CoCalc Public Fileswww / talks / MSRI-touching / tex / mentor.texOpen with one click!
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}
18
Let $N$ be a positive integer, and consider the group
19
$$\Gamma_0(N) := \left\{ \mtwo{a}{b}{c}{d}:
20
ad-bc=1 \text{ and } N\mid c\right\}.$$
21
It acts on the upper half plane
22
$$\h:=\{z\in \C : \Im(z)>0\}$$
23
by
24
$$z \mapsto \frac{az+b}{cz+d}.$$
25
\end{slide}
26
27
\begin{slide}
28
A {\em modular form} for $\Gamma_0(N)$ (of weight~$2$)
29
is a holomorphic function $f(z)$ on $\h$
30
such that
31
$$f(\gamma(z))d(\gamma(z)) = f(z)dz$$
32
for all $\gamma\in\Gamma_0(N)$.
33
34
Since $\gamma=\mtwo{1}{1}{0}{1}\in\Gamma_0(N)$,
35
$$f(z+1) = f(z),$$
36
so $f(z)$ has a {\em Fourier expansion}
37
$$f(z) = a_0 + a_1q + a_2 q^2 + a_3 q^3 + \cdots$$
38
where $q = e^{2\pi i z}$.
39
\end{slide}
40
41
\begin{slide}
42
The space of {\em cusp forms} $S_2(N)$ is the subspace of
43
modular forms that ``vanish at the cusps'' (essentially, $a_0=0$).
44
45
Use \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}
61
There is a family
62
$$T_2,\,\, T_3,\,\, T_5,\,\, T_7,\,\, \ldots$$
63
of commuting linear transformations of $S_2(N)$.
64
For example, if $p\nmid N$, then
65
$$T_p(f) = \sum_{n=1}^{\infty} (a_{np} q^n + a_n q^{np}).$$
66
The $T_p$ generate the {\em Hecke algebra} $\T$.
67
\end{slide}
68
69
\begin{slide}
70
Use \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
$$
76
T_2 = \left(
77
\begin{array}{rrr}
78
-2&0&\,\,1\\
79
-1&-2&2\\
80
-2 &0&2
81
\end{array}\right)$$
82
The 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
88
the characteristic polynomial of every $T_p$ has
89
a double root mod $N$?
90
91
I first searched using a {\em slow} trace formula.
92
93
Learned from Joe Wetherell at Arizona Winter School to
94
use {\em modular symbols}.
95
96
Found the only known example: $N=389$.
97
98
Hooked! 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
106
A {\em modular form} of weight~$k$
107
is a holomorphic function $f(z)$ on $\h$
108
such that
109
$$f\left(\frac{az+b}{cz+d}\right) = (cz+d)^kf(z)$$
110
whenever $a,b,c,d$ are integers with $ad-bc=1$.
111
\end{slide}
112
113
\begin{slide}
114
Use \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
$$
121
f_{12} = q - 24q^2 + 252q^3 - 1472q^4 + 4830q^5 + \cdots
122
$$
123
The $2$-adic {\em slope} of $f$ is $\ord_2(24)=3$.
124
\end{slide}
125
126
\begin{slide}
127
Let's look for eigenforms of weight $2$-adically
128
close to $12$ and slope $3$.
129
Try 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
$$f_{20}-f_{12} = 480q^2 + 50400q^3 - 314880q^4 - 2382240q^5 + \cdots$$
141
Maybe $f_{16} \con f_{12}\pmod {2^4}$
142
and
143
$f_{20} \con f_{12}\pmod {2^5}$?
144
\end{slide}
145
146
\begin{slide}
147
For the form of weight $12+16$, we must consider only the slope~$3$
148
form.
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*}
156
f_{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*}
162
f_{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
167
Robert is the world's expert in understanding how these families
168
of $f_k$ fit together.
169
\end{slope}
170
171
172
173
\end{document}
174
175