Paper XII: Patching Support, Hecke Packets, and Conditional Automorphic Completion

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1 \begin{tikzpicture}[x=1cm,y=1cm,font=\small,>=Stealth]
2 \node[anchor=west,font=\sffamily\small\bfseries] at (0,5.10) {Before imposing the ideals};
3 \draw[fill=white,draw=black!35] (0.0,0.0) rectangle ++(.69,.69);
4 \node[font=\scriptsize] at (0.345,0.345) {$1$};
5 \draw[fill=white,draw=black!35] (0.75,0.0) rectangle ++(.69,.69);
6 \node[font=\scriptsize] at (1.095,0.345) {$x$};
7 \draw[fill=white,draw=black!35] (1.5,0.0) rectangle ++(.69,.69);
8 \node[font=\scriptsize] at (1.845,0.345) {$x^{2}$};
9 \draw[fill=white,draw=black!35] (2.25,0.0) rectangle ++(.69,.69);
10 \node[font=\scriptsize] at (2.5949999999999998,0.345) {$x^{3}$};
11 \draw[fill=white,draw=black!35] (3.0,0.0) rectangle ++(.69,.69);
12 \node[font=\scriptsize] at (3.3449999999999998,0.345) {$x^{4}$};
13 \draw[fill=white,draw=black!35] (3.75,0.0) rectangle ++(.69,.69);
14 \node[font=\scriptsize] at (4.095,0.345) {$x^{5}$};
15 \draw[fill=white,draw=black!35] (0.0,0.75) rectangle ++(.69,.69);
16 \node[font=\scriptsize] at (0.345,1.095) {$y$};
17 \draw[fill=white,draw=black!35] (0.75,0.75) rectangle ++(.69,.69);
18 \node[font=\scriptsize] at (1.095,1.095) {$xy$};
19 \draw[fill=white,draw=black!35] (1.5,0.75) rectangle ++(.69,.69);
20 \node[font=\scriptsize] at (1.845,1.095) {$x^{2}y$};
21 \draw[fill=white,draw=black!35] (2.25,0.75) rectangle ++(.69,.69);
22 \node[font=\scriptsize] at (2.5949999999999998,1.095) {$x^{3}y$};
23 \draw[fill=white,draw=black!35] (3.0,0.75) rectangle ++(.69,.69);
24 \node[font=\scriptsize] at (3.3449999999999998,1.095) {$x^{4}y$};
25 \draw[fill=white,draw=black!35] (0.0,1.5) rectangle ++(.69,.69);
26 \node[font=\scriptsize] at (0.345,1.845) {$y^{2}$};
27 \draw[fill=white,draw=black!35] (0.75,1.5) rectangle ++(.69,.69);
28 \node[font=\scriptsize] at (1.095,1.845) {$xy^{2}$};
29 \draw[fill=white,draw=black!35] (1.5,1.5) rectangle ++(.69,.69);
30 \node[font=\scriptsize] at (1.845,1.845) {$x^{2}y^{2}$};
31 \draw[fill=white,draw=black!35] (2.25,1.5) rectangle ++(.69,.69);
32 \node[font=\scriptsize] at (2.5949999999999998,1.845) {$x^{3}y^{2}$};
33 \draw[fill=white,draw=black!35] (0.0,2.25) rectangle ++(.69,.69);
34 \node[font=\scriptsize] at (0.345,2.5949999999999998) {$y^{3}$};
35 \draw[fill=white,draw=black!35] (0.75,2.25) rectangle ++(.69,.69);
36 \node[font=\scriptsize] at (1.095,2.5949999999999998) {$xy^{3}$};
37 \draw[fill=white,draw=black!35] (1.5,2.25) rectangle ++(.69,.69);
38 \node[font=\scriptsize] at (1.845,2.5949999999999998) {$x^{2}y^{3}$};
39 \draw[fill=white,draw=black!35] (0.0,3.0) rectangle ++(.69,.69);
40 \node[font=\scriptsize] at (0.345,3.3449999999999998) {$y^{4}$};
41 \draw[fill=white,draw=black!35] (0.75,3.0) rectangle ++(.69,.69);
42 \node[font=\scriptsize] at (1.095,3.3449999999999998) {$xy^{4}$};
43 \draw[fill=white,draw=black!35] (0.0,3.75) rectangle ++(.69,.69);
44 \node[font=\scriptsize] at (0.345,4.095) {$y^{5}$};
45 \node[anchor=west,font=\footnotesize] at (0,-.46) {$i+j<6$ shown; coefficients lie in $\mathcal O$.};
46 \node[anchor=west,font=\sffamily\small\bfseries] at (6.35,5.10) {After imposing $(x^2,y)$};
47 \draw[fill=black!12,draw=black!35] (6.35,0.0) rectangle ++(.69,.69);
48 \node[font=\scriptsize] at (6.694999999999999,0.345) {$1$};
49 \draw[fill=black!12,draw=black!35] (7.1,0.0) rectangle ++(.69,.69);
50 \node[font=\scriptsize] at (7.444999999999999,0.345) {$x$};
51 \draw[fill=white,draw=black!35] (7.85,0.0) rectangle ++(.69,.69);
