\documentclass{article}1\title{Lawren Smithline's NTS talk, March 22, 2000}2\include{macros}3\begin{document}4\maketitle56Newton polygon of $U_p$.78{\bf Wan's result.} For $p\geq 5$, for all weights $k\in\Z$,9there is a parabola below the Newton polygon for $U_p$.1011More precise statement about Wan's result.12$$\det(1-t U_p) = \sum_{i=0}^{\infty} a_i t^i.$$131415{\bf New Theorem.} In level $N$, $p\nmid N$, the newton polygon16of $U_p$ is bounded below by a parabola with quadratic term17depending on18$$X(\Gamma_1(N)\intersect \Gamma_0(p))=X(\Gamma).$$19The dimensions $\dim S_k(\Gamma)$ grow essentially linearly in20terms of $k$, say roughly $s_\Gamma\cdot k$.2122\renewcommand{\M}{\mathcal{M}}23Fix $p\geq 5$.24What is our space of modular forms?25Overconvergent $p$-adic modular forms of weight $k$, level $N$, $p\nmid N$,26coefficients in~$R$, growth condition~$r$:27$$\M_k(N,R)$$2829Here's how to build the above spaces from the classical spaces30$M_k(N,R)$. My~$R$ is $\O_K$, where $[K:\Q_p]<\infty$.31Katz observed that32$$M_{k+p-1} = E_{p-1}\cdot M_k + W_{k+p-1}(N,B).$$33Then34$$\M_k =\left\{ b_0 + \sum_{j=1}^{\infty} r^j \frac{b_j}{E_{p-1}^j} \right\}$$35where $b_0 \in M_k(N,R)$ and $b_j\in W_{k+(p-1)j}$.36When $v_p(-r)>0$, these are overconvergent modular forms.3738{\bf Action of $U_p$.}\\39$U_p$ acts on $q$-expansions by40$$U(\sum b_n q^n) = \sum b_{np}q^n.$$41It turns out that42$$U\in \End(\M_k(N,R,p^{\frac{1}{p+1}})\tensor K).$$43Let44$$P_k(t)=\det(1-tU) = \sum_{i=0}^{\infty} a_i t^i45= \sum \tr \Lambda^i U_p\cdot t^i.$$4647{\bf Theorem (Wan).} The Newton polygon of $U_p$ (i.e., $P_k(t)$) is48bounded below by a parabola, independent of $k$, dependent on49$N$ and $p$. \\50{\bf Coleman's Twist:}51$$U_{((k)+(p-1)j)} \sim U_{(k)}\left(\frac{E_{p-1}(q)}{E_{p-1}(q^p)}\right)^j$$52Note53$$G = \frac{E_{p-1}(q)}{E_{p-1}(q^p)} \in54\M_0(N,R,p^{\frac{1}{p+1}}),$$55so multiplying by $G$ doesn't change the weight.56Here we abuse notation: the $p$ is ``understood.''57``Similiar'' is in the sense that there is an isomorphism of $K$-vector58spaces so that the appropriate diagram commutes.59``Abstractly, the spaces of various weights are all isomorphic.''60Using this we make $k+(p-1)j$, in range $0\ldots p-1$.6162Let $b_{i,s}$ be a basis for $W_{k+j(p-1})(N,R)$.63Let64$$e_{i,s} = \frac{r^i}{E_{p-1}} b_{i,s}.$$65Then66$$U\cdot G^j(e_{i,s}) =67\frac{1}{p}68\sum (\frac{r^p}{E_{p-1}})^u b_u(i,s,j)$$69for $b_u \in W_u(N,R)$ from classical weight $k+(p-1)u$.70Here little ``$u$'' is an index; it should be $i,j,\ldots$, but he71ran out.72$$U\cdot G^j(e_{i,s}) \frac{1}{p}73\sum r^{p-1}(\frac{r^p}{E_{p-1}})^a b_u(i,s,j)$$74$$U\cdot G^j(e_{i,s}) \frac{1}{p}75= \sum A^{a,v}_{i,s} e_{u,v}$$76$$v_p(A_{i,s}^{u,v})\geq u(p-1) v_p(r) - 1$$77$$v_p(A_{i,s}^{u,v})\geq \frac{u(p-1)}{p+1} - 1$$78Now let $d_{\ell} = \dim M_{a(p-1)\ell}(N,B)$. Then79$$v_p(a_n(k+j(p-1)))80\geq \frac{p-1}{p+1}81\sum_{0}^{\ell}82u m_u + (\ell? -1)(n-d_\ell)-n$$83for $d_{\ell} \leq n \leq d_{\ell+1}$,84where $m_\ell = d_\ell - d_{\ell - 1}$.8586{\bf Ogus attacks!!}: ``Draw the diagram which describes the87Coleman Twist!''8889Lawren responds:90$$U\cdot f(q^p)=f(q)\qquad\text{``Frobenius linear''}$$91$$U_p \sim U_p \frac{E_{p-1}}{V(E_{p-1})}$$92$$(E_p - 1)^{-1} U_p E_{p-1}$$9394{\bf $p=2,3$}95Katz expansion in powers of $E_4$.9697``The message of this talk is that $p=2$ and $p=3$98are not so special as they think they are.''99100E.g., $N=1$, $p\con 1\pmod{12}$.101In this case we have102$$\dim M_k(1,R) =103\begin{cases}104\text{floor of } k/12\\105\text{floor of }(k/12) + 1 & \text{ if $k\con 2 \pmod{12}$}106\end{cases}$$107$m_j = \frac{p-1}{12}$ so108``parabola with quadratic term $6/(p+1)$.109110{\bf Ogus attacks again!!} The important part must be to go111from overconvergent to classical forms! How do you do that?112Lawren responds: ``I'm not going to tell you.''113What is the conjecture!?114115The conjecture of GM says: ``close weights imply close slopes''116117\end{document}118