%PDF-1.4 5 0 obj << /S /GoTo /D (section.1) >> endobj 8 0 obj (1. Introduction) endobj 9 0 obj << /S /GoTo /D (subsection.1.1) >> endobj 12 0 obj (1.1. Main result) endobj 13 0 obj << /S /GoTo /D (subsection.1.2) >> endobj 16 0 obj (1.2. Previous work on generalized Fermat equations) endobj 17 0 obj << /S /GoTo /D (subsection.1.3) >> endobj 20 0 obj (1.3. Why x2+y3=z7 is particularly difficult) endobj 21 0 obj << /S /GoTo /D (section.2) >> endobj 24 0 obj (2. Notation) endobj 25 0 obj << /S /GoTo /D (section.3) >> endobj 28 0 obj (3. The theory behind the initial descent) endobj 29 0 obj << /S /GoTo /D (subsection.3.1) >> endobj 32 0 obj (3.1. Scheme-theoretic interpretation of the problem) endobj 33 0 obj << /S /GoTo /D (subsection.3.2) >> endobj 36 0 obj (3.2. An \351tale cover of SC) endobj 37 0 obj << /S /GoTo /D (subsection.3.3) >> endobj 40 0 obj (3.3. An \351tale cover of SZ[1/42]) endobj 41 0 obj << /S /GoTo /D (subsection.3.4) >> endobj 44 0 obj (3.4. Descent) endobj 45 0 obj << /S /GoTo /D (section.4) >> endobj 48 0 obj (4. The modular interpretation) endobj 49 0 obj << /S /GoTo /D (subsection.4.1) >> endobj 52 0 obj (4.1. Definition of X\(7\)) endobj 53 0 obj << /S /GoTo /D (subsection.4.2) >> endobj 56 0 obj (4.2. Automorphisms of X\(7\) over Q) endobj 57 0 obj << /S /GoTo /D (subsection.4.3) >> endobj 60 0 obj (4.3. The Klein quartic as X\(7\)) endobj 61 0 obj << /S /GoTo /D (subsection.4.4) >> endobj 64 0 obj (4.4. Twists of X\(7\)) endobj 65 0 obj << /S /GoTo /D (subsection.4.5) >> endobj 68 0 obj (4.5. Twists of X\(7\) associated to elliptic curves) endobj 69 0 obj << /S /GoTo /D (subsection.4.6) >> endobj 72 0 obj (4.6. Primitive solutions and elliptic curves) endobj 73 0 obj << /S /GoTo /D (section.5) >> endobj 76 0 obj (5. Reducible 7-torsion) endobj 77 0 obj << /S /GoTo /D (subsection.5.1) >> endobj 80 0 obj (5.1. Intermediate modular curves) endobj 81 0 obj << /S /GoTo /D (subsection.5.2) >> endobj 84 0 obj (5.2. The Galois action on G and its subgroups) endobj 85 0 obj << /S /GoTo /D (subsection.5.3) >> endobj 88 0 obj (5.3. Nonabelian cohomology of B) endobj 89 0 obj << /S /GoTo /D (subsection.5.4) >> endobj 92 0 obj (5.4. Twists of X\(7\) corresponding to reducible E[7]) endobj 93 0 obj << /S /GoTo /D (subsection.5.5) >> endobj 96 0 obj (5.5. Ramification conditions) endobj 97 0 obj << /S /GoTo /D (section.6) >> endobj 100 0 obj (6. Irreducible 7-torsion: level lowering) endobj 101 0 obj << /S /GoTo /D (section.7) >> endobj 104 0 obj (7. Explicit equations) endobj 105 0 obj << /S /GoTo /D (subsection.7.1) >> endobj 108 0 obj (7.1. Covariants of ternary quartic forms) endobj 109 0 obj << /S /GoTo /D (subsection.7.2) >> endobj 112 0 obj (7.2. Equations for XE\(7\) and XE-\(7\)) endobj 113 0 obj << /S /GoTo /D (subsection.7.3) >> endobj 116 0 obj (7.3. Equations for the degree-168 morphism to P1) endobj 117 0 obj << /S /GoTo /D (subsection.7.4) >> endobj 120 0 obj (7.4. The local test) endobj 121 0 obj << /S /GoTo /D (subsection.7.5) >> endobj 124 0 obj (7.5. The list) endobj 125 0 obj << /S /GoTo /D (subsection.7.6) >> endobj 128 0 obj (7.6. Local conditions for C5) endobj 129 0 obj << /S /GoTo /D (section.8) >> endobj 132 0 obj (8. Known rational points on the 10 curves) endobj 133 0 obj << /S /GoTo /D (section.9) >> endobj 136 0 obj (9. Overview of the strategy for determining the rational points) endobj 137 0 obj << /S /GoTo /D (section.10) >> endobj 140 0 obj (10. \(1-\)-descent on J1, J2, and J3) endobj 141 0 obj << /S /GoTo /D (subsection.10.1) >> endobj 144 0 obj (10.1. Side application to twists of the degree-7 Fermat curve) endobj 145 0 obj << /S /GoTo /D (section.11) >> endobj 148 0 obj (11. 2-descent on Jacobians of twists of X) endobj 149 0 obj << /S /GoTo /D (subsection.11.1) >> endobj 152 0 obj (11.1. Theory) endobj 153 0 obj << /S /GoTo /D (subsection.11.2) >> endobj 156 0 obj (11.2. Results) endobj 157 0 obj << /S /GoTo /D (section.12) >> endobj 160 0 obj (12. Chabauty and Mordell-Weil sieve) endobj 161 0 obj << /S /GoTo /D (subsection.12.1) >> endobj 164 0 obj (12.1. Mordell-Weil sieve theory) endobj 165 0 obj << /S /GoTo /D (subsection.12.2) >> endobj 168 0 obj (12.2. Chabauty theory) endobj 169 0 obj << /S /GoTo /D (subsection.12.3) >> endobj 172 0 obj (12.3. Results) endobj 173 0 obj << /S /GoTo /D (section.13) >> endobj 176 0 obj (13. Mordell-Weil sieve for C5) endobj 177 0 obj << /S /GoTo /D (subsection.13.1) >> endobj 180 0 obj (13.1. The strategy for C5) endobj 181 0 obj << /S /GoTo /D (subsection.13.2) >> endobj 184 0 obj (13.2. Sieve information at 3) endobj 185 0 obj << /S /GoTo /D (subsection.13.3) >> endobj 188 0 obj (13.3. Sieve information at 2) endobj 189 0 obj << /S /GoTo /D (subsection.13.4) >> endobj 192 0 obj (13.4. Sieve information at 23) endobj 193 0 obj << /S /GoTo /D (subsection.13.5) >> endobj 196 0 obj (13.5. 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