52 \draw[black!25] (7.9799999999999995,0.13) -- (8.41,0.56);\draw[black!25] (7.9799999999999995,0.56) -- (8.41,0.13);
53 \draw[fill=white,draw=black!35] (8.6,0.0) rectangle ++(.69,.69);
54 \draw[black!25] (8.73,0.13) -- (9.16,0.56);\draw[black!25] (8.73,0.56) -- (9.16,0.13);
55 \draw[fill=white,draw=black!35] (9.35,0.0) rectangle ++(.69,.69);
56 \draw[black!25] (9.48,0.13) -- (9.91,0.56);\draw[black!25] (9.48,0.56) -- (9.91,0.13);
57 \draw[fill=white,draw=black!35] (10.1,0.0) rectangle ++(.69,.69);
58 \draw[black!25] (10.23,0.13) -- (10.66,0.56);\draw[black!25] (10.23,0.56) -- (10.66,0.13);
59 \draw[fill=white,draw=black!35] (6.35,0.75) rectangle ++(.69,.69);
60 \draw[black!25] (6.4799999999999995,0.88) -- (6.91,1.31);\draw[black!25] (6.4799999999999995,1.31) -- (6.91,0.88);
61 \draw[fill=white,draw=black!35] (7.1,0.75) rectangle ++(.69,.69);
62 \draw[black!25] (7.2299999999999995,0.88) -- (7.66,1.31);\draw[black!25] (7.2299999999999995,1.31) -- (7.66,0.88);
63 \draw[fill=white,draw=black!35] (7.85,0.75) rectangle ++(.69,.69);
64 \draw[black!25] (7.9799999999999995,0.88) -- (8.41,1.31);\draw[black!25] (7.9799999999999995,1.31) -- (8.41,0.88);
65 \draw[fill=white,draw=black!35] (8.6,0.75) rectangle ++(.69,.69);
66 \draw[black!25] (8.73,0.88) -- (9.16,1.31);\draw[black!25] (8.73,1.31) -- (9.16,0.88);
67 \draw[fill=white,draw=black!35] (9.35,0.75) rectangle ++(.69,.69);
68 \draw[black!25] (9.48,0.88) -- (9.91,1.31);\draw[black!25] (9.48,1.31) -- (9.91,0.88);
69 \draw[fill=white,draw=black!35] (6.35,1.5) rectangle ++(.69,.69);
70 \draw[black!25] (6.4799999999999995,1.63) -- (6.91,2.06);\draw[black!25] (6.4799999999999995,2.06) -- (6.91,1.63);
71 \draw[fill=white,draw=black!35] (7.1,1.5) rectangle ++(.69,.69);
72 \draw[black!25] (7.2299999999999995,1.63) -- (7.66,2.06);\draw[black!25] (7.2299999999999995,2.06) -- (7.66,1.63);
73 \draw[fill=white,draw=black!35] (7.85,1.5) rectangle ++(.69,.69);
74 \draw[black!25] (7.9799999999999995,1.63) -- (8.41,2.06);\draw[black!25] (7.9799999999999995,2.06) -- (8.41,1.63);
75 \draw[fill=white,draw=black!35] (8.6,1.5) rectangle ++(.69,.69);
76 \draw[black!25] (8.73,1.63) -- (9.16,2.06);\draw[black!25] (8.73,2.06) -- (9.16,1.63);
77 \draw[fill=white,draw=black!35] (6.35,2.25) rectangle ++(.69,.69);
78 \draw[black!25] (6.4799999999999995,2.38) -- (6.91,2.81);\draw[black!25] (6.4799999999999995,2.81) -- (6.91,2.38);
79 \draw[fill=white,draw=black!35] (7.1,2.25) rectangle ++(.69,.69);
80 \draw[black!25] (7.2299999999999995,2.38) -- (7.66,2.81);\draw[black!25] (7.2299999999999995,2.81) -- (7.66,2.38);
81 \draw[fill=white,draw=black!35] (7.85,2.25) rectangle ++(.69,.69);
82 \draw[black!25] (7.9799999999999995,2.38) -- (8.41,2.81);\draw[black!25] (7.9799999999999995,2.81) -- (8.41,2.38);
83 \draw[fill=white,draw=black!35] (6.35,3.0) rectangle ++(.69,.69);
84 \draw[black!25] (6.4799999999999995,3.13) -- (6.91,3.56);\draw[black!25] (6.4799999999999995,3.56) -- (6.91,3.13);
85 \draw[fill=white,draw=black!35] (7.1,3.0) rectangle ++(.69,.69);
86 \draw[black!25] (7.2299999999999995,3.13) -- (7.66,3.56);\draw[black!25] (7.2299999999999995,3.56) -- (7.66,3.13);
87 \draw[fill=white,draw=black!35] (6.35,3.75) rectangle ++(.69,.69);
88 \draw[black!25] (6.4799999999999995,3.88) -- (6.91,4.3100000000000005);\draw[black!25] (6.4799999999999995,4.3100000000000005) -- (6.91,3.88);
89 \node[anchor=west,font=\footnotesize] at (6.35,-.46) {$i+j<6$ shown; coefficients lie in $\mathcal O$.};
90 \end{tikzpicture}

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