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\begin{slide}{}

\begin{itemize}

  \item The theory of $p$-adic modular forms initiated by Serre,

  Dwork, and Katz "lives" on the complement (in the in the $p$-adic

  completion of the appropriate modular curve)of the zero locus of

  the Eisenstein form $E_{p-1}$.



  \item The most interesting phenomena in the theory of

  differential modular forms initiated by Buium take place

  on the complement of the zero locus of a fundamental differential

  modular form called $f_{jet}$.



\end{itemize}



\end{slide}



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\begin{slide}{}

QUESTION:  Are the cases where both theories are applicable the same?



%This question is in fact

%An interesting question then arises as

%to the relation between the two zero loci. In particular it is

%important to establish that the zero locus of $E_{p-1}$ is not contained

%in the zero locus of $f_{jet}$; this would imply that Buium's theory

%covers situations not covered by the theory of $p$-adic modular forms.



\end{slide}



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\begin{slide}{}

\begin{thm}

Let $p$ be a prime greater than $3$.  If $p \equiv 2$ modulo $3$

or if $p \equiv 3 $ modulo $4$, then the zero locus of $E_{p-1}$

is not contained in the zero locus of $f_{jet}$

\end{thm}

\end{slide}



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\begin{slide}{\underline {An Arithmetic Analog of a Derivation}}



Assume that $p>3$ is a prime number.



\begin{defn}

If $A$ is a ring and $B$ is an $A$-algebra, then

$p$-derivation is a set theoretic map $\delta :A

\to B $ such that

\begin{align*}

\delta (x+y) &= \delta x + \delta y + C_p(x,y) \\

\delta (xy) &= y^p \delta x + x^p \delta y + p \delta x \delta y

\end{align*}

for all $x,y \in A$ where $C_p (X,Y) = {\frac {X^p + Y^p - (X+Y)^p}

{p}}$.

\end{defn}



This definition is analogous to the well known definition:

\begin{defn}

If $A$ is a ring and $B$ is an $A$-algebra, then a derivation is a

map $d:A \to B$ such that

\begin{align*}

d(x+y) &= dx + dy \\

d(xy)  &= ydx + xdy.

\end{align*}

\end{defn}



\end{slide}









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\begin{slide}{\underline {Example of a $p$-derivation}}



{\leftskip .5truein  {

Let

\begin{align} \nonumber

M^0 &= \Z _p [a_4, a_6, \Delta ^{-1} ]\comp \\

\nonumber

M^1 &= \Z _p [a_4, a_6, \delta a_4 , \delta a_6 , \Delta ^{-1} ]\comp

\end{align}





where $\Delta = - 2^4 (4a_4^3 + 27 a_6^2 )$ and the $\delta a_i $ are new variables.

Then $\delta :M^0 \to M^1 $ is the unique $p$-derivation such that



\begin{align} \nonumber

\delta (a_4) &= \delta a_4 \\

\nonumber

\delta (a_6) &= \delta a_6 \\

\nonumber

\delta ( \alpha) &= \frac {\alpha  - \alpha ^p }{p} {\rm \  for \ } \alpha

\in \Z _p.

\end{align}



}}



%\begin{defn}

%A modular $\delta $ function is an element in $M^1$.

%\end{defn}



\end{slide}

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\begin{slide}{}



Let $R$ have the following properties

\begin{itemize} %\itemsep -.1truein \partopsep -5pt

  \item Complete, discrete valuation ring



  \item Maximal ideal generated by $p$



  \item Algebraically closed residue field $k$



\end{itemize}





\end{slide}



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\begin{slide}{}



Let

\begin{itemize}

  \item $\phi $ be the unique lifting of the Frobenius on $k$.



  \item $\displaystyle \delta (x) = \frac {\phi(x) - x^p}{p}$ be

  a (unique) $p$-derivation on $R$.



  \item $M(R) = \{ (a,b) \in R^2 | 4 a^3 + 27 b^2 \in R^* \}.$

\end{itemize}



\bigskip

\bigskip



\begin{rem}

Recall that to any $(a,b)\in M(R)$ we can associate the elliptic

curve $y^2 = x^3 + ax +b$

\end{rem}





\end{slide}



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\begin{slide}{\underline {Differential Modular Functions}}



\begin{defn}

A $\delta $ modular function of order $\le n$ is a

function $f:M(R) \to R $ such that

$$f(a,b) = F (a, b, \delta a, \delta b, ... , \delta ^n a, \delta ^n b,

(4a^3 + 27 b^2)^{-1} )$$

where $F$ is a restricted power series in $2n +3 $ variables with $\Z _p$

coefficients.

