\td \alpha) = \kappa(\omega_{\td\alpha},
\omega_{\alpha})\ .$$




\pr  Suppose $A$ is an ordinary abelian variety over $F$ and $R$ is 
the coordinate ring of the moduli space of the  universal deformation $
\cal A$ of $A/F$.   Then, if $G$
 is the $p$-divisible group of $\cal A$, this is [K2, Theorem 3.7.1].
  It then follows for an abelian scheme over arbitrary $R$ with 
ordinary reduction by functoriality. 
The full result  follows from this and functoriality since we can 
embed an arbitrary ordinary $p$-divisible group
 in the Tate-module of an abelian scheme over $R$ with 
ordinary reduction.  
Indeed, by the theorem
of Serre-Tate [K2, Theorem 1.2.1] it suffices to embed the special fibre of 
$G$ into the $p$-divisible group of an ordinary abelian variety.
By adding the appropriate number of copies of $\Q_p/\Z_p$ or $\mu_{p^\infty}$ to $G$
 we can assume that the \'etale
and connected parts of $G$ have same dimension. Its reduction is then 
the $p$-divisible group of an ordinary abelian variety since
 there is only one $p$-divisible group with \'etale
and connected parts of the same given  dimension over the residue field of $R$
 as $F$ is algebraically closed and it is the $p$-divisible group of an ordinary abelian variety.

\proclaim Note.   Suppose $R=\Z_p[\zeta_p]$ where $\zeta_p$ is a primitive $p$-th root of unity.
 Then $W(F)=\Z_p$ and $\Omega_{R/W(F)}\isom R/(1-\zeta_p)^{p-2}R$ 
and is generated by $d\zeta_p$.  For $a\in 1+(1-\zeta_p)R$,
 $d\log(a)$ does not necessarily equal $d(\log(a))$.  (Note that$\log(a)$ is the the maximal ideal of $R$
in this case as $e<p-1$).  For example,
$d(\log(\zeta_p)) =0$ while $d\log(\zeta_p) = -d\pi$ where 
 $\pi\in R$ such that $\pi^{p-1}=-p$ and $(1-\zeta_p)/\pi\con 1\mod \pi$. (This point is  
crucial for us and perhaps explains why Gross's proof did not
 handle the exceptional case. )
Actually, the above equality is a  consequence of the stronger congruence:
$$(1-\zeta_p)+{(1-\zeta_p)^2\over 2}+\dots+{(1-\zeta_p)^{p-1}\over p-1}
\con \pi \mod \pi^{p+1}$$
which the reader may work out as a pleasant exercise.  

 Now let $R$ denote the ring of integers in the completion of
a maximal unramified extension of $\Q_p(\mu_p)$.
We will apply the above result when $G$ is a $p$-divisible subgroup with
ordinary  reduction over $R$ of the p-divisible
group of the  jacobian of $X_1 (pN)$.
The Kodaira-Spencer map can be calculated on the curve
and this will be done in  \S 3 extending some work of
Friedman-Smith [FS] and Fay [F].




\beginsection 2.  de Rham cohomology of curves.  

Suppose $X$ is a smooth irreducible complete curve over $S=Spec(F)
$  and $D$ is a non-trivial
 reduced effective divisor on $X$. Let $U=X-D$.  
Let $\eta$ denote the generic point of $X$.  
Consider the complex of groups
$$\ca D_{X,D}: \O X(U)\ra \Omega^1_{X/S}(U)\oplus \O {X,\eta}/\O {X,D}\ra
\Omega^1_{X,\eta}/\Omega^1_{X/S}(\log D)_D\eqno (2.1)$$ 
where the first arrow takes a section $h$ to
$(dh,h)$ and the second  takes a pair $(\omega,g)$ to $\omega-dg$.
 This complex computes the cohomology of the de Rham complex with 
log poles on $D$ (i .e. the hypercohomology of the complex $\Omega\per_{X/S}(\log D)$).  We can prove this as follows:

 Consider the bi-complex $\ca S$ defined by
$$ \ca S^{i,j}=\cases{\Omega^i_{X/S}(U),&if $j=0$\cr
\Omega^i_{X/S,\eta}/\Omega^i_{X/S}(\log D)_{D},&
if $j=1$.\cr}$$  
Then clearly the associated simple complex is $\ca D_{X,D}$.  Now if 
$V$ is an affine neighborhood of $D$, the bi-complex $\ca T_V$
$$\ca T^{i,j}_V = \cases{\Omega^i_{X/S}(U)\oplus\Omega^i_{X/S}
(\log D)(V),&if $j=0$\cr
\Omega^i_{X/S}(U\cap V), &if $j=1$\cr}$$
computes de Rham cohomology.  Now
$\dirlim_V \ca T_V$ is the bi-complex $\ca T$ where
$$\ca T^{i,j}=\cases
{\Omega^i_{X/S}(U)\oplus \Omega^i_{X/S}(\log D)_{D},&if $j=0$\cr
\Omega^{i}_{X/S,\eta},&if {$j=1$}.\cr}$$
As this is clearly quasi-isomorphic to $\ca S$ we get 
what we want.  

 Suppose $\omega$ is an element of $\Omega^1_{X/S}(nD)(X)$ and $(n-1)!$ is invertible.  Then, there exists a 
section $h$ of $\O X((n-1)D)_{D}$ 
such that $\omega-dh$ has at worst simple poles on $D$.  Moreover, $h$ is well 
defined $\hbox{mod}\, \O {X,D}$. Hence, associated to such an $\omega$ we have a 
well  defined class $[\omega]$ in the ${\bf H}^1(X,\Omega\per_{X/S}(\log D))$.  If $\omega$ has zero
residues, this class lies in the image of ${\bf H}^1(X,\Omega\per_{X/S})$.



\def\hsk#1{\hskip#1pt.}
\beginsection 3.  Kodaira-Spencer for semi-stable curves.

 Let $R=F[t]/(t^{b+1})$ where $b\ge 0$, $F$ is a field of characteristic $p$ 
and $b<p$ if $p\ne 0$.
Let $X$ be  a semi-stable curve over $R$   and let $s\colon X\ra Spec(R)$
 denote the structural
morphism.   


Let $R\t$ denote the log-scheme 
associated to the pre-log-structure $\N\ra R, 1\mapsto t$.   (See [Ko] for the 
foundations of log-schemes.)
Let  
$M_R$ denote the corresponding monoid.    
  The reduction of $R^{\times}$ to $F$, the 
``punctured point,'' we denote by $F^{\times}$.
We put the trivial log-structure on $F$ and denote 
this log-scheme by $F$ as well.  It follows that there 
is an element $T
\in M_R$ which maps to $t$ and  $\Omega^1_{R^{\times}/F}$ is a free 
$R$ module generated by $d\log T$.

We will put   a  log-structure on $X$ which is smooth  over $R\t$,
 in the following way:
We suppose  $\tilde X$ a lifting of $X$ to a 
semi-stable curve over $\tilde R$ 
where $\tilde R$
is a discrete valuation ring such that the generic fiber of $\tilde X$ is smooth over the 
generic point of
$\tilde R$. 
On $\tilde X$ and $\tilde R$ we have a natural log-structure which is
  the subsheaf of  the structure sheaf whose sections  become 
invertible upon the removal of the special fiber.  We  take $X^{\times}$ 
to be the
 reduction of this log-scheme to 
$R$.  This actually only depends on $\tilde X\mod { m}^{b+2}$ where 
$ m$ 
is the maximal ideal of $\tilde R$.  We let
 $\alpha\colon M_X\ra \O {X}$ be the corresponding log-structure.
   Denote the corresponding log-scheme $X\t$.  
 
