Number-theoretic properties of hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves over the rational number field

V. P. Platonov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 1, pp. 1-34

Résumé

In the past four years a theory has been developed for finding fundamental units in hyperelliptic fields, and on basis of this theory innovative and efficient algorithms for computing them have been constructed and implemented. A new local-global principle was discovered which gives a criterion for the existence of non-trivial units in hyperelliptic fields. The natural connection between the problem of computing fundamental units and the problem of torsion in Jacobian varieties of hyperelliptic curves over the rational number field has led to breakthrough results in the solution of this problem. The main results in the present survey were largely obtained using a symbiosis of deep theory, efficient algorithms, and supercomputing. Such a symbiosis will play an ever increasing role in the mathematics of the 21st century. Bibliography: 27 titles.

Détail

Keywords: fundamental units, hyperelliptic fields, local-global principle, Jacobian varieties, hyperelliptic curves, fast algorithms, continued fractions.
Mots-clés : torsion problem in Jacobians

Bibliographie

[1] J.-P. Serre, Groupes algébriques et corps de classes, Publ. de l'Institut de mathématique de l'Université de Nancago, VII, Hermann, Paris, 1959, 202 pp. | MR | Zbl | Zbl

[2] S. Lang, Abelian varieties, Reprint of the 1959 original, Springer-Verlag, New York–Berlin, 1983, xii+256 pp. | MR | Zbl

[3] F. Leprévost, “Famille de courbes de genre 2 munies d'une classe de diviseurs rationnels d'ordre 13”, C. R. Acad. Sci. Paris Sér. I Math., 313:7 (1991), 451–454 | MR | Zbl

[4] F. Leprévost, “Familles de courbes de genre 2 munies d'une classe de diviseurs rationnels d'ordre 15, 17, 19 ou 21”, C. R. Acad. Sci. Paris Sér. I Math., 313:11 (1991), 771–774 | MR | Zbl

[5] F. Leprévost, “Points rationnels de torsion de jacobiennes de certaines courbes de genre 2”, C. R. Acad. Sci. Paris Sér. I Math., 316:8 (1993), 819–821 | MR | Zbl

[6] F. Leprévost, “Jacobiennes de certaines courbes de genre 2: torsion et simplicité”, J. Théor. Nombres Bordeaux, 7:1 (1995), 283–306 | DOI | MR | Zbl

[7] E. W. Howe, F. Leprévost, B. Poonen, “Large torsion subgroups of split Jacobians of curves of genus two or three”, Forum Math., 12:3 (2000), 315–364 | DOI | MR | Zbl

[8] H. Ogawa, “Curves of genus 2 with a rational torsion divisor of order 23”, Proc. Japan Acad. Ser. A Math. Sci., 70:9 (1994), 295–298 | DOI | MR | Zbl

[9] N. D. Elkies, Curves of genus 2 over Q whose Jacobians are absolutely simple abelian surfaces with torsion points of high order, Preprint, Harvard Univ., 2010

[10] E. V. Flynn, “Large rational torsion on Abelian varieties”, J. Number Theory, 36:3 (1990), 257–265 | DOI | MR | Zbl

[11] J. W. S. Cassels, E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus 2, London Math. Soc. Lecture Note Ser., 230, Cambridge Univ. Press, Cambridge, 1996, xiv+219 pp. | DOI | MR | Zbl

[12] N. Bernard, F. Leprévost, M. Pohst, “Jacobians of genus-2 curves with a rational point of order 11”, Experiment. Math., 18:1 (2009), 65–70 | DOI | MR | Zbl

[13] V. P. Platonov, “Arithmetic of quadratic fields and torsion in Jacobians”, Dokl. Math., 81:1 (2010), 55–57 | DOI | MR | Zbl

[14] V. V. Benyash-Krivets, V. P. Platonov, “A new local-global principle for quadratic function fields”, Dokl. Math., 82:1 (2010), 531–534 | DOI | Zbl

[15] V. P. Platonov, M. M. Petrunin, “New orders of torsion points in Jacobians of curves of genus 2 over the rational number field”, Dokl. Math., 85:2 (2012), 286–288 | DOI | MR | Zbl

[16] V. P. Platonov, M. M. Petrunin, “On the torsion problem in Jacobians of curves of genus 2 over the rational number field”, Dokl. Math., 86:2 (2012), 642–643 | DOI | MR | Zbl

[17] V. V. Benyash-Krivets, V. P. Platonov, “$S$-units in hyperelliptic fields”, Russian Math. Surveys, 62:4 (2007), 784–786 | DOI | DOI | MR | Zbl

[18] V. V. Benyash-Krivets, V. P. Platonov, “Continued fractions and $S$-units in hyperelliptic fields”, Russian Math. Surveys, 63:2 (2008), 357–359 | DOI | DOI | MR | Zbl

[19] V. V. Benyash-Krivets, V. P. Platonov, “Groups of $S$-units in hyperelliptic fields”, Dokl. Math., 76:3 (2007), 886–890 | DOI | MR | Zbl

[20] V. V. Benyash-Krivets, V. P. Platonov, “Continued fractions and $S$-units in function fields”, Dokl. Math., 78:3 (2008), 833–838 | DOI | MR | Zbl

[21] V. V. Benyash-Krivets, V. P. Platonov, “Groups of $S$-units in hyperelliptic fields and continued fractions”, Sb. Math., 200:11 (2009), 1587–1615 | DOI | DOI | MR | Zbl

[22] E. Artin, “Quadratische Körper im Gebiete der höheren Kongruenzen. I”, Math. Z., 19:1 (1924), 153–206 | DOI | MR | Zbl

[23] V. P. Platonov, V. S. Zhgun, M. M. Petrunin, “On the simplicity of Jacobians for hyperelliptic curves of genus 2 over the field of rational numbers with torsion points of high order”, Dokl. Math., 87:3 (2013), 318–321 | DOI | DOI | Zbl

[24] W. W. Adams, M. J. Razar, “Multiples of points on elliptic curves and continued fractions”, Proc. Lond. Math. Soc. (3), 41:3 (1980), 481–498 | DOI | MR | Zbl

[25] B. Mazur, “Rational isogenies of prime degree”, with an appendix by D. Goldfeld, Invent. Math., 44:2 (1978), 129–162 | DOI | MR | Zbl

[26] L. Merel, “Borne pour la torsion des courbes elliptiques sur les cops de nombres”, Invent. Math., 124:1-3 (1996), 437–449 | DOI | MR | Zbl

[27] S. Lang, Introduction to diophantine approximations, Addison-Wesley Publishing Co., Reading, MA–London–Don Mills, ON, 1966, viii+83 pp. | MR | Zbl | Zbl