Siegel’s theorem and the Shafarevich conjecture

 Access to full text

 Full (PDF)

1. A. Baker, Bounds for the solutions of the hyperelliptic equation. Proc. Cambridge Philos. Soc. 65 (1969), 439–444. Zbl0174.33803MR234912
2. A. Baker, Transcendental number theory. Second ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Zbl0715.11032MR1074572
3. A. Baker and J. Coates, Integer points on curves of genus 1. Proc. Cambridge Philos. Soc. 67 (1970), 595–602. Zbl0194.07601MR256983
4. Yu. Bilu (Belotserkovskiĭ), Effective analysis of a new class of Diophantine equations. Vestsī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk (1988), no. 6, 34–39, 125. Zbl0669.10037MR984113
5. Yu. Bilu, Effective analysis of integral points on algebraic curves. Israel J. Math. 90 (1995), no. 1-3, 235–252. Zbl0840.11028MR1336325
6. Yu. Bilu, Quantitative Siegel’s theorem for Galois coverings. Compositio Math. 106 (1997), no. 2, 125–158. Zbl1044.11593MR1457336
7. S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. Zbl0705.14001MR1045822
8. S. G. Dalaljan, The Prym variety of an unramified double covering of a hyperelliptic curve. Uspehi Mat. Nauk 29 (1974), no. 6(180), 165–166. MR404270
9. S. G. Dalaljan, The Prym variety of a two-sheeted covering of a hyperelliptic curve with two branch points. Mat. Sb. (N.S.) 98(140) (1975), no. 2 (10), 255–267, 334. Zbl0322.14013MR399108
10. R. Dvornicich and U. Zannier, Fields containing values of algebraic functions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), no. 3, 421–443. Zbl0819.12003MR1310635
11. J.-H. Evertse and K. Győry, Effective finiteness results for binary forms with given discriminant. Compositio Math. 79 (1991), no. 2, 169–204. Zbl0746.11020MR1117339
12. G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73 (1983), no. 3, 349–366. Zbl0588.14026MR718935
13. D. Grant, Integer points on curves of genus two and their Jacobians. Trans. Amer. Math. Soc. 344 (1994), no. 1, 79–100. Zbl0828.11032MR1184116
14. M. Hindry and J. H. Silverman, Diophantine geometry. Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000, An introduction. Zbl0948.11023MR1745599
15. R. von Känel, An effective proof of the hyperelliptic Shafarevich conjecture and applications. PhD thesis, Eidgenössische Technische Hochschule Zürich, 2010. 
16. H. Kleiman, On the Diophantine equation $f(x,y)=0$. J. Reine Angew. Math. 286/287 (1976), 124–131. Zbl0332.10010MR417049
17. Q. Liu, Modèles minimaux des courbes de genre deux. J. Reine Angew. Math. 453 (1994), 137–164. Zbl0805.14013MR1285783
18. Q. Liu, Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète. Trans. Amer. Math. Soc. 348 (1996), no. 11, 4577–4610. Zbl0926.11043MR1363944
19. J. R. Merriman, Binary forms and the reduction of curves. 1970, D.Phil, thesis, Oxford University. 
20. J. R. Merriman and N. P. Smart, Curves of genus $2$ with good reduction away from $2$ with a rational Weierstrass point. Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 2, 203–214. Zbl0805.14018MR1230127
21. D. Mumford, Prym varieties. I. Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325–350. Zbl0299.14018MR379510
22. A. P. Ogg, On pencils of curves of genus two. Topology 5 (1966), 355–362. Zbl0145.17802MR201437
23. F. Oort, Hyperelliptic curves over number fields. Classification of algebraic varieties and compact complex manifolds, Springer, Berlin, 1974, pp. 211–218. Lecture Notes in Math., Vol. 412. Zbl0299.14017MR354676
24. A. N. Parshin, Minimal models of curves of genus $2$. and homomorphisms of abelian varieties defined over a field of finite characteristic. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 67–109. Zbl0246.14007MR316456
25. B. Poonen, Computational aspects of curves of genus at least $2$. Algorithmic number theory (Talence, 1996), Lecture Notes in Comput. Sci., vol. 1122, Springer, Berlin, 1996, pp. 283–306. Zbl0891.11037MR1446520
26. D. Poulakis, Points entiers et modèles des courbes algébriques. Monatsh. Math. 118 (1994), no. 1-2, 111–143. Zbl0811.11023MR1289852
27. G. Rémond, Hauteurs thêta et construction de Kodaira. J. Number Theory 78 (1999), no. 2, 287–311. Zbl0947.14016MR1713465
28. J.-P. Serre and J. Tate, Good reduction of abelian varieties. Ann. of Math. (2) 88 (1968), 492–517. Zbl0172.46101MR236190
29. I. R. Shafarevich, Algebraic number fields. Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 163–176. Zbl0133.29303MR202709
30. C.L. Siegel, Über einege Anwendungen Diophantischer Approximationen. Abh. Preuss. Akad. Wiss. Phys. Math. Kl. (1929), 41–69. 
31. J. H. Silverman, Integral points on abelian varieties. Invent. Math. 81 (1985), no. 2, 341–346. Zbl0576.14041MR799270
32. J. H. Silverman, Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. Zbl0911.14015MR1312368
33. N. P. Smart, $S$-unit equations, binary forms and curves of genus $2$. Proc. London Math. Soc. (3) 75 (1997), no. 2, 271–307. Zbl0885.11031MR1455857
34. U. Zannier, Roth’s theorem, integral points and certain ramified covers of ${\mathbb{P}}_1$. Analytic number theory, Cambridge Univ. Press, Cambridge, 2009, pp. 471–491. Zbl1231.11069MR2508664

1. Rafael von Känel, An effective proof of the hyperelliptic Shafarevich conjecture