We may view $M_R$ as a submonoid of $M_X(X)$.  Moreover,
$H^1(X,\Om{X\t}{R\t})\isom R$.  (Probably the results in this section will remain valid for 
any smooth log-structure on $X$ over $R\t$ with this property.)
We also have an exact sequence of sheaves (see [Ko, Proposition 3.12])
$$0\ra s^*\Omega^1_{R^{\times}/F}
\ra \Omega^1_{X^{\times}/F}\ra\Omega^1_{X^{\times}/R^{\times}}\ra 0\ \ .\eqno (3.1)$$ 
Let $Kod\colon H^0(X,\Om {X\t}{R\t})\ra H^1(X,s^*\Om {R\t}{F})\isom H^1(X,\O{X})\otimes 
\Om{R\t}{F}$ denote the boundary map in the long exact sequence obtained by taking cohomology 
of the above exact sequence.  This sequence and $Kod$ are functorial in maps of 
smooth log-structures.  We let  $d$ denote the boundary map for the complex
 $\Omega\per_{X^{\times}/R^{\times}}$ and
$d'$ the boundary map for the complex
$\Omega\per_{X^{\times}/F}$ .   Moreover, when $X$ is smooth over $R$, $\Om {X\t}{R\t}\isom
\Om {X}{R}$,  the 
sequence (3.2) below is exact and this $Kod$ is the composition of the obvious generalization 
of the map $Kod$ 
discussed in \S1  and the
natural map from $H^1(X,\O{X})\otimes\Om RF$ into $H^1(X,\O{X})\otimes 
\Om{R\t}{F}$ (which is an injection when $p=0$ or $b<p-1$).  




Now suppose $X$ is a semi-stable curve over $R$ (i.e. locally isomorphic to 
$xy= t$ in the \'etale topology)whose  reduction$\mod t$ is 
$\bar X =\colon  C_1\cup C_2$ where $C_1$ and $C_2$
 are smooth irreducible curves.  Let $\iota_i$ be the inclusion map
$C_i\ra X$ and $D=C_1\cap
C_2$.  Let $U_1=X-C_2$ and $U_2=X-C_1$.





 



\proclaim Theorem 3.1.   Suppose $\omega$ is in the image
of the natural map from $H^0(X,\Om XR)$ to $ H^0(X,\Om {X\t}{R\t})$
 and $\omega|_{U_2} = t^{b}
\eta$ for $\eta\in \Om X{R}(U_2)$.  Then $$\bar\eta\colon=
\eta|_{C_2}\in H^0(C_2,\Om {C_2}{F}((b+1)D))\hsk5$$
In particular, we 
obtain a cohomology class $[\bar\eta]\in {\bf H}^1(C_2,\Omega\per_{C_2/F}(\log D))$ by the 
methods of \S 2.  
If
$\nu$ is in the image of $H^0(X,\Om XR)$ in $ H^0(X,\Om {X\t}{R\t})$
 and $\nu|_{U_1}
\in t^b\Om {X\t}{R\t}(U_1)$ then
$$\nu.Kod(\omega)=([\nu|_{C_2}],[\eta|_{C_2}])_{C_2}b\,t^bd\log T
\hskip5pt ,$$
where $(\ ,\ )_{C_2}$ is the cup product pairing between ${\bf H}^1(C_2,{\cal I}_D\ra \Om{C_2}{F})$
and ${\bf H}^1(C_2,\Omega\per_{C_2/F}(\log D))$.


As we shall see, the hypothesis
that $\omega$ is in the image of $H^0(X,\Om XR)$  follows from the other 
hypotheses if $b>0$.

\pr  We may suppose $F$ is algebraically 
closed.
We will first 
compute $Kod(\omega)$.   
For each $Q\in D$, let $U_Q$ be an affine 
open neighborhood of $Q$. 



 Using the exactness of  (3.1), we can lift $\omega|_{U_i}$ to $
\omega_i\in\Omega^1_{X\t/F}(U_i)$  for\pen $i\in \{1,2\}\cup D$.  Then 
$\omega_i-\omega_j = f_{i,j} d'\log T$ for some $f_{i,j}\in \O X(U_{i,j})$ where
$U_{i,j}=U_i\cap U_j$.
It follows that the class $Kod(\omega)$ is represented by the 
one-cocycle $U_{i,j}\mapsto f_{i,j} d'\log T$.  

Now we will be a little 
more careful about our choices.    First, 
we may assume that there are elements $x_Q, y_Q\in M_X(U_Q)$ 
such that $x_Qy_Q = T$, $\alpha(x_Q)$ and $\alpha(y_Q)$ are local parameters 
at $Q$ and $\alpha(x_Q)$ vanishes on $C_2\cap U_Q$.   We also suppose that $Q'\not
\in U_Q$ for $Q'$ different from  $Q$ in $D$.   
  By the exactness 
of the sequence of sheaves:
$$s^*\Omega^1_{R/F}
\ra \Omega^1_{X/F}\ra\Omega^1_{X/R}\ra 0\ \ .\eqno (3.2)$$
there exists an $\tilde \eta \in \Om X{F}(U_2)$ lifting $\eta$. 
The image of the form $ t^b\tilde\eta$ in $\Om {X\t}{F}(U_2)$ lifts $\omega|_{U_2}$
and is independent of  
choices.  We take this to be $\omega_2$. 
We may also suppose $\omega_1$ is in the image of $ \Om X{F}(U_2)$.
Now fix $Q$ and set $x=\alpha(x_Q)$.  We will  write $d\log x_Q$ as
$d\log x$ and $d\log y_Q$ as $d\log y$ (and similarly with $d'$ in place of $d$).  Note that 
$$d\log x +d\log y = 0\quad \hb{and}\quad d'\log x +d'\log y = d'\log T\hsk5 $$
   We may expand $\omega$ in $x$ and $y$
at $Q$ and  write
$$\omega=f(x)d\log x + g(y)d\log y$$
at $Q$, where $f(x)$ and $g(y)$ are power series  in $x$ and $y$ , 
respectively, over $R$ and $f(0)=0$.
  Using the fact that $\omega|_{U_2} \con 0 \mod t^b$ we see that
$f(x)-g(y) \con 0 \mod (x,t)^b$ and this in turn implies
$f(x)\in (x,t)^{b}F[[x]]$
 {and}
$g(y)\in t^bF[[y]]$.  
Suppose $f(x)=\sum_{n=1}^{\infty} a_nx^n$.  Let $r(x)=\sum_{n=1}^{b}a_n
x^n/n$.  Then
$${t\,r'(t/y)\con 0 \mod t^b
}\hs {5pt}$$
  Write  $$\omega-dr(x)=x^{b+1}kd\log x+t^bhd\log y$$ where $h$ and
$k$ are elements of  $\O X(U_Q)
$ and
$t^bh\in y\O X(U_Q)$. (This is where we use the hypothesis that 
$\omega$ is in the image of $H^0(X,\Om XR)$ and as we remarked above it 
is only needed when $b=0$.)  This implies
that $\omega_2$  equals $-xr'(x)d'\log y+t^bhd'\log y$ on $U_{2,Q}$
since  $xr'(x)$ is divisible by $t^b$ on $U_{2,Q}$, $x^{b+1}=0$ on $U_{2,Q}
$ and $
d'\log y \in \Om X{F}(U_{2,Q})$.  
  Set $\omega_Q = d'r(x)+x^{b+1}kd'\log x+t^bhd'\log y$.  Then $\omega_Q$ lies in the image of
$\Om XF(U_Q)$.  

Then,
$$\eqalign{\omega_Q-\omega_2&= d'r(x)+xr'(x)d'\log y\cr
&=xr'(x)d'\log T+r^{(d')}(x)d't\hs5pt.\cr}$$
(Here, $r^{(d')}$ is the polynomial obtained from 
$r$ by applying $d'/d't$ to its coefficients.)
Now $$\eqalignno{f_{Q,2}&=xr'(x)+t r^{(d')}(x)\cr
&=t/y\cdot r'(t/y)+t r^{(d')}(t/y)\cr&
=t^bu(y\iv)\cr}$$ where $u=:u_Q$ is a 
polynomial of degree at most $b$.  
  Moreover, computing each side of the relation
$$d'\omega_Q- d'\omega_2 =d'(t^bu(y\iv)d'\log T)$$
independently, yields
$$\bar\eta -{1\over b}d\bar u(y\iv)=vd\log y\hskip3pt$$  
where $v=:\bar h|_{U_2} \in \O {C_2}(U_{2,Q}\cap C_2)$.  (Here we used 
that $x^{b+1}=0$ and $t^bd'h\wedge d'\log y = 0$ on $U_2$.)
 This yields the  the first part of the theorem
In fact, this means $(\{\bar\eta,v_Qd\log y\},b\iv\{u_Q(y_Q\iv)\})$ is a hypercocycle representing 
the class  $[\bar\eta]$.  We don't have to worry  about $U_{Q,Q'}$
because it is contained in  $U_1\cup U_2$.

Now we finish the proof of the theorem.  The class $\nu. Kod(\omega)$ 
is represented by the cocycle $U_{i,j}\mapsto r_{i,j}=\colon f_{i,j}\nu \otimes
d'\log T$.   Since, $f_{1,Q}\in t\O X(U_{1,Q})$ by the exactness of (3.2) it
follows that $r_{1,Q} =0$.  In fact, it follows that $\nu.
Kod(\omega)=t^b\gamma\otimes d'\log T$  where $\gamma\in H^1(X,
\Om {X\t}{R\t})$.  Moreover, $\bar\gamma$ is the image of 
$b(\iota_2^*\nu. [\bar\eta]
)$ under the  homomorphism
$$H^1(C_2,\Om {C_2}F)\ra H^1(\bar X,\Om {
\bar X\t}{F\t})$$ coming from the natural map $\iota_{2*}\Om {C_2}{F}\ra \Om
{X\t}{F\t}$.
  The theorem follows from the naturality of the 
trace map. \bx


There is an  exact sequence of complexes
$$0\ra s^*\Omega^1_{R^{\times}/F}\otimes\Omega\per_{X^{\times}/R^{\times}}[-1]
\ra \Omega\per_{X^{\times}/F}\ra\Omega\per_{X^{\times}/R^{\times}}\ra 
0\ \ .$$
 Taking cohomology, we get a log-connection
$$\nabla: {{\bf H}^1(X,\Omega\per_{X^{\times}/R^{\times}})}
\ra {\bf H}^1(X,\Omega\per_{X^{\times}/R^{\times}})\otimes \Omega^1_{R^{\times}/F}
\ .$$  



The log-structure induced on $C_2$ from the map $C_2\ra X$ is the log-structure
associated to the divisor $D$ on $C_2$.  We denote the corresponding log-scheme by $C_2\t$.
Now fix a divided power structure on $R$ so that $(T)$ is a divided power ideal (this determines it when 
$b<p-1$).
Then ${{\bf H}^1(X,\Omega\per_{X^{\times}/R^{\times}})}$ is canonically 
isomorphic to $H^1_{Cris}(\bar X^{\times}/R\t)$ and hence we have a 
natural map from  ${{\bf H}^1(X,\Omega\per_{X^{\times}/R^{\times}})}$ 
to  $H^1_{Cris}(C_2\t/R\t)$.
Let $U_i = X-C_i$.  The following underlies the above theorem:

\proclaim Theorem 3.2.  Suppose $p>0$, $\omega\in H^0(X,\Om X{/R})$ and 
$\omega|_{U_2} = t^{b}
\eta$ for $\eta\in \Om X{R}(U_2)$.  Then $\bar\eta\colon=
\eta|_{C_2}\in H^0(C_2,\Om {C_2}{F}((b+1)D))
$ and if $b+1\le p$ the image of $\nabla {[\omega]}$ in
 $H^1_{Cris}(C_2\t/F\t)\otimes \Om{R\t}{F}$
is $bt^b[\bar\eta]d\log T$.

This may be proved by first observing that $\iota_2$ is an exact closed immersion and $X$ 
is log-smooth.  This means one can compute the
crystalline cohomology of $C_2\t$ over $R\t$ by considering the 
divided power log-de Rham complex
on the log-divided power neighborhood of $C_2\t$ in $X\t$.    Making 
this explicit and using the arguments in the proof of the above theorem 
yields this theorem.






\beginsection 4. Applications to modular forms. 

Suppose $N$ is a positive integer and $p>2$ is a prime not dividing  $N$. 
In this section we will apply the results of the preceeding sections to modular forms of weights
$2<k\le p$ on $X_1(N)\mod p$.  

We will follow the notation of [G].  
  In particular, suppose $P$ and $M$ are relatively prime positve integers.
 Then we have automorphisms $\dia d_P$ and $w_\xi$ of $X_1(PM)$, 
 where $d\in(\Z/P\Z)^*$ and $\xi$ is a $P$-th root of unity,
 described on points as follows:
Suppose $E$ is an elliptic curve and  $\alpha\colon\mu_{PM}\ra E$ is an embedding  
.  Write $\alpha = \beta\cdot\gamma$ 
where $\beta\colon \mu_M\ra E$ and $\gamma\colon \mu_P\ra E$ are 
embeddings and let $\phi$ be the natural isogeny $E\ra E/Im(\gamma)$.  Let 
$Q$ be the point of order $P$ on $E$ such that under the Weil pairing 
$(\gamma(\xi),Q)=\xi$. Then,
$\dia d_P(E,\alpha)=(E,\beta\cdot d\gamma)$ and $w_{\xi}(E,\alpha)
=(E',(\phi\circ\beta)\cdot\gamma')$ where $E'=E/Im(\beta)$ and $\gamma'\colon\mu_P\
\ra E'$ is the embedding which takes $\xi$ to $\phi(Q)$.  (See [G,
Proposition 6.7] for relations among these automorphisms.)  We will let $\dia{\ }$ denote
$\dia{\ }_{PM}$.  When we 
speak of the $q$-expansion of a form on $X_1(PM)$, we mean the 
$q$-expansion at the  cusp corresponding to the inclusion $\mu_{PM}\ra {\bf G}_m$.

Fix a primitive $p$-th 
root of unity $\zeta=: \zeta_p\in \bar \Q_p$ and let $\pi$ be the $(p-1)$-st root 
{$\pi$} of $-p$ in $\Q_p{(\zeta)}$ 
such that ${(1-\zeta)/ \pi}\con 1 \mod\pi$.     By 
$R$ we will henceforth mean the ring of integers in the completion of a maximal unramified 
extension $K$ of $\Q_p(\zeta)$ and $\F$ will denote the residue field of $R$. 

 Let $X$ denote the base change to $R$ of canonical  model for $X_1(pN)
$ over $\Z_p[\zeta_p]$ 
described by Deligne-Rapoport [DR, V \S 2] (see also [G, \S 7]).  It is semi-stable in the sense of 
\S3 if $N\ge4$. As 
discussed in [G, Proposition 7.1],  the reduction $\bar X$  of  $X$ 
consists of two components $I$ and $I'$ crossing normally at a finite 
set of points.    The curve $I$ is canonically isomorphic to the Igusa 
curve $I_1(N)$ and the singular points of $\bar X$ are the 
supersingular points $SS$ on $I_1(N)$.   If $\xi$ is a primitive $p$-th
or $pN$-th root of unity the reduction of $w_{\xi}$ is an automorphism
 of $\bar X$ which interchanges $I$ and $I'$.  Let $w=:w_{\zeta}$.
If $\xi$ 
is a primitive $N$-th root of unity or $d\in(\Z/pN\Z)^*$  then the 
reductions of $w_{\xi}$ and $\dia d$ are automorphisms of $
\bar X$ which preserve $I$ and $I'$.  

