CoCalc Public Fileswww / talks / MSRI-touching / tex / mentor.tex
Author: William A. Stein
1\documentclass{slides}
2\usepackage{amsmath}
3\usepackage{amssymb}
4\newcommand{\C}{\mathbf{C}}
5\newcommand{\Z}{\mathbf{Z}}
6\newcommand{\T}{\mathbf{T}}
7\newcommand{\h}{\mathfrak{h}}
8\newcommand{\mtwo}[4]{\left(
9        \begin{matrix}#1&#2\\#3&#4
10        \end{matrix}\right)}
11\newcommand{\magma}{{\sc Magma}}
12\DeclareMathOperator{\ord}{ord}
13\newcommand{\con}{\equiv}
14
15\begin{document}
16
17\begin{slide}
18Let $N$ be a positive integer, and consider the group
19$$\Gamma_0(N) := \left\{ \mtwo{a}{b}{c}{d}: 20ad-bc=1 \text{ and } N\mid c\right\}.$$
21It acts on the upper half plane
22$$\h:=\{z\in \C : \Im(z)>0\}$$
23by
24$$z \mapsto \frac{az+b}{cz+d}.$$
25\end{slide}
26
27\begin{slide}
28A {\em modular form} for $\Gamma_0(N)$ (of weight~$2$)
29is a holomorphic function $f(z)$ on $\h$
30such that
31$$f(\gamma(z))d(\gamma(z)) = f(z)dz$$
32for all $\gamma\in\Gamma_0(N)$.
33
34Since $\gamma=\mtwo{1}{1}{0}{1}\in\Gamma_0(N)$,
35$$f(z+1) = f(z),$$
36so $f(z)$ has a {\em Fourier expansion}
37$$f(z) = a_0 + a_1q + a_2 q^2 + a_3 q^3 + \cdots$$
38where $q = e^{2\pi i z}$.
39\end{slide}
40
41\begin{slide}
42The space of {\em cusp forms} $S_2(N)$ is the subspace of
43modular forms that vanish at the cusps'' (essentially, $a_0=0$).
44
45Use \magma{} to compute some modular forms:
46\begin{verbatim}
47  > qEigenform(ModularSymbols("11A"),7);
48  > qEigenform(ModularSymbols("23A"),6);
49  > qEigenform(ModularSymbols("37A"),7);
50\end{verbatim}
51$q - 2q^2 - q^3 + 2q^4 + q^5 + 2q^6 + \cdots$
52
53$q \, +\, \frac{\sqrt{5} - 1}{2}q^2 \,-\, \sqrt{5}q^3 \,-\, 54 \frac{\sqrt{5} + 1}{2}q^4 \, +\, (\sqrt{5} - 1)q^5 + \cdots$
55
56$q - 2q^2 - 3q^3 + 2q^4 - 2q^5 + 6q^6 + \cdots$
57
58\end{slide}
59
60\begin{slide}
61There is a family
62  $$T_2,\,\, T_3,\,\, T_5,\,\, T_7,\,\, \ldots$$
63of commuting linear transformations of $S_2(N)$.
64For example, if $p\nmid N$, then
65$$T_p(f) = \sum_{n=1}^{\infty} (a_{np} q^n + a_n q^{np}).$$
66The $T_p$ generate the {\em Hecke algebra} $\T$.
67\end{slide}
68
69\begin{slide}
70Use \magma{} to compute some $T_p$:
71\begin{verbatim}
72  > M := ModularSymbols(43,2,+1);
73  > Restrict(Tn(M,2),SkZ(M));
74\end{verbatim}
75$$76T_2 = \left( 77\begin{array}{rrr} 78-2&0&\,\,1\\ 79-1&-2&2\\ 80-2 &0&2 81\end{array}\right)$$
82The characteristic polynomial of $T_2$ is
83$$(x+2)(x^2-2).$$
84\end{slide}
85
86\begin{slide}
87{\bf Question of Ribet}. Is there a prime~$N$ so that
88the characteristic polynomial of every $T_p$ has
89a double root mod $N$?
90
91I first searched using a {\em slow} trace formula.
92
93Learned from Joe Wetherell at Arizona Winter School to
94use {\em modular symbols}.
95
96Found the only known example: $N=389$.
97
98Hooked!  This is the tip of an iceberg.
99\end{slide}
100
101\begin{slide}
102\begin{center}
103{\bf Higher weight modular forms}
104\end{center}
105
106A {\em modular form} of weight~$k$
107is a holomorphic function $f(z)$ on $\h$
108such that
109$$f\left(\frac{az+b}{cz+d}\right) = (cz+d)^kf(z)$$
110whenever $a,b,c,d$ are integers with $ad-bc=1$.
111\end{slide}
112
113\begin{slide}
114Use \magma{} to find an example:
115\begin{verbatim}
116  > M   := ModularSymbols(1,12);
117  > D   := Decomposition(M);
118  > f12 := qEigenform(D[1],5);
119\end{verbatim}
120$$121f_{12} = q - 24q^2 + 252q^3 - 1472q^4 + 4830q^5 + \cdots 122$$
123The $2$-adic {\em slope} of $f$ is $\ord_2(24)=3$.
124\end{slide}
125
126\begin{slide}
127Let's look for eigenforms of weight $2$-adically
128close to $12$ and slope $3$.
129Try weights
130$$12+4,\quad 12+8,\quad\text{and } 12+16.$$
131\begin{verbatim}
132  > f16 := qEigenform(Decomposition(
133               ModularSymbols(1,12+4))[1],7);
134  > f20 := qEigenform(Decomposition(
135               ModularSymbols(1,12+8))[1],7);
136  > f28 := qEigenform(Decomposition(
137               ModularSymbols(1,12+16))[1],7);
138\end{verbatim}
139$$f_{16}-f_{12} = 40q^2 - 3600q^3 + 15360q^4 + 47280q^5 + \cdots$$
140$$f_{20}-f_{12} = 480q^2 + 50400q^3 - 314880q^4 - 2382240q^5 + \cdots$$
141Maybe $f_{16} \con f_{12}\pmod {2^4}$
142and
143$f_{20} \con f_{12}\pmod {2^5}$?
144\end{slide}
145
146\begin{slide}
147For the form of weight $12+16$, we must consider only the slope~$3$
148form.
149\begin{verbatim}
150 > f28 := SlopeAlphaNewforms(
151              ModularSymbols(1,12+16),
152              3,2,30,7)[1];  // slope, p, precision
153 > f28 - CastPowerSeries(Parent(f28),f12);
154\end{verbatim}
155\begin{align*}
156f_{28} &=
157         q + (2^3 + 2^5 + 2^7 + \cdots) q^2 \\
158   &\qquad  + (2^2 + 2^3 + 2^4 +
159         2^5 + 2^7 +\cdots)q^3 + \cdots\\
160%\end{align*}
161%\begin{align*}
162f_{28}-f_{12} &= (2^6 + 2^8 + 2^9 + \cdots)q^2 \\
163               &\qquad +(2^6 + 2^9 + \cdots )q^3  + \cdots
164\end{align*}
165$$f_{28} \con f_{12}\pmod {2^6}?$$
166
167Robert is the world's expert in understanding how these families
168of $f_k$ fit together.
169\end{slope}
170
171
172
173\end{document}
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