ABC: source and evidence reader

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1 \clearpage
2 \section{Sources and theorem boundaries}
3 The external source check distinguishes a stated theorem, a contextual reference and an unresolved comparison. The critical conjecture quantifiers, known unconditional bound, polynomial theorem, modified Szpiro equivalence, modularity, polylogarithm formulas and current IUT status have authenticated primary-source leaves in the evidence package.
4 
5 The prime contact and determinant calculations are derived in this text. Pasten's arithmetic derivative work supplies an established base for that branch. Its small-derivative conjecture and its weak $abc$ equivalences are not silently identified with the stronger all-$\eps$ packet target stated here. The Wiles-defect and motivic references explain relevant established frameworks. They do not construct the project-specific persistence-to-Hecke or integral period transfer maps.
6 
7 Otsubo and Yamazaki's motivic identities have precise coefficient-field and base-field hypotheses. Some Gauss-sum statements are over finite fields, while the multiplication relation extends to fields containing the needed roots of unity. None gives the missing integral radical-scale norm comparison in this work. Borel's rank input is used only for the rational weight motivation explicitly described in the path-packet section.
8 
9 Bibliographic entries inherited solely as mathematical background are retained for that purpose. The exact source-check report states which leaves were freshly inspected. A failed search or a citation title is not treated as proof of a theorem or of historical novelty.
10 
11 \begingroup
12 \begin{thebibliography}{99}
13 \raggedright
14 \bibitem{FaltingsIsogeny1986}
15 G. Faltings,
16 ``Finiteness theorems for abelian varieties over number fields,''
17 in G. Cornell and J. H. Silverman (eds.), \emph{Arithmetic Geometry},
18 Springer, 1986, Chapter II, 9--26.
19 \href{https://doi.org/10.1007/978-1-4613-8655-1_2}{doi:10.1007/978-1-4613-8655-1\_2}.
20 Translation of the 1983 German original with its 1984 erratum.
21 
22 
23 \bibitem{SerreAbelian1989}
24 J.-P. Serre,
25 \emph{Abelian $\ell$-adic representations and elliptic curves},
26 with the collaboration of W. Kuyk and J. Labute,
27 Addison--Wesley, 1989, reprint of the 1968 original.
28 Chapter IV, Appendix A.1.2--A.1.3; Special Preface, p.~ix.
29 
30 
31 % Verified publication metadata and exact consulted versions, 2026-10-08.
32 % Bibitem fragments only: insert inside an existing thebibliography environment.
33 % Do not duplicate existing keys. See LITERATURE_CHECK.md for dependency scope.
34 
35 \bibitem{StewartYu2001}
36 C. L. Stewart and K. R. Yu,
37 ``On the $abc$ conjecture, II,''
38 \emph{Duke Mathematical Journal} 108, no.~1 (2001), 169--181.
39 \href{https://doi.org/10.1215/S0012-7094-01-10815-6}{doi:10.1215/S0012-7094-01-10815-6}.
40 
41 \bibitem{Snyder2000}
42 N. Snyder,
43 ``An alternate proof of Mason's theorem,''
44 \emph{Elemente der Mathematik} 55, no.~3 (2000), 93--94.
45 \href{https://doi.org/10.1007/S000170050074}{doi:10.1007/S000170050074}.
46 
47 \bibitem{Oesterle1988}
48 J. Oesterl\'e,
49 ``Nouvelles approches du `th\'eor\`eme' de Fermat,''
50 S\'eminaire Bourbaki, expos\'e 694,
51 \emph{Ast\'erisque} 161--162 (1988), 165--186.
52 \url{https://www.numdam.org/item/SB_1987-1988__30__165_0/}.
53 
54 \bibitem{PastenDerivatives}
55 H. Pasten,
56 ``Arithmetic derivatives through geometry of numbers,''
57 \emph{Canadian Mathematical Bulletin} 65, no.~4 (2022), 906--923.
58 \href{https://doi.org/10.4153/S0008439521000990}{doi:10.4153/S0008439521000990}.
59 Consulted author version: arXiv:2106.16165v3, 11 December 2021.
60 
61 \bibitem{VojtaABC}
62 P. Vojta,
63 ``A more general $ABC$ conjecture,''
64 \emph{International Mathematics Research Notices} 1998, no.~21, 1103--1116.