\end{defn}



\begin{rem}

In the case of order $1$, any element $f\in M^1 = \Z _p [a_4, a_6,

\delta a_4 , \delta a_6 , \Delta ^{-1} ]\comp $ defines a $\delta $

modular function (still denoted by) $f: M(R) \to R $ by

substituting $a, b, \delta a , \delta b$ in for $a_4, a_6, \delta

a_4, \delta a_6 $.

\end{rem}



\end{slide}



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\begin{slide}{}



\begin{defn}

A $\delta $ character of order $\le 1$ is a group homomorphism

$\chi: R^* \to R^*$ of the form $\chi = \chi _{m,n}$ where

$$\chi_{m,n} (\lambda) = \lambda ^m \left ({\frac {\phi (\lambda )

} {\lambda ^p }}\right )^n .$$

\end{defn}



\begin{rem}

We can define $\delta $ characters of any order in a similar way.

\end{rem}





\end{slide}



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\begin{slide}{\underline {Differential Modular Forms }}



\begin{defn}

A modular $\delta $ function of order 1 has weight $\chi $ if for any $\lambda \in

R^* $

$$f(\lambda ^4 a, \lambda ^6 b) = \chi (\lambda ) f(a,b)$$

for all $(a,b) \in M(R)$. A $\delta $ modular form of order 1 is a

modular $\delta $ function of order 1 with a weight.

\end{defn}





\end{slide}



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\begin{slide}{\underline {Isogeny Covariance}}



\begin{defn}

A $\delta $ modular form is isogeny-covariant if for any two pairs

$(a,b)$ and $(\tilde a ,\tilde b )$ such that there is an etale

isogeny of degree $N$ between the corresponding elliptic curves

that pulls back $\frac {dx}{y}$ to $\frac {dx}{y}$

$$f(a,b) = N^{-k/2} f(\tilde a, \tilde b )$$

where $k= m + n(1-p) $ if $\chi = \chi _{m,n}$. For

$\chi = \chi _{-p-1,-1}$ the constant is $k = -2$.

\end{defn}



\end{slide}



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\begin{slide}{}



\begin{thm}[Buium]

For all $\delta $-characters, $\chi$, of order $\le 1$ there is up

to multiplication by a constant in $\Z _p$ a unique isogeny

covariant $\delta $ modular form of weight $\chi $.

\end{thm}



From now on 'unique' will mean 'up to multiplication of a non-zero

constant in $\Z _p$'.



The 'reduction modulo $p$' of a $\delta $ modular form $f \in M^1 =

\Z _p [a_4, a_6, \delta a_4, \delta a_6, \Delta ^{-1}]\comp $ is the

image of $f$ in $M_0^1 = \F _p [a_4, a_6, \delta a_4, \delta a_6,

\Delta ^{-1}]$.



\end{slide}



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\begin{slide}{}



The fundamental differential modular form

$\fjetnobar $ is the unique isogeny covariant differential modular

form of weight $\chi _{-p-1,-1}$ and $\fjet $ is its

reduction modulo $p$.



\end{slide}



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\begin{slide}{\underline {Formula for $\gamma _{a,b}$}}







{\leftskip .5 truein

Let $a=3m + n$ for $m$ and $n$ integers with $n \in \{ 0,1,2\}$.

Then if $b$ is even, $\gamma _{a,b} = 0$.  If $b$ is odd,

\begin{align} \nonumber

\gamma _{a,b} & =  \\

\nonumber

&\sum _{k=0}^\infty {{\alpha _1}\choose {\alpha _2}}

{{\alpha _3}\choose {\alpha _4}} (-1)^{\alpha _1 - \alpha _4 }(a_4)^{\alpha _2}

(a_6)^{\alpha _3 -{\alpha _4}}

\end{align}

where $\alpha _1 = m+k$, $\alpha _2 = 3k +2 -n$, $\alpha _3 =

m-2k-2+n$, and $\alpha _4 = {\frac {b-1} {2}} $.



}



\begin{rem}

The $\gamma _{2p,p}$ is the Hasse invariant and Deligne's

congruence tells us the Hasse invariant is the reduction modulo $p$

of $E_{p-1}$, the normalized Eisenstein form of weight $p-1$.

