CoCalc Public Fileswww / tables / Notes / smithline.tex
Author: William A. Stein
1\documentclass{article}
2\title{Lawren Smithline's NTS talk, March 22, 2000}
3\include{macros}
4\begin{document}
5\maketitle
6
7Newton polygon of $U_p$.
8
9{\bf Wan's result.}  For $p\geq 5$, for all weights $k\in\Z$,
10there is a parabola below the Newton polygon for $U_p$.
11
12More precise statement about Wan's result.
13   $$\det(1-t U_p) = \sum_{i=0}^{\infty} a_i t^i.$$
14
15
16{\bf New Theorem.}  In level $N$, $p\nmid N$, the newton polygon
17of $U_p$ is bounded below by a parabola with quadratic term
18depending on
19    $$X(\Gamma_1(N)\intersect \Gamma_0(p))=X(\Gamma).$$
20The dimensions $\dim S_k(\Gamma)$ grow essentially linearly in
21terms of $k$, say roughly $s_\Gamma\cdot k$.
22
23\renewcommand{\M}{\mathcal{M}}
24Fix $p\geq 5$.
25What is our space of modular forms?
26Overconvergent $p$-adic modular forms of weight $k$, level $N$, $p\nmid N$,
27coefficients in~$R$, growth condition~$r$:
28 $$\M_k(N,R)$$
29
30Here's how to build the above spaces from the classical spaces
31$M_k(N,R)$.  My~$R$ is $\O_K$, where $[K:\Q_p]<\infty$.
32Katz observed that
33  $$M_{k+p-1} = E_{p-1}\cdot M_k + W_{k+p-1}(N,B).$$
34Then
35$$\M_k =\left\{ b_0 + \sum_{j=1}^{\infty} r^j \frac{b_j}{E_{p-1}^j} \right\}$$
36where $b_0 \in M_k(N,R)$ and $b_j\in W_{k+(p-1)j}$.
37When $v_p(-r)>0$, these are overconvergent modular forms.
38
39{\bf Action of $U_p$.}\\
40$U_p$ acts on $q$-expansions by
41    $$U(\sum b_n q^n) = \sum b_{np}q^n.$$
42It turns out that
43$$U\in \End(\M_k(N,R,p^{\frac{1}{p+1}})\tensor K).$$
44Let
45   $$P_k(t)=\det(1-tU) = \sum_{i=0}^{\infty} a_i t^i 46 = \sum \tr \Lambda^i U_p\cdot t^i.$$
47
48{\bf Theorem (Wan).}  The Newton polygon of $U_p$ (i.e., $P_k(t)$) is
49bounded below by a parabola, independent of $k$, dependent on
50$N$ and $p$. \\
51{\bf Coleman's Twist:}
52    $$U_{((k)+(p-1)j)} \sim U_{(k)}\left(\frac{E_{p-1}(q)}{E_{p-1}(q^p)}\right)^j$$
53Note
54   $$G = \frac{E_{p-1}(q)}{E_{p-1}(q^p)} \in 55 \M_0(N,R,p^{\frac{1}{p+1}}),$$
56so multiplying by $G$ doesn't change the weight.
57Here we abuse notation: the $p$ is understood.''
58Similiar'' is in the sense that there is an isomorphism of $K$-vector
59spaces so that the appropriate diagram commutes.
60Abstractly, the spaces of various weights are all isomorphic.''
61Using this we make $k+(p-1)j$, in range $0\ldots p-1$.
62
63Let $b_{i,s}$ be a basis for $W_{k+j(p-1})(N,R)$.
64Let
65     $$e_{i,s} = \frac{r^i}{E_{p-1}} b_{i,s}.$$
66Then
67 $$U\cdot G^j(e_{i,s}) = 68\frac{1}{p} 69\sum (\frac{r^p}{E_{p-1}})^u b_u(i,s,j)$$
70  for $b_u \in W_u(N,R)$ from classical weight $k+(p-1)u$.
71Here little $u$'' is an index; it should be $i,j,\ldots$, but he
72ran out.
73$$U\cdot G^j(e_{i,s}) \frac{1}{p} 74\sum r^{p-1}(\frac{r^p}{E_{p-1}})^a b_u(i,s,j)$$
75$$U\cdot G^j(e_{i,s}) \frac{1}{p} 76 = \sum A^{a,v}_{i,s} e_{u,v}$$
77$$v_p(A_{i,s}^{u,v})\geq u(p-1) v_p(r) - 1$$
78$$v_p(A_{i,s}^{u,v})\geq \frac{u(p-1)}{p+1} - 1$$
79Now let $d_{\ell} = \dim M_{a(p-1)\ell}(N,B)$. Then
80$$v_p(a_n(k+j(p-1))) 81 \geq \frac{p-1}{p+1} 82 \sum_{0}^{\ell} 83 u m_u + (\ell? -1)(n-d_\ell)-n$$
84for $d_{\ell} \leq n \leq d_{\ell+1}$,
85where $m_\ell = d_\ell - d_{\ell - 1}$.
86
87{\bf Ogus attacks!!}: Draw the diagram which describes the
88Coleman Twist!''
89
90Lawren responds:
91$$U\cdot f(q^p)=f(q)\qquad\text{Frobenius linear''}$$
92$$U_p \sim U_p \frac{E_{p-1}}{V(E_{p-1})}$$
93$$(E_p - 1)^{-1} U_p E_{p-1}$$
94
95{\bf $p=2,3$}
96Katz expansion in powers of $E_4$.
97
98The message of this talk is that $p=2$ and $p=3$
99are not so special as they think they are.''
100
101E.g., $N=1$, $p\con 1\pmod{12}$.
102In this case we have
103$$\dim M_k(1,R) = 104\begin{cases} 105\text{floor of } k/12\\ 106\text{floor of }(k/12) + 1 & \text{ if k\con 2 \pmod{12}} 107\end{cases}$$
108$m_j = \frac{p-1}{12}$ so
109parabola with quadratic term $6/(p+1)$.
110
111{\bf Ogus attacks again!!} The important part must be to go
112from overconvergent to classical forms!  How do you do that?
113Lawren responds: I'm not going to tell you.''
114What is the conjecture!?
115
116The conjecture of GM says: close weights imply close slopes''
117
118\end{document}