Siegel’s theorem and the Shafarevich conjecture
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, p. 705-727

It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field k and any finite set of places S of k, one can effectively compute the set of isomorphism classes of hyperelliptic curves over k with good reduction outside S. We show here that an extension of this result to an effective Shafarevich conjecture for Jacobians of hyperelliptic curves of genus g would imply an effective version of Siegel’s theorem for integral points on hyperelliptic curves of genus g.

Il est connu que dans le cas des courbes hyperelliptiques la conjecture de Shafarevich peut être rendue effective, c’est à dire, pour tout corps de nombres k et tout ensemble fini de places S de k, on peut effectivement calculer l’ensemble des classes d’isomorphisme des courbes hyperelliptiques sur k ayant bonne réduction en dehors de S. Nous montrons ici qu’une extension de ce résultat à une version effective de la conjecture de Shafarevich pour les Jacobiennes de courbes hyperelliptiques de genre g impliquerait une version effective du théorème de Siegel pour les points entiers sur les courbes hyperelliptiques de genre g.

@article{JTNB_2012__24_3_705_0,
     author = {Levin, Aaron},
     title = {Siegel's theorem and the Shafarevich conjecture},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {3},
     year = {2012},
     pages = {705-727},
     doi = {10.5802/jtnb.818},
     mrnumber = {3010636},
     zbl = {1271.11065},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2012__24_3_705_0}
}
Levin, Aaron. Siegel’s theorem and the Shafarevich conjecture. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 705-727. doi : 10.5802/jtnb.818. http://www.numdam.org/item/JTNB_2012__24_3_705_0/

[1] A. Baker, Bounds for the solutions of the hyperelliptic equation. Proc. Cambridge Philos. Soc. 65 (1969), 439–444. | MR 234912 | Zbl 0174.33803

[2] A. Baker, Transcendental number theory. Second ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. | MR 1074572 | Zbl 0715.11032

[3] A. Baker and J. Coates, Integer points on curves of genus 1. Proc. Cambridge Philos. Soc. 67 (1970), 595–602. | MR 256983 | Zbl 0194.07601

[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. | MR 984113 | Zbl 0649.10014 | Zbl 0669.10037

[5] Yu. Bilu, Effective analysis of integral points on algebraic curves. Israel J. Math. 90 (1995), no. 1-3, 235–252. | MR 1336325 | Zbl 0840.11028

[6] Yu. Bilu, Quantitative Siegel’s theorem for Galois coverings. Compositio Math. 106 (1997), no. 2, 125–158. | MR 1457336 | Zbl 1044.11593

[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. | MR 1045822 | Zbl 0705.14001

[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. | MR 404270

[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. | MR 399108 | Zbl 0322.14013

[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. | Numdam | MR 1310635 | Zbl 0819.12003

[11] J.-H. Evertse and K. Győry, Effective finiteness results for binary forms with given discriminant. Compositio Math. 79 (1991), no. 2, 169–204. | Numdam | MR 1117339 | Zbl 0746.11020

[12] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73 (1983), no. 3, 349–366. | MR 718935 | Zbl 0588.14026

[13] D. Grant, Integer points on curves of genus two and their Jacobians. Trans. Amer. Math. Soc. 344 (1994), no. 1, 79–100. | MR 1184116 | Zbl 0828.11032

[14] M. Hindry and J. H. Silverman, Diophantine geometry. Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000, An introduction. | MR 1745599 | Zbl 0948.11023

[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. | MR 417049 | Zbl 0332.10010

[17] Q. Liu, Modèles minimaux des courbes de genre deux. J. Reine Angew. Math. 453 (1994), 137–164. | MR 1285783 | Zbl 0805.14013

[18] Q. Liu, Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète. Trans. Amer. Math. Soc. 348 (1996), no. 11, 4577–4610. | MR 1363944 | Zbl 0926.11043

[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. | MR 1230127 | Zbl 0805.14018

[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. | MR 379510 | Zbl 0299.14018

[22] A. P. Ogg, On pencils of curves of genus two. Topology 5 (1966), 355–362. | MR 201437 | Zbl 0145.17802

[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. | MR 354676 | Zbl 0299.14017

[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. | MR 316456 | Zbl 0246.14007

[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. | MR 1446520 | Zbl 0891.11037

[26] D. Poulakis, Points entiers et modèles des courbes algébriques. Monatsh. Math. 118 (1994), no. 1-2, 111–143. | MR 1289852 | Zbl 0811.11023

[27] G. Rémond, Hauteurs thêta et construction de Kodaira. J. Number Theory 78 (1999), no. 2, 287–311. | MR 1713465 | Zbl 0947.14016

[28] J.-P. Serre and J. Tate, Good reduction of abelian varieties. Ann. of Math. (2) 88 (1968), 492–517. | MR 236190 | Zbl 0172.46101

[29] I. R. Shafarevich, Algebraic number fields. Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 163–176. | MR 202709 | Zbl 0133.29303

[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. | MR 799270 | Zbl 0576.14041

[32] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. | MR 1312368 | Zbl 0911.14015

[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. | MR 1455857 | Zbl 0885.11031

[34] U. Zannier, Roth’s theorem, integral points and certain ramified covers of 1 . Analytic number theory, Cambridge Univ. Press, Cambridge, 2009, pp. 471–491. | MR 2508664 | Zbl 1231.11069