\end{rem}



\end{slide}



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\begin{slide}{\underline {Formula for $\fjet $}}

\begin{thm}[H]

The class $\fjet $ is

$$\fjet ={\frac {-E_{p-1} 9 a_6^p \delta a_4 +E_{p-1} 6 a_4^p \delta a_6 }

{2^{-4}\Delta ^p}} + f_0

$$

where $f_0 \in \F _p [a_4 , a_6 , \Delta^{-1}]$ is given by

{\small

$$ \displaylines {

f_0 = \sum _{k=1}^{p-1}  {\frac {(-1)^k} {k2^{-4}\Delta ^p}} \,

\left [ 6a_4^{p+k} a_6^{p-k}\gamma _{2p+k,p} - 9 a_4^k a_6^{2p-k}

\gamma_{p+k,p} \right ]

\cr  \hskip .75truein +

\sum _{k=1}^{p-1} \sum _{i=0}^{p-k} {\binom {p-k}{i}}

{\frac {(-1)^k}{k2^{-4}\Delta ^p}}  \,   a_4^i a_6^{p-k-i}

\hfill \cr \hfill

\left  [ 4a_4^{2p} \gamma _{3k+i,p} + 6 a_4^p\gamma _{2p+3k+i,p}

- 9a_6^p \gamma_{p+3k+i,p} \right ]

\cr +

\sum _{k=1}^{p-1} \sum _{i=0}^{p-k} {\binom {p-k}{i}}

{\frac {(-1)^k} {k}} \,

a_4^i a_6^{p-k-i} \gamma _{p+3k+i,3p} \cr }

$$ }

and $E_{p-1} = \gamma _{2p,p}$ is the Hasse invariant.

\end{thm}



\end{slide}



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\begin{slide}{}



\begin{lem}

If $a_4 = 0$, then

\smallskip

$$

\gamma _{a,b} = \begin{cases}

0 \hfill {\rm \ \ if \ } n=0\\

\hfill \\

0 \hfill {\rm \ \ if \ } n=1\\

\hfill \\

{\binom {m}{\frac {b-1}{2}}} (-a_6)^{m-{\frac {b-1}{2}}}

\hfill {\rm \ \ if \ } n=2\\

%{\binom {m} {2 -n}}

%{\binom {m-2+n}{\frac {b-1}{2}}} (-1)^{m-{\frac {b-1}{2}}}(a_6)^{m-2+n-{\frac {b-1}{2}}}

\end{cases}

$$

\end{lem}



\end{slide}



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\begin{slide}{}



\begin{lem}

If $a_6 = 0$ and $m+n - \frac {b-1}{2} - 2 \ge 0$, then

$$

\gamma _{a,b} = \begin{cases}

0 \hfill \\

\hfill {\rm \ if \ } a {\rm

\ is \ odd \ and \ } b \equiv 1 {\rm \ modulo \ } 4\\

\hfill \\

{\binom {\frac {2a -3 -b}{4}}{\frac {2a -1 -3b}{4}}} (-a_4)^{\frac {2a -1 -3b}{4}}

\hfill \\

\hfill {\rm \ if \ } a {\rm

\ is \ odd \ and \ } b \equiv 3 {\rm \ modulo \ } 4\\

\hfill \\

{\binom {\frac {2a -3 -b}{4}}{\frac {2a -1 -3b}{4}}} (-a_4)^{\frac {2a -1 -3b}{4}}

\hfill \\

\hfill {\rm  \ \  if \ } a {\rm

\ is \ even \ and \ } b \equiv 1 {\rm \ modulo \ } 4\\

\hfill \\

0 \hfill \\

\hfill {\rm  \ \  if \ } a {\rm

\ is \ even \ and \ } b \equiv 3 {\rm \ modulo \ } 4\\

\end{cases}

$$

\end{lem}



\end{slide}



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\begin{slide}{}



\begin{rem}

If $a_4 = 0$, then the Hasse invariant $E_{p-1} = \gamma _{2p,p} =

0$ if and only if $p\equiv 2$ modulo $3$. If $a_6 = 0$, then

$E_{p-1} = \gamma _{2p,p} = 0$ if and only if $p \equiv 3$ modulo

$4$.

\end{rem}



\end{slide}



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\begin{slide}{}



\begin{thm}[H]

Suppose $p \equiv 2$ modulo $3$ and let $q$ be an integer such that $p = 3q +2$.

Then if $a_4 = 0$,

$$\fjet  = {\frac {(-1)^{\frac{p-1}{2}}}{ 2^{-4}\Delta ^p}}9 a_6^{\frac {11q + 7}{2}}

\sum _{k=1}^{p-1}

{\frac {1}{k }}  \,

{\binom {q+ k}{\frac {p-1}{2}}}

 $$

\end{thm}



\end{slide}



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\begin{slide}{}



\begin{thm}[H]

Suppose $p \equiv 3$ modulo $4$.

Then if $a_6 = 0$, the Hasse invariant is 0 and

$$\fjet  = {\frac {(-1)^{\frac {p+1}{4}}}{2^{-4}\Delta ^p}} (a_4)^{\frac {11p-1}{4}}

\left ( -2 \sum _{k={\frac {p+1}{4}}}^{p-1} \frac{1}{k}

{\binom {k + \frac {p-3}{4}}{\frac{p-1}{2}}}

\right )$$

\end{thm}



\end{slide}



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