65 \href{https://doi.org/10.1155/S1073792898000658}{doi:10.1155/S1073792898000658}.
66 
67 \bibitem{MochizukiIUTIII}
68 S. Mochizuki,
69 ``Inter-universal Teichm\"uller theory III: Canonical splittings of the log-theta-lattice,''
70 \emph{Publications of the Research Institute for Mathematical Sciences}
71 57, no.~1/2 (2021), 403--626.
72 \href{https://doi.org/10.4171/PRIMS/57-1-3}{doi:10.4171/PRIMS/57-1-3}.
73 Consulted author PDF: May 2020.
74 
75 \bibitem{MochizukiIUTIV}
76 S. Mochizuki,
77 ``Inter-universal Teichm\"uller theory IV: Log-volume computations and set-theoretic foundations,''
78 \emph{Publications of the Research Institute for Mathematical Sciences}
79 57, no.~1/2 (2021), 627--723.
80 \href{https://doi.org/10.4171/PRIMS/57-1-4}{doi:10.4171/PRIMS/57-1-4}.
81 Consulted author PDF: April 2020.
82 
83 \bibitem{ScholzeStix2018}
84 P. Scholze and J. Stix,
85 ``Why $abc$ is still a conjecture,'' August 2018 manuscript.
86 \url{https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf}.
87 
88 \bibitem{MochizukiComments2018}
89 S. Mochizuki,
90 ``Comments on the manuscript (2018-08 version) by Scholze--Stix concerning inter-universal Teichm\"uller theory (IUTch),'' September 2018.
91 \url{https://www.kurims.kyoto-u.ac.jp/~motizuki/Cmt2018-08.pdf}.
92 
93 \bibitem{MochizukiReport2025}
94 S. Mochizuki,
95 ``Report on the current situation surrounding inter-universal Teichm\"uller theory (IUT),'' October 2025.
96 \url{https://www.kurims.kyoto-u.ac.jp/~motizuki/IUT-report-2025-10.pdf}.
97 
98 \bibitem{LANA2026}
99 LANA Project,
100 \emph{Project LANA interim report on IUT theory}, last updated 20 July 2026.
101 Author report: \url{https://github.com/katobungen/LANA_report_202607}.
102 Official announcement: \url{https://zen.ac.jp/news/zmcpostevent0717e}.
103 
104 \bibitem{DLMFPolylog}
105 NIST Digital Library of Mathematical Functions,
106 \S25.12, ``Polylogarithms,'' equations 25.12.4, 25.12.6, 25.12.10, and 25.12.13.
107 and \S25.11, equation 25.11.14. Accessed 8 October 2026.
108 \url{https://dlmf.nist.gov/25.12}. \url{https://dlmf.nist.gov/25.11}.
109 
110 \bibitem{Wiles}
111 A. Wiles,
112 ``Modular elliptic curves and Fermat's Last Theorem,''
113 \emph{Annals of Mathematics} (2) 141, no.~3 (1995), 443--551.
114 \href{https://doi.org/10.2307/2118559}{doi:10.2307/2118559}.
115 
116 \bibitem{TaylorWiles}
117 R. Taylor and A. Wiles,
118 ``Ring-theoretic properties of certain Hecke algebras,''
119 \emph{Annals of Mathematics} (2) 141, no.~3 (1995), 553--572.
120 \href{https://doi.org/10.2307/2118560}{doi:10.2307/2118560}.
121 
122 \bibitem{BCDT}
123 C. Breuil, B. Conrad, F. Diamond, and R. Taylor,
124 ``On the modularity of elliptic curves over $\mathbb Q$: wild 3-adic exercises,''
125 \emph{Journal of the American Mathematical Society} 14, no.~4 (2001), 843--939.
126 \href{https://doi.org/10.1090/S0894-0347-01-00370-8}{doi:10.1090/S0894-0347-01-00370-8}.
127 
128 \bibitem{Agashe}
129 A. Agashe,
130 ``The modular number, congruence number, and multiplicity one,''
131 arXiv:0810.5176v2, 5 October 2025 (first version 2008).
132 \url{https://arxiv.org/abs/0810.5176v2}.
