It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field and any finite set of places of , one can effectively compute the set of isomorphism classes of hyperelliptic curves over with good reduction outside . We show here that an extension of this result to an effective Shafarevich conjecture for Jacobians of hyperelliptic curves of genus would imply an effective version of Siegel’s theorem for integral points on hyperelliptic curves of genus .
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 et tout ensemble fini de places de , on peut effectivement calculer l’ensemble des classes d’isomorphisme des courbes hyperelliptiques sur ayant bonne réduction en dehors de . 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 impliquerait une version effective du théorème de Siegel pour les points entiers sur les courbes hyperelliptiques de genre .
@article{JTNB_2012__24_3_705_0, author = {Levin, Aaron}, title = {Siegel{\textquoteright}s theorem and the {Shafarevich} conjecture}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {705--727}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {24}, number = {3}, year = {2012}, doi = {10.5802/jtnb.818}, zbl = {1271.11065}, mrnumber = {3010636}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.818/} }
TY - JOUR AU - Levin, Aaron TI - Siegel’s theorem and the Shafarevich conjecture JO - Journal de théorie des nombres de Bordeaux PY - 2012 SP - 705 EP - 727 VL - 24 IS - 3 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.818/ DO - 10.5802/jtnb.818 LA - en ID - JTNB_2012__24_3_705_0 ER -
%0 Journal Article %A Levin, Aaron %T Siegel’s theorem and the Shafarevich conjecture %J Journal de théorie des nombres de Bordeaux %D 2012 %P 705-727 %V 24 %N 3 %I Société Arithmétique de Bordeaux %U http://archive.numdam.org/articles/10.5802/jtnb.818/ %R 10.5802/jtnb.818 %G en %F 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://archive.numdam.org/articles/10.5802/jtnb.818/
[1] A. Baker, Bounds for the solutions of the hyperelliptic equation. Proc. Cambridge Philos. Soc. 65 (1969), 439–444. | MR | Zbl
[2] A. Baker, Transcendental number theory. Second ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. | MR | Zbl
[3] A. Baker and J. Coates, Integer points on curves of genus 1. Proc. Cambridge Philos. Soc. 67 (1970), 595–602. | MR | Zbl
[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 | Zbl
[5] Yu. Bilu, Effective analysis of integral points on algebraic curves. Israel J. Math. 90 (1995), no. 1-3, 235–252. | MR | Zbl
[6] Yu. Bilu, Quantitative Siegel’s theorem for Galois coverings. Compositio Math. 106 (1997), no. 2, 125–158. | MR | Zbl
[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 | Zbl
[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
[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 | Zbl
[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 | Zbl
[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 | Zbl
[12] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73 (1983), no. 3, 349–366. | MR | Zbl
[13] D. Grant, Integer points on curves of genus two and their Jacobians. Trans. Amer. Math. Soc. 344 (1994), no. 1, 79–100. | MR | Zbl
[14] M. Hindry and J. H. Silverman, Diophantine geometry. Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000, An introduction. | MR | Zbl
[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 . J. Reine Angew. Math. 286/287 (1976), 124–131. | MR | Zbl
[17] Q. Liu, Modèles minimaux des courbes de genre deux. J. Reine Angew. Math. 453 (1994), 137–164. | MR | Zbl
[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 | Zbl
[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 with good reduction away from with a rational Weierstrass point. Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 2, 203–214. | MR | Zbl
[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 | Zbl
[22] A. P. Ogg, On pencils of curves of genus two. Topology 5 (1966), 355–362. | MR | Zbl
[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 | Zbl
[24] A. N. Parshin, Minimal models of curves of genus . and homomorphisms of abelian varieties defined over a field of finite characteristic. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 67–109. | MR | Zbl
[25] B. Poonen, Computational aspects of curves of genus at least . Algorithmic number theory (Talence, 1996), Lecture Notes in Comput. Sci., vol. 1122, Springer, Berlin, 1996, pp. 283–306. | MR | Zbl
[26] D. Poulakis, Points entiers et modèles des courbes algébriques. Monatsh. Math. 118 (1994), no. 1-2, 111–143. | MR | Zbl
[27] G. Rémond, Hauteurs thêta et construction de Kodaira. J. Number Theory 78 (1999), no. 2, 287–311. | MR | Zbl
[28] J.-P. Serre and J. Tate, Good reduction of abelian varieties. Ann. of Math. (2) 88 (1968), 492–517. | MR | Zbl
[29] I. R. Shafarevich, Algebraic number fields. Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 163–176. | MR | Zbl
[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 | Zbl
[32] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. | MR | Zbl
[33] N. P. Smart, -unit equations, binary forms and curves of genus . Proc. London Math. Soc. (3) 75 (1997), no. 2, 271–307. | MR | Zbl
[34] U. Zannier, Roth’s theorem, integral points and certain ramified covers of . Analytic number theory, Cambridge Univ. Press, Cambridge, 2009, pp. 471–491. | MR | Zbl
Cited by Sources: