\def\bd{\bff d}

\sect{Gouv\^ea-Mazur (type) Conjectures.}

Let $N_{>0}\in \Z$, $p$ be a  prime  such 
that $(N,p)=1$ and $v=:v_p$ the valuation on $\C_p$ such that
$v(\alp)=1$ and $|r|=p^{-v(r)}$.
  Let $\alp_{\ge0}\in\Q$.  Then a normalized 
eigenform $F$,
$$F(q)=\sum_{n\ge 0}a_nq^n$$
on $X_0(Np)$ over $\C_p$ is said to have slope $\alp$ if $v(a_p)=\alp$.  For an integer $k$, let $\bff d (k,\alp)$ be the number of distinct such eigenforms of weight $k$.  Suppose $p$ is odd for now.\sk
\def\bd{\bff d}

\proclaim Conjecture 1.  If $k,k'>\alp+1$ and $k\con k' \mod p^\alp(p-1)$ then
$$\bd(	k,\alp)=\bd(k',\alp).$$\vfil

Assume this for now and keep the notation.\sk

\proclaim Conjecture 2.  Suppose that $\bd(k,\alp)=1$.  Let $F_k$ and $F_{k'}$ be the normalized forms predicted by Conjecture 1.  Then
$$F_k(q)\con F_{k'}(q)\mod p(k-k')	$$\sk


\proclaim Conjecture 3.  Suppose $\bd(k,\alp)=d$ and for $j>\alp+1$,
$j\con k\mod p^\alp(p-1)$
$F_{j,1},\dots, F_{j,d}$ are the $d$ forms of slope $\alp$ and weight
$j$. 
$$\sum_{i=1}^dF_{j,i}(q)\con \sum_{i=1}^dF_{k,i}(q)\ \mod p(j-k).$$\sk

\proclaim Conjecture 4.  There is a geometric/ring theoretic explanation
of all this.

Hida theory establishes all 4 conjectures when $\alp =0$.\sk


\sect{Conjecture 1 and the Eigencurve}

 $p$ may be 2 now. Let $\cW$ be the space of
continuous characters
 $\Z_p^*\ra\C_p^*$.  If $k\in\Z$ define
$\chi_k\in \cW$ by
$$\chi_k\colon a\to a^k.$$
If $\chi,\psi\in\cW$,
set $$d_\cW(\chi,\psi)=\sup_{a\in\Z_p^*}|\chi(a)-\psi(a)|.$$\sk
There is a rigid analytic curve $X=:X_N$
 whose points correspond to
overconvergent eigenforms of tame level $N$ and finite slope.  In particular, for each
$x\in X$ there is a $q$-expansion
$$F_x(q)=\sum_{n\ge0}\cA_n(x)q^n.$$
In fact the 
$\cA_n$'s are rigid analytic functions on $X$.  Set
$\alp(x)=v(\cA_p(x))$. \sk If $x,y\in X(\C_p)$,
set
$$d_X(x,y)=\sup_{n\ge 0}|\cA_n(x)-\cA_n(y)|.$$
There is also a natural map $\kappa\colon X\ra \cW$.
  Moreover,
$$d_{\cW}(\kappa(x),\kappa(y))\le d_{X}(x,y).$$\sk

For $\bet_{\ge0}\in\Q$ and $\chi\in \cW$, let
$$B_{\bet}(\chi)=\stt{\psi\in\cW\colon
d(\psi,\chi)< |p|^{\bet}},$$
$$X_{\alp}(\chi)=\stt{x\in X\colon \kappa(x)\in B_\alp(\chi), \alp(x)=\alp}.$$

Then $X_{\alp}(\chi)$ is  a rigid subspace of $X$
and Conjecture 1 follows from

\proclaim Conjecture 5. $X_{\alp}(\chi)$ is finite and flat over
$B_\alp(\chi)$.\sk

\beginsection Lipshitz functions


Suppose $(Y,d)$ is a rigid space over $K$ with a metric on
$Y(\C_p)$.  A {\bf Lipshitz
function} on $Y$ with respect to $d$, is a rigid analytic 
function $f$ such that for $x,y\in Y(\C_p)$
$$|f(x)|\le 1\ \hba\ |f(x)-f(y)|\le d(x,y).$$
Let $\Lambda_K(Y,d)$ (or $\Lambda(Y)$)
denote the ring of
Lipshitz functions on $Y$ over $K$ wrt $d$.\sk

\exs  (i) $\Lambda(\cW)=\Z_p[[\Z_p^*]]$\vskip.5cm

\ni Suppose $r,R\in \Z_p$ $0<|r|\le|R|\le 1$.  Then $$\Lambda (A(|r|,|R|))=\Z_p\oplus r\Z_p{[[r/T]]}\oplus R\Z_p[[T/R]].$$


All previous conjectures follow from Conjecture 5 and,

\proclaim Suspicion 6. If $Y\subset X$ is an affinoid
finite and flat over its image $Z$ in $\cW$, is $\Lambda(Y)$ 
integral over $\Lambda(Z)$.\sk

 When there is only one normalized
eigenform on  $X_1(N\bq)$ ($\bq=4$ for $p=2$ and $p$ otherwise) of weight $k$, character
$\eps$ for the action of $(\Z_p/\bq\Z_p)^*$ and
slope $\alp$, then  
Suspicion 6 implies
$\kappa\colon (X_\alp(\eps\chi_k),d_X)\ra (B_\alp(\eps\chi_k),d_\cW)$
is an isometry.\sk

\beginsection Evidence

When $\alp=0$ these conjectures and suspicions follow from Hida theory.