 Let $\theta$ be the operator 
on$\mod p$ modular forms on $X_1(N)$ which acts on $q-$expansion as $qd/dq$.  
If $k$ is a weight, we let $k'=p+1-k$.  
If $f$ is a cusp form of weight $2< k\le p$ on $X_1(N)$ we let $f'=\theta^{k'}
f$.  This  is a cusp form of filtration $2(p+1)-k$ by [G, Propostion 4.10].
By pullback, we identify forms of weight $k$ on $X_1(N)$
 with forms of weight $k$ on $I$.   If $N>2$, let $a$ be the weight one modular form on $I$ 
whose $q$-expansion is $1$.  Otherwise, let $a^2$ denote the weight $2$ modular form whose $q$
expansion is $1$.
Then $a^{p-1}=(a^2)^{(p-1)/2}$ is 
the Hasse invariant regarded as a weight $p-1$ modular form on $I$.
  For a modular form $f$ of weight $k$ on $X_1(N)$, 
we let $\omega_f$  denote the differential form on $I$ whose 
\b{$q$-expansion} is $(f/a^{k-2})dq/q$.  (We note that $k$ must be even if $N\le2$.)
It follows, by [G, Thm. 5.8], if $2 < k\le 
p$, $\omega_{f'}$ has poles only at the supersingular points of order 
at most $k'+1 \le p$ if $N> 2$ and $(k'+1)/2 \le p$ if $N\le2$. 
(This will also follow from 
the next proposition and
Lemma 3.3.)    
Now $\omega_{f'}|\dia d_p = d^{k'}\omega_{f'}$ so it follows from
 [G, Propositon 5.2] 
that it is of the second kind 
if $k\ne 2$.  Thus, by the discussion in 
Section 2, it  defines a de Rham 
cohomology class {$[f']=\colon [\omega_{f'}]$} on $I$.   

     Let $t\colon (\zmp)^*\ra \Z_p^*$ be the Teichm\"uller character. 
We call a cusp form $F$ of weight $2$ on $X_1(pN)$ such 
that the 
Fourier coefficients of $F$ and $F|w$ lie in $R$ regular.  If $f$ is a 
form of weight $k$ on $X_1(N)$$\mod p$ we say that $F$ is a lifting of $f$ 
if $F$ is regular  $\bar F(q)= f(q)$ and $F|\dia d_p=t(d)^{-k'}$.    
This implies that the reduction of the regular differential form on $X$ $\omega_F=: F(q)dq/
q$ restricted to $I$ equals $\omega_f=: (f/a^{k-2})dq/q$.  

Let $T_l$, $U_s$ and $U_p'$ where $l$ and $s$ are primes, $l\,\not | \, pN$ and $s\, |\, pN$,
be the Hecke operators on $X_1(pN)$ defined
 in [G].  We set $T_l=U_l$.   We let $H$ denote the subalgebra of $End(J_1(pN))$
generated by the $T_l$ and $\dia d$ for $d \in (\Z/pN\Z)^*$.


 
 



\proclaim  Proposition 4.1.  Suppose $f$ is a cusp form of weight $k$, $2<k\le p$ on $X_1(N)
$ of nebentypus $\eps$.  Let $F$ be a form 
of weight $2$ on $X_1(pN)$ lifting $f$.  Then,
$$\big(F|U_pw\big)(q) \con -\eps(p)f'(q)(k'-1)!\pi^{k'} \mod \pi^{k'+1} .$$

\pr  First,  $F|U_pw=G|U_p'$ where $G=F|w$.  Next, 
by [G, Props. 6.7 and 6.10], 
$$\eqalign{G|U_p'(q) &\con\sum_{d\in \F_p^*} G|w_{\zeta^d}(\zeta^dq)\cr
&\con \sum_{d\in \F_p^*}F|\dia p_{_{\scriptstyle N}}
\dia {-d}_p(\zeta^dq)\cr
&\con\sum_{d\in \F_p^*} t(-d)^{-k'}F|\dia p_{_{\scriptstyle N}}(\zeta^dq) 
\mod \pi^{k'+1}\mod p,\cr}$$
and $p\con 0 \mod \pi^{k'+1}$. 
Now, if $F|\dia p_{_{\scriptstyle N}}(q) = \sum_{n\ge 1} A_nq^n$,
$$\eqalign{\sum_{d\in \F_p^*} t(-d)^{-k'}F|\dia p_{_{\scriptstyle N}}(\zeta^dq)  
&= \sum_{n\ge 1}\sum_{d\in \F_p^*} t(-d)^{-k'}\zeta^{dn}A_nq^n\cr
&=\bigg(\sum_{d\in \F_p^*} t(-d)^{-k'}\zeta^{d}\bigg )\bigg(\sum_{n\not\con 0 
\hb{\sevenrm mod}\, p}
t(n)^{k'}
A_nq^n\bigg)  .\cr}$$
By Stickelberger's Theorem,  $\sum_{d\in \F_p^*} t(-d)^{-k'}\zeta^{d} 
\con -(k'-1)!\pi^{k'}  \mod \pi^{k'+1} $.  (In fact, this
congruence is true modulo $p\pi^{k'}$.)  Also 
 $F|\dia p_{_{\scriptstyle N}}(q) \con \eps(p)
f \mod \pi$ so $\sum_{n\not\con 0\hb{\sevenrm mod}\, p}t(n)^{k'}
A_nq^n \con \eps(p)\theta^{k'}f\mod\pi$ .  Putting all this together we obtain the result.\bx

Let $\omega_{X/R}$ denote the sheaf of regular differentials on $X$ over $R$.  In particular, if 
$F$ is a weight two cusp form then $F$ is regular \ff $\omega_F$ is a regular diferential ([G, Prop. 8.4]).
Let  $Z$ denote the correspondence $\sum_{d\in \F_p^*}\dia d_p \in H$. 
Let
$$\displaylines{W^{ord}=\bigcap_mU_p^mH^0(X,\omega_{X/R})^Z\quad\hb{ 
and}\cr
W^{anti-ord}=wW^{ord}=\bigcap_mU_p'^mH^0(X,\omega_{X/R})^Z\hs{5pt}\cr}$$   We call the elements of
$W^{ord}$ ordinary forms.



\proclaim Corollary 4.2.   Suppose $\omega=\omega_{F|U_p}$ where $F$ is a regular form
of weight $2$ and 
$\omega|\dia d _p= t(d)^{-b}
\omega$, $0 < b< p-1$
then  $\omega|_{X_1(pN)-I}\con 0 \mod \pi^{b}$.  






Let $TJ_1(pN)$ denote the $p$-adic Tate module of $J_1(pN)$.  Then, there
exists dual ordinary $p$-divisible groups $G$ and $G'$ over $R$ such that

$$
{ TG=\bigcap_n U_p^n(T(J_1(pN)^Z)\quad{\hbox{and}}\quad  
TG'=\bigcap_n {U_p'^n}(T(J_1(pN)^Z).}$$
Moreover, $w_{\xi}TG=TG'$ if $\xi$ is any primitive $p$-th 
or $pN$-th root of unity.
 In fact, if $\xi$ is a primitive $pN$-th root of unity and
 $\alpha\in H$, $w_{\xi}\circ
\alpha = ros(\alpha)\circ w_{\xi}$ where $ros$ is the Rosati 
involution.  In particular, $H$ acts on $G$ and $H'=ros(H)$ acts on $G'$.

