ABC: source and evidence reader

Download original file

This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.

Read the typeset paper

1 \clearpage
2 \section{Filtered Cusp Contact and Witt Carry Structure}
3 
4 \subsection{The three-cusp section}
5 View a primitive triple as the integral section
6 \[
7  \sigma_{a,b,c}:
8  \Spec\Z\longrightarrow\mathbb P^1_\Z,
9  \qquad
10  \sigma=[a:c].
11 \]
12 The pullbacks of the boundary divisors $0$, $1$, and $\infty$ are
13 \[
14  \operatorname{div}(a),
15  \qquad
16  \operatorname{div}(b),
17  \qquad
18  \operatorname{div}(c).
19 \]
20 For the output cusp, let
21 \[
22  C=\operatorname{div}(c),
23  \qquad
24  C_{\mathrm{red}}=\operatorname{div}(R_c).
25 \]
26 Place the two-term contact complex in degrees $-1,0$ and define
27 \[
28  \boxed{
29  \mathbf J_c
30  =
31  [\mathcal O(-C)\longrightarrow\mathcal O(-C_{\mathrm{red}})].
32  }
33 \]
34 On global sections,
35 \[
36  \mathbf J_c
37  =
38  [c\Z\hookrightarrow R_c\Z]
39  \simeq
40  [\Z\xrightarrow{c/R_c}\Z].
41 \]
42 
43 \begin{theorem}[Filtered contact quotient]
44 \[
45  \boxed{
46  H^0(\mathbf J_c)
47  =
48  R_c\Z/c\Z
49  \cong
50  \Z/(c/R_c)\Z.
51  }
52 \]
53 \end{theorem}
54 
55 The output persistence is cyclic before grading.
56 
57 \subsection{Associated graded and carry extensions}
58 At $p^e\parallel c$,
59 \[
60  \Jcal_{p,e}
61  =
62  p\Z_p/p^e\Z_p
63  \cong
64  \Z/p^{e-1}\Z.
65 \]
66 Its filtration
67 \[
68  p^j\Z_p/p^e\Z_p
69 \]
70 has successive quotients $\F_p$. Thus
71 \[
72  \gr\Jcal_{p,e}\cong(\F_p)^{e-1}.
73 \]
74 The exact sequences
75 \[
76  0\longrightarrow
77  \Z/p^{j-1}\Z
78  \xrightarrow{\times p}
79  \Z/p^j\Z
80  \longrightarrow
81  \F_p
82  \longrightarrow0
83 \]
84 are nonsplit for $j\ge2$. The $j=1$ sequence has zero left term and is split. These carry extensions are precisely what the split persistence packet forgot.
85 
86 \begin{corollary}[Correction of the cyclic-rank barrier]
87 The $p$-rank obstruction applies to the associated graded source. It does not obstruct a filtered map from $\Z/p^{e-1}\Z$ to a cyclic target of matching depth.
88 \end{corollary}
89 
90 \subsection{Exact cyclic transducer}
91 Let $m,\eta$ be positive integers and put
92 \[
93  C(m)=[\Z\xrightarrow{m}\Z]
94 \]
95 and
96 \[
97  C(\eta)=[\Z\xrightarrow{\eta}\Z].
98 \]
99 Set $g=\gcd(m,\eta)$. Multiplication by $m/g$ in degree $-1$ and by $\eta/g$ in degree $0$ defines a chain map. On cohomology,
100 \[
101  \boxed{
102  |\ker|=m/g,
103  \qquad
104  |\im|=g,
105  \qquad
106  |\coker|=\eta/g.
107  }
108 \]
109 The maximal transferable packet is the common prime-power depth.
110 
111 For the output source $m=c/R_c$, the corrected algebraic defect against a cyclic target of order $\eta_f$ is
112 \[
113  \boxed{
114  \delta_{\mathrm{alg}}(c,f)
115  =
116  \frac{c/R_c}{\gcd(c/R_c,\eta_f)}.
117  }
118 \]
119 Prime by prime,
120 \[
121  \delta_{\mathrm{alg}}(c,f)
122  =
123  \prod_{p^e\parallel c}
124  p^{\max(e-1-v_p(\eta_f),0)}.
125 \]
126 The problem is missing depth, not missing rank.
127 
128 \subsection{Witt realization}
129 For $e\ge2$, the local packet has a natural additive identification
130 \[
131  \boxed{
132  \Jcal_{p,e}
133  \cong
134  W_{e-1}(\F_p).
135  }
136 \]
137 Division by $p$ identifies the source with $\Z_p/p^{e-1}\Z_p$, and the standard additive identification $W_n(\F_p)\cong\Z/p^n\Z$ gives the result. For $e=1$, take $W_0=0$. The Verschiebung filtration matches multiplication by successive powers of $p$. This is the typed local object underlying the carry chain.
138 
139 At odd $p$, the $p$-adic logarithm identifies
140 \[
141  (1+p\Z_p)/(1+p^e\Z_p)
142  \cong
143  p\Z_p/p^e\Z_p.
144 \]
145 At $2$, the additive contact packet remains uniform, while the multiplicative coordinate begins on $1+4\Z_2$ and therefore carries the familiar weighted dyadic shift.
146 
147 \subsection{The filtered local-global target}
148 The intended transducer is a filtered chain
149 \[
150  \mathbf J_c
151  \longrightarrow
152  \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}}
153  \longrightarrow
154  \mathbf L_{R^{\mathrm{glob}}}
155  \xrightarrow{R=\mathbb T}
156  \mathbf L_{\mathbb T_f}
157  \longrightarrow
158  \mathbf C_f.
159 \]
160 It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
161 
162 \begin{theorem}[Conditional filtered-transfer criterion]
163 Suppose every primitive positive triple admits a canonical map
164 \[
165  H^0(\Theta_f):\Z/(c/R_c)\Z\longrightarrow H^0(\mathbf C_f)
166 \]
167 satisfies, for every $\eps>0$, with constants independent of the triple,
168 \[
169  |\ker H^0(\Theta_f)|\le C_\eps R^\eps
170 \]
171 and
172 \[
173  |H^0(\mathbf C_f)|\le C'_\eps R_aR_bR^\eps.
174 \]
175 Then the strong $abc$ conjecture follows.
176 \end{theorem}
177 
178 \begin{proof}
179 The image has order
180 \[
181  \frac{c/R_c}{|\ker|}
182 \]
183 and is bounded by the target order. Therefore
184 \[
185  \frac{c/R_c}{C_\eps R^\eps}
186  \le
187  C'_\eps R_aR_bR^\eps.
188 \]
189 Multiplying by $R_c$ and replacing $\eps$ by $\eps/2$ gives the strong bound.
190 \end{proof}

Back to source controls

Back to ABC: source and evidence reader