   Suppose $p=2$ and $N=1$. In this case Conjecture $5$ was proven
by Emerton when $d_\cW(\chi,\chi_{14})\le 1/8$
and $\alp=\min\stt{6,-\log_2 d(\chi,\chi_{14})}$.\sk
Moreover, 
Emerton also proves if
$$Y_i=\stt{x\in X:
\cases{ 2^{-\alp(x)}=d(\kappa(x),\chi_{14})>1/64,& $i=1$\cr
\alp(x)=6, d(\kappa(x),\chi_{14})\le 14& $i=2$}},$$
the map from $Y_i$ to its image in $\cW$ is finite and flat
of degree $i$ and one verify Suspicion 6 affirmatively for affinoids
contained in
$Y_i$  (all the
ones I've tried so far anyway).\sk

On $X_0(3)$ for $k_n=10+18n$, $0\le n\le 10$, Stein has shown there are exactly two normalized eigenforms of weight $k_n$
 of slope $4$ for $p=3$ and if $A_{k_n}$ denotes the sum of their $U_3$ eigenvalues,
$$v_3(A_{10}-A_k) = 1+v_3(k_n-10).$$\sk

But, Suspicion 6 can't be true in general.  On $X_0(6)$, for
$k_n=8+8n$, $0\le n\le 8$, Stein has shown that there
are three forms of weight $k_n$ of slope $3$ for $p=2$
and if $A_k$ denotes the sum of their $U_2$ eigenvalues
$$v_2(A_8-A_{k_n})=1+v_2(k_n-8).$$\vfil
                                    
This is a problem?  \sk

\end

\sect{ Infinite Slope}

There are eigenforms with $a_p=0$.  We know that if $F$ is a
classical eigenform of
weight $k$, tame level $N$ and finite slope, there exists a sequence
of classical eigenforms $F_n$     of tame level $N$ and weight $k_n$ such that $k_n\ra\infty$ and \pad ally
$$F_n(q)\ra F(q).$$

\proclaim Question.  What happens if $F$ has infinite slope?\sk

Suppose $\omega\in \cW$ is the Teichm\"uller character.  Let $F^{\omega}$ the form with \qe 
$$\sums{n\ge 1}_{p\not|n}a_n\omega(n)q^n.$$
Then $F^{\omega}$ is a classical  eigenform of infinite slope.

In response to a question of Jochnowitz, we've proven,

\proclaim Theorem.  If $F$ has finite slope there exists classical eigenforms $F_n$ of finite slope on $X_1(N)$ such that \pad ally,
$$F_n(q)\ra F^\omega(q).$$

But how big must the weights be to get a given closeness?\sk

Let $$\Delta(q)=q\prod(1-q^n)^{24}.$$
 Stein found  newforms of weights $12+2\cdot 3^{n-1}$ and on
$\Gamma_0(3)$
congruent to $\Delta^{\omega_3}(q)$ modulo $3^n$
for $n=1, 2,3,4$.

What about non-twist  infinite slope forms?\sk
  

The form
$$    f = q + 6q^2 + 4q^4 - 6q^5 - 40q^7+\cdots$$ is a non-CM, non-twist eigenform  in $S_6(\Gamma_0(9))$.
Stein found for $n=1,2,3,4$ there is a newform in $S_{6+2\cdot3^{n-1}}(\Gamma_0(3))$ that
  is congruent to $f$ modulo $3^n$.\sk

On the other hand\dots

   Let
   $$ f = q - 2q^4 - q^7 + 5q^{13}  + \cdots$$
be the weight $2$ normalized form on $X_0(27)$. Then if $g$ is a normalized
eigenform of weight $2+2\cdot3^{n-1}$, $n\le 4$, on $\Gamma_0(3)$
               $v_3(g(q)-f(q)) \le 1$.\sk


\beginsection Fixed weight

Suppose you fix the weight and let the slope tend to $\infty$.
This only makes sense when one talks about \pad\ modular forms.

Lawren  Smithline observed:
 ``Let $R_n$ be the number of congruence classes mod $3^n$ after
3-deprivation [of overconvergent $3$-adic weight 0 eigenforms of tame level 1.]  (One can also avoid the 3-deprivation issue by throwing
out a few forms of lowest slope.)"  From, computations involving the first $33$ such forms he observed,

``The sequence ${R_1, R_2, ...}$ [seems to] begin $1, 2, 2, 4, 8.$"
and conjectured, $$R_n = R_{n-2} + 2.3^{n-4}.$$  What this suggests 
is that there are an uncountable number of overconvergent modular
eigenforms of weight 0 of tame level 1.  

\beginsection Other Questions\dots