\proclaim Lemma 4.3.  There are natural isomorphisms
$$T\bar G'\otimes_{\Z_p} R\ra 
W^{ord}\ra \Omega_{G}\quad\hb{ and}\quad 
T\bar G\otimes_{\Z_p}R\ra W^{anti-ord}\ra \Omega_{G'}\hs{5pt}$$


\pr  The isomorphism $T\bar G'\otimes_{\Z_p}R \cong \Omega_{G}$
was described in \S1 (see also [K2 \S 3.3]). Let $J\t$ denote a semi-stable model of the
Jacobian of $X_1(pN)$ with the log-structure over $R\t$ coming from the singular divisor.
  Let $\R_p\t$ denote the ring of integers in the completion $\C_p$ of an algebraic 
closure of $K$ with the log-structure extending that on $R$.
By Hodge-Tate theory, we have a natural map $h\colon T(J)\otimes_{\Z_p}\R_p\ra H^0(J, \Om {J\t}{\R_p\t})$ whose 
cokernel is torsion and whose kernel is spanned over $\R_p$ by the elements
 on which $Gal_{cont}(\C_p/K)$ acts via the 
cyclotomic character.
Also, the following diagram commutes
$$\matrix{
TG'&\ra&T(J)\cr
\downarrow&&\downarrow\cr
\Omega_{G}\otimes\R_p&\leftarrow &H^0(J, \Om {J\t}{\R_p\t})&\cr}$$
where $TG'\ra\Omega_{G}$ is the natural 
map which factors through $T\bar G' \ra \Omega_{G}$.
As the kernel of $TG'\otimes\R_p\ra \Omega_{G'}\otimes\R_p$ is 
also spanned by the elements on which galois acts via the cyclotomic 
character, the image of $TG'\otimes\R_p$  in $H^0(J, \Om {J\t}{\R_p\t})$ maps isomorphically
onto $\Omega_{G}\otimes\R_p$.

Now by functoriality
$$h(e(\alpha))=ros(e)h(\alpha),\eqno (4.1)$$
where $\alpha\in T(J)$ and $e$ is an endomorphism of $J$.  It follows the image of
$TG'\otimes\R_p$
 is contained 
in $W^{ord}\otimes \R_p$ with a torsion quotient.  As this 
module is torsion free, 
above assertion about  implies that $W^{ord}\otimes \R_p$ 
is naturally isomorphic to $\Omega_{G}\otimes\R_p$.  Taking $Gal_{cont}(\C_p/K)
$ invariants yields the first set of isomorphisms.  The second set 
follows similarly.  \bx

 

We, henceforth, identify $W^{ord}$ with $\Omega_{G}$  and $W^{anti-ord}$ with 
$\Omega_{G'}$.   If $\alpha$ is in the Tate module of $G$ or $G'$ we let $ \omega_{\alpha}
$ denote its image via the respective map discussed in the lemma. 
 
Moreover, we have the \hb{Serre-\kern-1.8ptTate} pairing
$$q\colon T\bar G\times T\bar G'\ra 1+\pi R$$ described in Section 1. 
This gives us a pairing $$(d\log)\circ q: T\bar G\times T\bar
G'\ra \Omega_{R/\Z_p^{unr}}\ $$  
where $\Z_p^{unr}$ is the completion of the maximal
unramifed extension of $\Z_p$ in $R$ and $d\log\colon a\mapsto da/a$.  We extend $d\log q$  by scalars to obtain a pairing,
$$\bigl(T\bar G(\F)\otimes_{\Z_p} {R}\bigr)\times\bigl( T\bar G'(\F)\otimes_{\Z_p
}{R}\bigr)\ra \Omega_{{R}/\Z_p^{unr}}\ .$$





  

 For an integer $j$, let $\bar G(j)$ denote the subgroup on which $(\Z/p\Z)^*$ acts via $t^j$. 






\proclaim Theorem 4.4.  Suppose $f$ is a cusp form
of weight $2<k\le p$ on $X_1(N)
\mod p$ of nebentypus $\eps$ such that $f|U_p = a_pf$ and $a_p\ne0$. If 
$\alpha \in T\bar G({-k'})\otimes R$, $
\beta \in T\bar G'(k')\otimes R$ and
$\omega_{\alpha}|_{I} = \omega_f$,   then
$$d\log q(\alpha,\beta) =
(\eps(p)/a_p)\bigl(w^*\omega_{\beta}|_I,[f']
\bigr )_{I}k'!\pi^{k'-1}d\pi+\dots$$ 




\pr  
First suppose $N\ge 4$ so that $X$ is semi-stable.  Let $F$ be a weight $2$ cusp form such that  $\omega_\beta=\omega_F$.  Then, $F$ is a lifting of
$f$ and $\omega_{ros(U_p)\beta}=\omega_{F|U_p}$.  It follows from 
Proposition 4.1 that 
$w^*\omega_{ros(U_p)\beta}|_{X-I'}$ equals
$$-\eps(p)\omega_{f'}(k'-1)!\pi^{k'}+\dots $$ 
Now, $ros(U_p)\beta \con a_p\beta \mod \pi T\bar G'(k')\otimes R$ as the map 
from $\bar G'[p]\otimes \F$ to $H^0(I,\Om{I}{F})$ induced from 
$\gamma \mapsto \omega_{\gamma}|_{I}$
 is an injection which commutes with Hecke after it is twisted by the
Rosati involution (see (4.1)). 
Hence the theorem
follows in this case from  Theorem 1.1,  Corollary 4.2 and Theorem 3.1.

Now suppose $N\le 3$.  Then, we may lso suppose $p>3$ as there are no forms of odd
weight on $X_1(1)$ or $X_1(2)$.  Let $l\ne p$ be a prime and let $d$ denote the degree of 
$X_1(lN)$ over $X_1(N)$.  Passing from level $N$ to level $Nl$ multiplies both sides of the formula
by $d$.  Hence, its truth for level $N$ follows from its truth for level $Nl$ as long as $(p,d)=1$.  
Since,
$d=(l+1)(l-1)$ if $N\ge 3$ and $(l+1)(l-1)/2$ if $N$ is 1 or 2 this concludes the proof.  \bx




 








\beginsection 5.  Frobenius. 

The results of this section were originally contained in [C1].  
Now suppose  $I:=I_1(N)$ is the complete Igusa curve over $Y$ the the modular curve $X_1(N)\mod p$.
  Let $\sigma$ 
denote the Frobenius automorphism of $\bar{ \F}_p$.  Since $I$ is defined
over $\F_p$  there is 
a natural action of $Gal(\bar \F_p/\F_p)$ on sections of $\O I$ and of $
\Omega^1_I$.  A reference for the results on Igusa curves used
in this section is [G, \S5]. There is a 
canonical differential with $q$-expansion $dq/q$  which has simple poles at all the cusps 
and zeros of order $p$ at the supersingular points and we will denote it $dq/q$. (Note
that $dq/q=\omega_A$
where $A$ is the Hasse invariant form on $X_1(N)$
in the notation of \S4).  Moreover,
$$\eqalignno{
{dq\over q}\big\vert\dia b_N&={dq\over q};&(5.1)\cr
{dq\over q}\big\vert\dia c_p &= c^{-2}{dq\over q}.&(5.2)\cr}$$

Define an operator $M$ on differentials by the formula
$$M\nu = d\Big({\nu\over {dq/q}}\Big).$$

 Suppose $\omega$ is a holomorphic differential such that
$$\eqalignno{
\omega |\dia d_N &=\epsilon(d)\omega;&(5.3)\cr
\omega|\dia c_p &= c^{-j}\omega;&(5.4)\cr
\sigma\ca C\omega &= a_p\omega,&(5.5)\cr}$$
where $1\le j \le p-1$, $\sigma$ is 
Frobenius and $\ca C$ is the Cartier operator.  Let $SS$ denote the supersingular locus on $I$.

\proclaim Theorem 5.6.  For all 
supersingular points $y$ $ord_yM^j(\omega) \le p$  and
$$\sigma\iv Frob[M^{j}\omega]={a_p\over \epsilon(p)}[M^{j}\omega].$$ 
where $[M^j\omega]$ is the cohomology class in ${\bf H}^1(I,\Omega\per_{I/\bar F_p}(\log SS))$
associated to $M^j\omega$ as in \S 2.  

The first part follows from  [G, Prop. 9.9], identifying $\omega$ with a form of 
weight $p+1-j$, $M$ with $\theta$ and $\sigma\ca C$ with $U_p$.  In 
the case $j=1$, it is obvious.
Also, we will only prove the second part for $j=1$ (weight p) in this paper, the remaining
 cases are dealt with in [C2] using rigid analysis.  

