Sharedwww / tables / Notes / smithline.texOpen in CoCalc
Author: William A. Stein
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\documentclass{article}
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\title{Lawren Smithline's NTS talk, March 22, 2000}
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\include{macros}
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\begin{document}
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\maketitle
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Newton polygon of $U_p$.
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{\bf Wan's result.} For $p\geq 5$, for all weights $k\in\Z$,
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there is a parabola below the Newton polygon for $U_p$.
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More precise statement about Wan's result.
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$$\det(1-t U_p) = \sum_{i=0}^{\infty} a_i t^i.$$
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{\bf New Theorem.} In level $N$, $p\nmid N$, the newton polygon
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of $U_p$ is bounded below by a parabola with quadratic term
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depending on
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$$X(\Gamma_1(N)\intersect \Gamma_0(p))=X(\Gamma).$$
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The dimensions $\dim S_k(\Gamma)$ grow essentially linearly in
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terms of $k$, say roughly $s_\Gamma\cdot k$.
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\renewcommand{\M}{\mathcal{M}}
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Fix $p\geq 5$.
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What is our space of modular forms?
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Overconvergent $p$-adic modular forms of weight $k$, level $N$, $p\nmid N$,
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coefficients in~$R$, growth condition~$r$:
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$$\M_k(N,R)$$
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Here's how to build the above spaces from the classical spaces
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$M_k(N,R)$. My~$R$ is $\O_K$, where $[K:\Q_p]<\infty$.
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Katz observed that
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$$M_{k+p-1} = E_{p-1}\cdot M_k + W_{k+p-1}(N,B).$$
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Then
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$$\M_k =\left\{ b_0 + \sum_{j=1}^{\infty} r^j \frac{b_j}{E_{p-1}^j} \right\}$$
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where $b_0 \in M_k(N,R)$ and $b_j\in W_{k+(p-1)j}$.
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When $v_p(-r)>0$, these are overconvergent modular forms.
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{\bf Action of $U_p$.}\\
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$U_p$ acts on $q$-expansions by
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$$U(\sum b_n q^n) = \sum b_{np}q^n.$$
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It turns out that
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$$U\in \End(\M_k(N,R,p^{\frac{1}{p+1}})\tensor K).$$
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Let
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$$P_k(t)=\det(1-tU) = \sum_{i=0}^{\infty} a_i t^i
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= \sum \tr \Lambda^i U_p\cdot t^i.$$
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{\bf Theorem (Wan).} The Newton polygon of $U_p$ (i.e., $P_k(t)$) is
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bounded below by a parabola, independent of $k$, dependent on
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$N$ and $p$. \\
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{\bf Coleman's Twist:}
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$$U_{((k)+(p-1)j)} \sim U_{(k)}\left(\frac{E_{p-1}(q)}{E_{p-1}(q^p)}\right)^j$$
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Note
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$$G = \frac{E_{p-1}(q)}{E_{p-1}(q^p)} \in
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\M_0(N,R,p^{\frac{1}{p+1}}),$$
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so multiplying by $G$ doesn't change the weight.
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Here we abuse notation: the $p$ is ``understood.''
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``Similiar'' is in the sense that there is an isomorphism of $K$-vector
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spaces so that the appropriate diagram commutes.
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``Abstractly, the spaces of various weights are all isomorphic.''
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Using this we make $k+(p-1)j$, in range $0\ldots p-1$.
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Let $b_{i,s}$ be a basis for $W_{k+j(p-1})(N,R)$.
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Let
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$$e_{i,s} = \frac{r^i}{E_{p-1}} b_{i,s}.$$
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Then
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$$U\cdot G^j(e_{i,s}) =
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\frac{1}{p}
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\sum (\frac{r^p}{E_{p-1}})^u b_u(i,s,j)$$
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for $b_u \in W_u(N,R)$ from classical weight $k+(p-1)u$.
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Here little ``$u$'' is an index; it should be $i,j,\ldots$, but he
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ran out.
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$$U\cdot G^j(e_{i,s}) \frac{1}{p}
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\sum r^{p-1}(\frac{r^p}{E_{p-1}})^a b_u(i,s,j)$$
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$$U\cdot G^j(e_{i,s}) \frac{1}{p}
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= \sum A^{a,v}_{i,s} e_{u,v}$$
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$$v_p(A_{i,s}^{u,v})\geq u(p-1) v_p(r) - 1$$
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$$v_p(A_{i,s}^{u,v})\geq \frac{u(p-1)}{p+1} - 1$$
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Now let $d_{\ell} = \dim M_{a(p-1)\ell}(N,B)$. Then
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$$v_p(a_n(k+j(p-1)))
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\geq \frac{p-1}{p+1}
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\sum_{0}^{\ell}
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u m_u + (\ell? -1)(n-d_\ell)-n$$
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for $d_{\ell} \leq n \leq d_{\ell+1}$,
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where $m_\ell = d_\ell - d_{\ell - 1}$.
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{\bf Ogus attacks!!}: ``Draw the diagram which describes the
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Coleman Twist!''
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Lawren responds:
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$$U\cdot f(q^p)=f(q)\qquad\text{``Frobenius linear''}$$
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$$U_p \sim U_p \frac{E_{p-1}}{V(E_{p-1})}$$
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$$(E_p - 1)^{-1} U_p E_{p-1}$$
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{\bf $p=2,3$}
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Katz expansion in powers of $E_4$.
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``The message of this talk is that $p=2$ and $p=3$
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are not so special as they think they are.''
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E.g., $N=1$, $p\con 1\pmod{12}$.
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In this case we have
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$$\dim M_k(1,R) =
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\begin{cases}
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\text{floor of } k/12\\
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\text{floor of }(k/12) + 1 & \text{ if $k\con 2 \pmod{12}$}
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\end{cases}$$
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$m_j = \frac{p-1}{12}$ so
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``parabola with quadratic term $6/(p+1)$.
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{\bf Ogus attacks again!!} The important part must be to go
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from overconvergent to classical forms! How do you do that?
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Lawren responds: ``I'm not going to tell you.''
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What is the conjecture!?
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The conjecture of GM says: ``close weights imply close slopes''
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\end{document}