% ---------------------------------------------------------------- % Slides ********************************************************* % ---------------------------------------------------------------- % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \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} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \begin{slide}{} \end{slide} % ---------------------------------------------------------------- % ---------------------------------------------------------------- \begin{slide}{} \end{slide} % ----------------------------------------------------------------