We may suppose $N\ge 4$ as the map in cohomology resulting from changing the
level from $N$ to $Nl$ where $l\ne p$ is a prime is a functorial injection as long as 
$p\,\not |\, (l^2-l)$ and if $p=2$ or $3$, $I_1(N)$ has genus zero if $N< 4$.  


\proclaim Lemma 5.7.  Let $y$ be a supersingular point on $I$ and let $v$ be a local
parameter at $y$. Then
$$Res_y\Big(v^{-(p+1)}{dq\over q}\Big)^{p-1} = {\Big({v\over v^{\sigma^2}| \dia
{-p}_N\iv}\Big)(y)}. $$

\proclaim Note. the differential on the left hand side  has a simple pole at $y$
since $N\ge 4$. This is false for $N<3$ or if $N=3$ and $j(y)=0$.  If one
replaces the exponents $p+1$ and $p-1$ by $ord_y(dq/q)+1$ and 
 $(p^2-1)/(ord_y(dq/q)+1)$ the statement should be still true for these
levels.  
  Also, $\dia {-p}_N$ acts like $\sigma^{-2}$ on the 
supersingular points. 

  We will prove this by first interpreting both sides as values of
modular forms  on $SS$, the supersingular locus on $Y$,
 at the image of $y$ on $Y$.

Let $x$ be a supersingular point on $Y$.  Let $\omega$ be a non-zero section
 of $\underline{ \omega}_x$.  Let $w$ be a 
local parameter at $x$ such that $dw\vert_x$ corresponds to $\omega^{\otimes 2}
$ via  Kodaira-Spencer (this is another manifestation of the Kodaira-Spencer map
different from that used elsewhere in the paper).  Let $v$ denote a parameter at the point $y$
above $x$ on $I$ such that $v^{p-1}\equiv w \ \hbox{mod}\,m_x^2$.  Then set
$$\eqalign{
r(x,\omega)&=Res_y(v^{-(p+1)}{dq\over q})^{p-1}\cr
s(x,\omega)&=
({v\over v^{\sigma^2}| \dia
{-p}_N\iv})(y).\cr}$$

Both $r$ and $s$ are both modular forms of weight $2(p+1)$ on $SS$.  
Of course, there is another well known modular form of weight $2(p+1)$ 
on $SS$, namely $B^2$ (see [S3]).  

\proclaim Proposition 5.8.  Both $r$ and $s$ are equal to $B^2$.

\pr We will first use the second definition of $B$ in [E, \S 7.2]
  which translates using the above notation into: 
let  $w={A/ \omega^{\otimes p-1}}$, which is a parameter at $x$.
Then $B(x,\omega)\omega^{\otimes 2}$ corresponds to $dw$ under via Kodaira-Spencer.  Let
$b=B(x,\omega)$. Now, let $\delta^{p-1} = b$ and $v=\delta\iv{a/ \omega}$.  Then,
$$\eqalign{
Res_y(v^{-(p+1)}{dq\over q})&=\delta^{p+1}Res_y(({\omega\over
a})^{p+1}{dq \over q})\cr
&=\delta^2bRes_y({{\omega}^{p+1}\over a^{p-1}})
\cr
&=\delta^2bRes({\ dw\over bw}))=-\delta^2.
\cr}$$
This takes care of $r$ and, in fact, can be used to give yet another definition of $B$ 
when $p$ is odd.

Now 
$$\eqalign{
s(x,\omega) &= 
\delta^{p^2-1}\Big({a/\omega\over
 (a/\omega)^{\sigma^2}|\dia {-p}_N\iv}\Big)(y)\cr
&= b^{p+1}\Bigl({\omega^{\sigma^2}|\dia {-p}_N\iv\over\omega}\Big)(x).
\cr}$$
If $E$ denotes the canonical model of the supersingular elliptic curve
corresponding to $x$ over $\F_{p^2}$ (i.e. with Frobenius 
endomorphism $-p$) and we think of $\omega$ as global section of $\Omega^1_{E}$,  
then $$\bigl({\omega^{\sigma^2}|\dia {-p}_N\iv\over\omega}\big)(x)
 = \omega^{\varphi}/\omega$$
where $\varphi$ is $\sigma^2$ on $E$.  But this is $b^{1-p}$ by a 
theorem in Serre's course [S3] .  (I.e. $B^{p-1}(
x) = (\omega^{\varphi}/\omega) \omega^{\otimes p^2-1}$).  
\medskip

\ni {\it Proof of  Theorem 5.6.}  

Let $h={\omega\over dq/q}$.  Let $v$ be a local parameter at a supersingular point $y$
 on $I$
such that $v|\dia d_p = d\iv v$. 
We may expand $dq/q$ and $h$ in $v$, using (5.2) and (5.4), to get
$$\eqalign{ 
{dq\over q} &= \sum_{n=1}^{\infty}c_n(v)v^{2+n(p-1)}{dv\over v}\cr
h &= \sum_{n=-1}^{\infty} b_n(v)v^{-1+n(p-1)}
\cr}$$
at $y$. Then $[M\omega]$ is represented by the cocycle $\alpha:=(\omega,f)$ 
where   $f(v) = b_0(v)v\iv$ for any $v$ as above. But since $dh=M\omega
$, this class is also represented by $\beta := (0,g)$ where
$g(v)=-b_{-1}(v)v^{-p}$.
 Finally, if $\phi$ is the Frobenius endomorphism of $I$,
$\sigma\iv Frob(M(\omega))$ is represented by $\phi^*\alpha = (0,\phi^*f)$ and
$\phi^*f(v)=b_0(v^{\sigma})v^{-p}$.

The fact that $\ca C (dq/q)=dq/q$ implies that $c_2(v)=0$. Hence,
$$\omega = (c_1(v)b_{-1}(v)v + c_{1}(v)b_0(v)v^{p}+\hbox{higher terms})dv/v.$$
Since $dq/q$ is defined over $\F_p$, $c_1(v^{\sigma})=c_1(v)^p$.  Hence,  
(5.5) implies  that
$$c_1(v)^pb_0(v^{\sigma})=a_pc_1(v)^{p^2}b_{-1}(v^{\sigma^2}).$$  And 
 (5.3) combined with (5.1) implies that 
$$b_{-1}(v^{\sigma^2})=-\epsilon(p)\iv b_{-1}(v)\Big({v^{\sigma^2}|\dia
{-p}\iv\over v}\Big)^p(y)  .$$  
Hence
$$\eqalign{
b_{0}(v^{\sigma})&=- {a_p\over\epsilon(p)}b_{-1}(v)
\Big(c_1(v)^{p-1}\Big({v^{\sigma^2}|\dia {-p}\iv\over v}\Big)(y)\Big)^p\cr
&= -{a_p\over \epsilon(p)}b_{-1}(v),
\cr}$$
by the above lemma as $c_1(v)=Res_y(v^{-(p+1)}dq/q)
$.  Hence,  $\phi^*\alpha = (a_p/\epsilon(p))\beta$ \pen which completes the 
proof.  \bx



\proclaim Corollary 5.9.  Suppose $a_p\ne 0$.  Then $([M^j\omega],\nu)_{I}=0$  for all $\nu \in
H^0(I,\Om I{\F})$ 
\ff $[M^j\omega]=0$.

\pr   Theorem 5.6 implies $[M^j\omega]$ lies in the unit root subspace of $H^1_{DR}(I/F)$.  
The corollary
follows since this subspace intersects trivially with the subspace 
spanned by the classes of global differentials on $I$.\bx





\beginsection 6.   End of the proof. 

Let $f=\sum_{n=1}^{\infty}a_nq^n$ be a normalized ordinary cuspidal eigenform 
on $X_1( N)$ of weight $2<k\le 
p$ and nebentypus $\eps$ defined over $\F$, the residue field of $R$
.  Let $E=\colon E_f$ denote the field generated by the 
coefficients of $f$.
Let $\rho_f\colon Gal(\bar \Q/\Q)\ra Gl_2(E_f)$ be the 
representation attached to $f$ as in  [G, Proposition 11.1].    