133 
134 \bibitem{CesnaviciusManin}
135 K. \v{C}esnavi\v{c}ius, M. Neururer, and A. Saha,
136 ``The Manin constant and the modular degree,''
137 \emph{Journal of the European Mathematical Society} 26, no.~2 (2024), 573--637.
138 \href{https://doi.org/10.4171/JEMS/1367}{doi:10.4171/JEMS/1367}.
139 Consulted arXiv:1911.09446v3, 2 November 2022.
140 
141 \bibitem{BKM1}
142 G. B\"ockle, C. B. Khare, and J. Manning,
143 ``Wiles defect for Hecke algebras that are not complete intersections,''
144 \emph{Compositio Mathematica} 157, no.~9 (2021), 2046--2088.
145 \href{https://doi.org/10.1112/S0010437X21007454}{doi:10.1112/S0010437X21007454}.
146 Consulted arXiv:1910.08507v3, 4 April 2021.
147 
148 \bibitem{BKM2}
149 G. B\"ockle, C. B. Khare, and J. Manning,
150 ``Wiles defect of Hecke algebras via local-global arguments,''
151 with an appendix by N. Fakhruddin and C. B. Khare,
152 \emph{Journal of the Institute of Mathematics of Jussieu} 23, no.~6 (2024), 2461--2541.
153 \href{https://doi.org/10.1017/S1474748024000021}{doi:10.1017/S1474748024000021}.
154 Consulted arXiv:2108.09729v2, 24 August 2021.
155 
156 \bibitem{OtsuboYamazaki}
157 N. Otsubo and T. Yamazaki,
158 ``Motivic Gauss and Jacobi sums,''
159 \emph{Annales de la Facult\'e des Sciences de Toulouse}, series 6,
160 35, no.~1 (2026), 95--135.
161 \href{https://doi.org/10.5802/afst.1842}{doi:10.5802/afst.1842}.
162 Consulted arXiv:2402.06072v3, 2 March 2025.
163 
164 \bibitem{Goncharov}
165 A. B. Goncharov,
166 ``Multiple polylogarithms and mixed Tate motives,''
167 arXiv:math/0103059v4, 15 May 2001.
168 \url{https://arxiv.org/abs/math/0103059v4}.
169 
170 \bibitem{ZagierPolylog}
171 D. Zagier,
172 ``Polylogarithms, Dedekind zeta functions, and the algebraic $K$-theory of fields,''
173 in \emph{Arithmetic Algebraic Geometry}, Progress in Mathematics 89,
174 Birkh\"auser, 1990, 391--430.
175 \href{https://doi.org/10.1007/978-1-4612-0457-2_19}{doi:10.1007/978-1-4612-0457-2\_19}.
176 
177 \bibitem{Borel}
178 A. Borel,
179 ``Stable real cohomology of arithmetic groups,''
180 \emph{Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, series 4,
181 7, no.~2 (1974), 235--272.
182 \href{https://doi.org/10.24033/asens.1269}{doi:10.24033/asens.1269}.
183 
184 \bibitem{SUnit1988}
185 J.-H. Evertse, K. Gy\H{o}ry, C. L. Stewart, and R. Tijdeman,
186 ``On $S$-unit equations in two unknowns,''
187 \emph{Inventiones Mathematicae} 92 (1988), 461--477.
188 \href{https://doi.org/10.1007/BF01393743}{doi:10.1007/BF01393743}.
189 
190 \bibitem{AokiFermat}
191 N. Aoki,
192 ``Simple factors of the Jacobian of a Fermat curve and the Picard number of a product of Fermat curves,''
193 \emph{American Journal of Mathematics} 113 (1991), 779--833.
194 \href{https://doi.org/10.2307/2374786}{doi:10.2307/2374786}.
195 
196 \bibitem{ColemanMcCallum}
197 R. Coleman and W. McCallum,
198 ``Stable reduction of Fermat curves and Jacobi sum Hecke characters,''
199 \emph{Journal f\"ur die reine und angewandte Mathematik} 385 (1988), 41--101.
200 \href{https://doi.org/10.1515/crll.1988.385.41}{doi:10.1515/crll.1988.385.41}.
201 
202 \end{thebibliography}
203 \endgroup

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