An effective proof of the hyperelliptic Shafarevich conjecture
Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 507-530.

Soit C une courbe hyperelliptique de genre g1 sur un corps de nombres K avec bonne réduction en dehors d’un ensemble fini S de places de K. Nous démontrons que C possède un modèle de Weierstrass sur l’anneau des entiers de K avec hauteur effectivement bornée en termes de g, S et K. En particulier, nous démontrons que pour tout corps de nombres K, tout ensemble fini S de places de K et tout entier g1, on peut déterminer en principe l’ensemble des classes d’isomorphisme de courbes hyperelliptiques de genre g sur K avec bonne réduction en dehors de S.

Let C be a hyperelliptic curve of genus g1 over a number field K with good reduction outside a finite set of places S of K. We prove that C has a Weierstrass model over the ring of integers of K with height effectively bounded only in terms of g, S and K. In particular, we obtain that for any given number field K, finite set of places S of K and integer g1 one can in principle determine the set of K-isomorphism classes of hyperelliptic curves over K of genus g with good reduction outside S.

DOI : 10.5802/jtnb.877
von Känel, Rafael 1

1 IHÉS 35 Route de Chartres 91440 Bures-sur-Yvette France
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von Känel, Rafael. An effective proof of the hyperelliptic Shafarevich conjecture. Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 507-530. doi : 10.5802/jtnb.877. http://www.numdam.org/articles/10.5802/jtnb.877/

[1] A. Baker and G. Wüstholz, Logarithmic forms and Diophantine geometry, New Mathematical Monographs, 9, Cambridge University Press, Cambridge, (2007). | MR | Zbl

[2] A. Bérczes, J.-H. Evertse, and K. Győry, Diophantine problems related to discriminants and resultants of binary forms, Diophantine geometry, CRM Series, 4, Ed. Norm., Pisa (2007), 45–63. | MR | Zbl

[3] E. Bombieri and W. Gubler, Heights in Diophantine geometry, New Mathematical Monographs, 4, Cambridge University Press, Cambridge, (2006). | MR | Zbl

[4] A. Brumer and J. H. Silverman, The number of elliptic curves over Q with conductor N, Manuscripta Math. 91, 1, (1996), 95–102. | MR | Zbl

[5] J. Coates, An effective p-adic analogue of a theorem of Thue. III. The diophantine equation y 2 =x 3 +k, Acta Arith., 16 (1969/1970), 425–435. | MR | Zbl

[6] R. de Jong and G. Rémond, Conjecture de Shafarevitch effective pour les revêtements cycliques, Algebra Number Theory 5,8 (2011), 1133–1143. | MR | Zbl

[7] J.-H. Evertse, On equations in S-units and the Thue-Mahler equation, Invent. Math. 75, 3 (1984), 561–584. | MR | Zbl

[8] J.-H. Evertse and K. Győry, Effective finiteness results for binary forms with given discriminant, Compositio Math., 79, 2 (1991), 169–204. | Numdam | MR | Zbl

[9] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73,3 (1983), 349–366. | MR | Zbl

[10] C. Fuchs, R. von Känel, and G. Wüstholz, An effective Shafarevich theorem for elliptic curves, Acta Arith., 148, 2 (2011), 189–203. | MR | Zbl

[11] K. Győry, Effective finiteness theorems for polynomials with given discriminant and integral elements with given discriminant over finitely domains, J. Reine Angew. Math., 346 (1984), 54–100. | MR | Zbl

[12] —, Polynomials and binary forms with given discriminant, Publ. Math. Debrecen, 69, 4, (2006), 473–499. | MR

[13] K. Győry and K. Yu, Bounds for the solutions of S-unit equations and decomposable form equations, Acta Arith., 123, 1 (2006), 9–41. | MR | Zbl

[14] H. A. Helfgott and A. Venkatesh, Integral points on elliptic curves and 3-torsion in class groups, J. Amer. Math. Soc., 19, 3 (2006), 527–550. | MR | Zbl

[15] H. W. Lenstra, Jr., Algorithms in algebraic number theory, Bull. Amer. Math. Soc. (N.S.), 26,2 (1992), 211–244. | MR | Zbl

[16] A. Levin, Siegel’s theorem and the Shafarevich conjecture, J. Théor. Nombres Bordeaux, 24, 3 (2012), 705–727. | Numdam | MR | Zbl

[17] Q. Liu, Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète, Trans. Amer. Math. Soc., 348, 11 (1996), 4577–4610. | MR | Zbl

[18] —, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, 6, Oxford University Press, Oxford, (2002), Oxford Science Publications. | MR | Zbl

[19] P. Lockhart, On the discriminant of a hyperelliptic curve, Trans. Amer., Math. Soc., 342, 2 (1994), 729–752. | MR | Zbl

[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, 2 (1993), 203–214, Corrigenda: [21]. | MR | Zbl

[21] —, Corrigenda: “Curves of genus 2 with good reduction away from 2 with a rational Weierstrass point”, Math. Proc. Cambridge Philos. Soc., 118, 1 (1995), 189. | MR

[22] F. Oort, Hyperelliptic curves over number fields, Classification of algebraic varieties and compact complex manifolds, Springer, Berlin, 412, (1974) 211–218. Lecture Notes in Math. | MR | Zbl

[23] A. N. Paršin, 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 | Zbl

[24] F. Pazuki, Theta height and Faltings height, Bull. Soc. Math. France, 140, 1 (2012), 19–49. | Numdam | MR | Zbl

[25] B. Poonen, Computational aspects of curves of genus at least 2, Algorithmic number theory (Talence, 1996), Lecture Notes in Comput. Sci., Springer, Berlin, 1122, (1996) pp. 283–306. | MR | Zbl

[26] G. Rémond, Hauteurs thêta et construction de Kodaira, J. Number Theory, 78, 2 (1999), 287–311. | MR | Zbl

[27] —, Nombre de points rationnels des courbes, Proc. Lond. Math. Soc. (3), 101, 3 (2010), 759–794. | MR

[28] T. Saito, Conductor, discriminant, and the Noether formula of arithmetic surfaces, Duke Math. J., 57, 1 (1988), 151–173. | MR | Zbl

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

[30] J. H. Silverman, The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, 106, Springer, Dordrecht, (2009). | MR | Zbl

[31] N. P. Smart, S-unit equations, binary forms and curves of genus 2, Proc. London Math. Soc. (3) 75, 2 (1997), 271–307. | MR | Zbl

[32] H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math., 23, (1974), 135–152. | MR | Zbl

[33] R. von Känel, On Szpiro’s Discriminant Conjecture, Internat. Math. Res. Notices, (2013), 1–35, Available online: doi: 10.1093/imrn/rnt079. | MR

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