  Let $\chi$ denote 
the cyclotomic character and if $a\in \F$ let $\lambda(a)$ denote the 
character on $Gal(\bar\Q_p/\Q_p)\ra \F^*$ which takes an element whose 
restriction to $K$ is $\sigma$ to the element $a$ of $\F^*$.  
As in  [G, Proposition 12.1], 
$\rho_{f,p}$ 
 in matrix form with respect to some basis is
$$\pmatrix{\chi^{k-1}\cdot\lambda(\eps(p)/a_p)&\ast\cr
0&\lambda(a_p)}\ \ .\eqno (6.1)$$



Let $m=:m_f$ denote the 
maximal ideal of $H$ associated to $f$ as in  [G, Proposition 12.4] and
$m'=Ros(m)$.  
 It follows from  [G, Prop. 12.9 4)] that over $\Q_p(\zeta_p)$ $B=\colon
B_f=\colon G'[m']$ has the structure of an 
$E$-vector space scheme and sits in a \ses of 
$E$-vector space schemes
$$0\ra B^0\ra B\ra B^e\ra 0.\eqno (6.2)$$ 
where $B^0$ is the maximal connected subgroup of $B$ and $B^e$ is the
maximal \'etale quotient group of $B$.
Moreover, the vector space schemes in (6.2) all have canonical flat 
extensions to $\Z_p[\zeta_p]$. The group $Gal(\bar\Q_p/\Q_p)$
acts on the semi-simplification of $B^0(\bar\Q_p)$ which has dimension at
least one by $\lambda(\eps(p)/a_p)\cdot
\chi^{k-1}$ and on $B^e(\bar\Q_p)$ which has dimension one 
by $\lambda(a_p)$.  From now on we will regard 
(6.2) as a sequence over $R$.  As such we get an $E$-bilinear
 pairing as in [G, \S 13]
$$q_f\colon (\td B)^e(\F)\times B^e(\F)\ra {(R^*}/{{R^*}}^p)\otimes_{\F_p} E^{\vee}.$$
Here $\td B$ is the Cartier dual of $B$.  It is canonically 
isomorphic to $G[p]/mG[p]$.  By [BLR, Theorem 1] (See also [BLR, Theorem 2]), 
if $\rho_f$ is irreducible $B(\bar\Q)$
 is a direct sum of copies of $\rho_f$.  
By [E, Thm. 9.2] if the number of copies
is strictly greater than one, i.e. the multiplicity of $\rho_f$ is greater than 
one, $\rho_f$ is unramified.  

\proclaim Lemma 6.1.  The image of $q_f$ lies in the $\chi^{k'}$\kern-0.95pt-eigenspace of  
${(R^*}/{{R^*}}^p)\otimes E^{\vee}$.

\pr If the multiplicity of $\rho_f$ is one this follows as in [G, 13.5 2)].
  If the multiplicity is greater than one this follows from the facts
that $\rho_f$ is unramified and $B(\bar\Q)$ is a direct sum of copies of $\rho_f
$.\bx

Let $tr\V$ denote the linear map from $E\V$ to $\F_p$, $h\mapsto h(1)$.  Let 
$d\log q_f$ denote the pairing  
$$(\td B)^e(\F)\times B^e(\F)\otimes_{\F_p}\F
\otimes_{\F_p}\F
\ra\Omega_{R/\Z_p}$$
obtained from $d\log\otimes tr\V$ by extension of scalars.


\proclaim Proposition 6.2.  \line{Suppose $\rho_f$ is irreducible  and $k>2$. 
Then, the following are equivalent:\hfil}
\halign{#\hfil &#\hfil\cr
(i) &The representation $\rho_f$ is tamely ramified above $p$.\cr  
 (ii)  &The pairing $d\log q_f\mod \pi^{k'}\Omega_{R/\Z_p}$ is trivial.\cr
 (iii) &The pairing $d\log q_f\mod \pi^{k'}\Omega_{R/\Z_p}$ is degenerate.\cr}

First we prove the following:
 
\proclaim Lemma 6.3.  \line{Suppose $2\le k\le p$ and $\rho_f$ is irreducible. \hfil}
\line{Then, the following are equivalent:\hfil}
\halign{#\hfil &#\hfil\cr
(i) &The representation $\rho_f$ is tamely ramified above $p$.\cr  
(ii)  &The restriction of $\rho_f$ to $Gal(\bar\Q_p/K)$ is trivial.\cr
   (iii)  &The action of $Gal(\bar\Q_p/K)$ on $B(\bar\Q_p)$ is trivial.\cr
  (iv)  &The sequence (6.2) splits over $K$.\cr
  (v)  &The sequence (6.2) splits over $R$.\cr
 (vi)  &The pairing $q_f$ is trivial.\cr
 (vii) &The pairing $q_f$ is degenerate.\cr}

\pr By (6.1) the restriction of $\rho_f$ to $Gal(\bar \Q_p/K)$ takes the form 
$$\pmatrix{1&\ast\cr
0&1}\ \ .\eqno (6.4)$$
Thus $\rho_f$ is tamely ramified above $p$ \ff $\ \ast$ is zero.  This proves the 
equivalence of (i) and (ii).  The equivalence of (ii) and (iii) 
follows from the result of [BLR] mentioned above. 
Since finite group schemes over $K$ are 
determined by their associated Galois representation (iii)  implies (iv) .  
The sequence splits over $R$ if and only if $q_f$ is trivial by 
the definition of $q_f$.  The equivalence of (iv)-(vi) follows because
 the sequence splits over $K$ \ff the composition of $q_f$ with the 
natural map from ${R^*}/{{R^*}}^p\otimes E\V$ into $K^*/{K^*}^p\otimes E\V$ 
is trivial and ${R^*}/{{R^*}}^p\ra K^*/{K^*}^p $ is an injection. Obviously (vi)
implies (vii). If, on the other hand,
 $q_f$ is degenerate and the multiplicity is one then it is 
trivial since in this case it is an $E$-linear pairing of two one-dimensional 
$E$-vector spaces so (6.1) splits over $K$ and so because $\rho_f$ has 
multiplicity one $\rho_f$ is tamely ramified.
 If, on the other hand, the multiplicity of $\rho_f$ is not one, as
 we stated above,
$\rho_f$ is unramified above $p$.  
Thus (vii) implies (i)-(vi) and this
completes the proof. \bx



 
The proposition now follows from:

\proclaim  Lemma 6.4.    If $k>2$ then $q_f$ is a non-degenerate 
(resp. trivial) pairing of $E$-vector spaces
if and only if  $d\log q_f\mod \pi^{k'}\Omega_{R/\Zp}$
 is a non-degenerate (resp. trivial) pairing of $\F$-vector spaces.  

\pr First we observe that $q_f$ is non-degenerate (resp. trivial) \ff it is non-degenerate
(resp. trivial)  modulo
$1+\pi^{k'+1}R\otimes E\V$ since the natural map from the $\chi^{k'}$\kern-0.95pt-eigenspace of 
$R^*/{R^*}^p$ to $(1+\pi R)/(1+\pi^{k'+1}R)$ is an isomorphism.  
 Second  the map $d\log$ yields an isomorphism from $(1+\pi R)/(1+\pi^{k'+1} R)$ onto
$\Omega_{R/\Z_p}/\pi^{k'}\Omega_{R/\Z_p}$ since $k'< p-1$.
 This together with the fact that $tr\V$ is a surjective linear map implies the lemma.\bx


The pairing $q$ induces a pairing $ G[p](\F)\times  G'[p](\F)\ra R^*/{R^*}^p$
(note $G[p](\F)$ is naturally isomorphic to $\bar G^e[p](\F)$) and it follows from the
defining properties of $q$ and $q_f$ that

\proclaim Lemma 6.5.  Suppose $\alpha\in G[p]( \F)$  and $\beta\in B(\F)$.
Then 
$$(1\otimes tr\V)q_f(\alpha\mod m, \beta)
=q(\alpha,\beta) \mod {{R^*}}^p.$$ 

\vbox{

\proclaim Lemma 6.6.  Suppose 
$l$ is a prime, $l\ne p$ and $d\in (\Z/pN\Z)^*$.  Then, 
$$[f']|T_l = a_l[f'],\quad
[f']|\dia{d}_N=\eps(d)[f']\quad
{\hbox{and}}\quad
[f']|\dia d_p=d^{2-k}[f']\hs5pt.$$
Moreover, $\sigma\iv Frob[f']=(a_p/\eps(p)) [f']$.

}

\pr  Observe that $\omega_f$
satisfies all the hypotheses of Theorem 5.6 with $j=k'$ and, in this case,
$M^j\omega_{f}=\omega_{f'}$.  The lemma then follows from this theorem
 together with the commutation relations of $\theta$
and the generators of $H$ (see [G, \S4]).\bx



The composition 
$$
T({G'}^e)\ra H^0(X,\Om X{R})\ra H^0(I,\Om I{\F})$$
 induces an isomorphism from
$B^e(\bar\Q_p)\otimes_{\F_p}\F$
 onto the $f$-eigenspace of \pen $H^0(I,\Om I{\F})$ (this map can also be described in terms of 
the Cartier-Serre isomorphism).   The following generalizes [G, Prop. 13.14 4)].  We note however 
that
the  the proof in [G] is incomplete.


\proclaim Proposition 6.7.  Let $\beta_f$ be the element of 
$B^e(\bar\Q_p)\otimes_{\F_p}\F$
which corresponds to $\omega_f
$ via the above map.  Then
a companion form exists \ff $d\log q_f(\alpha,\beta_f) = 0$ for all 
$\alpha\in (\td B)^e(\bar\Q_p)\otimes_{\F_p} \F$.




\pr Let $\xi$ be a primitive {$N$-th} root of unity.  First note that $H^0(I,\Om I{\F})$
breaks into a direct sum of isotypic components corresponding to maximal ideals of
 $w_{\xi}Hw_{\xi}\iv$.  Next, if $ros_I$ is the Rosati involution attached to $I$,
$$ros_I(T_l)=\dia l_p\iv w_{\xi} T_lw_{\xi}\iv$$
if $l\ne p$ and $$ros_{I}(\sigma\iv
Frob) =U_p=\dia p_N\iv w_{\xi} U_pw_{\xi}\iv\hsk5$$
  It follows from Lemma 6.6,  that $[f']$ is orthogonal to
the isotypic components corresponding to maximal ideals other than $w_{\xi} mw_{\xi}\iv$.
On the other hand, the image of 
 the $m$-adic completion of $TG'$ in $H^0(I,\Om I{\F})$ via the map 
$\gamma\mapsto \big((w_{\zeta}\iv)^*\omega_{\gamma}\big)|_{I}$ is the $w_{\xi}mw_{\xi}\iv$-isotypic 
component.
Moreover, it follows from Theorem 4.4 and  Lemma 6.5 that 
for $\gamma\in \bar G[p](\F)$
$$d\log q_f(\gamma\mod m,\beta_f) =
(\eps(p)/a_p)\big((w_{\zeta}^*\omega_{\gamma})|_I,[f']\big)_{I}k'!\pi^{k'-1}d\pi+\dots$$ 
Hence,
$d\log q_f(\alpha,\beta_f) = 0$ for all 
$\alpha\in (\td B)^e(\bar\Q_p)\otimes_{\F_p}\F$ \ff
$(\delta,[f'])_{I} = 0$
for all $\delta\in (w_{\xi} mw_{\xi}\iv)^* H^0(I,\Om I{\F})$\ff
$(\eta,[f'])_{I}=0$
for all $\eta\in H^{0}(I,\Om I{\F})$.
  Thus,  by \hbox{Corollary 5.9},
$d\log q_f(\alpha,\beta_f) = 0$ for all 
$\alpha\in (\td B)^e(\bar\Q_p)\otimes_{\F_p}\F$ \ff $[f']=0$. 
 Finally, by  [G, \hbox{Theorem 13.14 1)}], $[f']=0$ 
\ff $f$ has a companion form.  This completes the proof.\bx





We will now complete the proof of the  Theorem 0.1.
If $\rho_f $ is reducible the result follows from the theory of Eisenstein series as in [G].
Therefore, we will suppose $\rho_f$ is irreducible.
Suppose $\rho_f$ is tamely ramified.  Then, by Proposition 6.2, $q_f$ is 
trivial.  By the previous proposition, 
this means a companion form exists.  
 If, on the other hand, a companion form exists then by the same proposition 
$d\log q_f \mod 1+\pi^{k'}R$ is 
degenerate, as $\omega_f\ne 0$.   Proposition 6.2 then
implies that $\rho_f$ is tamely ramified.\bx

\ni {\sl Proof of  Corollary 0.2. } 

 Let $A$ and $V_p$ be as in [G, \S 4].  Let $W$ be the space of forms of weight $p$ spanned by
$g_1=Af$ and $g_2=f|V_p$.  The elements in $W$ are eigenforms for 
$T_l$, $l\ne p$, with  eigenvalue $a_l$ and they all have 
nebentypus $\eps$.  Now by [G, (4.7)] and the $q$-expansion principle,
$$\eqalign{g_1|U_p&=a_pg_1-\eps(p)g_2\cr
g_2|U_p&=g_1\hs{5pt}\cr}$$
It follows that $U_p$ restricts to an automorphism of $W$.  If $g$ is
 a normalized eigenvector for $U_p$ in $W$, $g$ is an 
eigenform with nebentypus $\eps$ and $f$ is a companion form of $g$.  
As $\rho_f=\rho_g$ the corollary follows immediately from Theorem 0.1.\bx

  

\proclaim Remarks.  
{  The pairing $q$ takes values in $1+\pi^{k'}R$ and 
our proof required knowledge  of the leading term of $q-1$.  This  is
contained in knowledge of $dq/q$ so long as $k'<p-1$.
The most patent 
reason our proof fails for $k=2$ is that  $db/b=0$ when $b\in 1+\pi^{p-1}R$.
 However, in [C2] a formula for this leading term in the spirit of Theorem 4.4 
valid even for $k=2$ will be given.   Unfortunately, at present, it
is only proven for $p>2$.}\vb

\beginsection Errata to [G].

I would like to thank Coleman and Voloch for giving me this opportunity
to correct some errors in [G].\par
pg.462, line 10. $I_1(N)$ should be $I_1(N)^h$.\par
pg.486, line -11. $2 \leq k \leq p$ should be $3 \leq k \leq p$.\par
pg.500.  Proposition 13.14 4). The statement is only correct when $\epsilon
= 1$, and the proof is incomplete. In general, one must replace the 
differential $\nu_f$ by $\nu_{f|w_N}$ to get a non-zero cup product with 
$\nu_{f'}$. A complete proof is given by Coleman and Voloch in Proposition 6.7
of this paper.\par
pg.514, lines 15-16. The statement that ``the local action on $p^n$-torsion
is diagonalizable if and only if $j_E \equiv j_0\ \mod 2p^{n+1}$" requires
the additional hypothesis that $j_0 \not\equiv 0, 1728 \mod p$.\par
Benedict H. Gross






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{\obeylines
R. F. Coleman 
Department of Mathematics
University of California
Berkeley, CA 94705
U. S. A.
\medskip
J. F. Voloch
IMPA
Est. D. Castorina, 110
Rio de Janeiro 22460
Brazil
current address:
Department of Mathematics
University of Texas at Austin
Austin, TX 78712
U. S. A.}







\bye