CALCULATION OF THE FUNDAMENTAL S-UNITS IN HYPERELLIPTIC FIELDS OF GENUS 2 AND THE TORSION PROBLEM IN THE JACOBIANS OF HYPERELLIPTIC CURVES

A new approach to the torsion problem in the Jacobians of hyperelliptic curves over the field of rational numbers was offered by Platonov. This new approach is based on the calculation of fundamental units in hyperelliptic fields. The existence of torsion points of new orders was proved with the help of this approach. The full details of the new method and related results are contained in [2]. Platonov conjectured that if we consider the S consisting of finite and infinite valuation and change accordingly definition of the degree of S-unit, the orders of torsion Q-points tend to be determined by the degree of fundamental S-units. The main result of this article is the proof of existence of the fundamental S-units of large degrees. The proof is based on the methods of continued fractions and matrix linearization based on Platonov’s approach. Efficient algorithms for computing S-units using method of continued fractions have been developed. Improved algorithms have allowed to construct the above-mentioned fundamental S-units of large degrees. As a corollary, alternative proof of the existence of torsion Q-points of some large orders in corresponding Jacobians of hyperelliptic curves was obtained.

1. Platonov, V. P. 2010, "Arithmetic of quadratic fields and torsion in Jacobians" , Dokl. Math, 81, 1, pp. 55–57

2. Platonov, V. P. 2014, "Number-theoretic properties of hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves over the rational number field" , Russian Math. Surveys 69, 1, pp. 1–34.

3. Platonov, V. P. & Petrunin, M. M. 2012, "New orders of torsion points in Jacobians of curves of genus 2 over the rational number field" , Dokl. Math, 85, 2, pp. 286–288.

4. Platonov, V. P. & Petrunin, M. M. 2012, "On the torsion problem in jacobians of curves of genus 2 over the rational number field" , Dokl. Math, 86, 2, pp. 642– 643.

5. Platonov, V. P. & Petrunin, M. M. 2015, "Fundamental S-units in hyperelliptic fields and the torsion problem in jacobians of hyperelliptic curves" , Dokl. Math, 92, 3, pp. 23–25.

6. Benyash-Krivets, V. V. & Platonov, V. P. 2009, "Groups of S-units in hyperelliptic fields and continued fractions" , Sb. Math, 200, 11, pp. 1587-1615.

7. Lang, S. 1966, "Introduction to diophantine approximations" , Reading, Mass., Addison-Wesley Pub. Co.

8. Flynn, E. V. 1990. "Large rational torsion on abelian varieties" , J. Number Theory, pp. 257–265.

9. Leprevost, F. 1991. "Famille de courbes de genre 2 munies dune classe de diviseurs rationnels dordre 13" , C. R. Acad. Sci. Paris Ser. I Math, 313, 7, pp. 451–454.

10. Leprevost, F. 1991b. Familles de courbes de genre 2 munies dune classe de diviseurs rationnels d’ordre. C. R. Acad. Sci. Paris Ser. I Math, 313, 11, pp. 771– 774.

11. Leprevost, F. 1993. "Points rationnels de torsion de jacobiennes de certaines courbes de genre 2" , C. R. Acad. Sci. Paris, 316, 8, pp. 819–821.

12. Ogawa, H. 1994. "Curves of genus 2 with a rational torsion divisor of order 23" , Proc. Japan Acad. Ser. A Math. Sci, 70, 9, pp. 295–298.

13. Leprevost, F. 1995. "Jacobiennes de certaines courbes de genre 2: torsion et simplicite" , Journal de Theorie des Nombres de Bordeaux, 7, 1, pp. 283–306.

14. W. S. Cassels, E. V. F. 1996. "Prolegomena to a middlebrow arithmetic of curves of genus 2" , Cambridge Univ. Press.

15. E. W. Howe, F. L. and Poonen, B. 2000. "Large torsion subgroups of split Jacobians of curves of genus two or three" , Forum Mathematicum, 12, pp. 315– 364.

16. Nicolas, B. et al. 2009. "Jacobians of genus-2 curves with a rational point of order 11" , Experiment. Math, 18, 1, pp. 65–70.

17. Elkies, N. D. 2010. "Curves of genus 2 over Q whose Jacobians are absolutely simple abelian surfaces with torsion points of high order" , preprint Harvard University.

18. Howe, E. W. 2015. "Genus-2 Jacobians with torsion points of large order" , Bulletin of the London Mathematical Society, 47, 1, pp. 127–135.

19. Hart W., Van Hoeij M.& Novocin A. 2011. "Practical polynomial factoring in polynomial time" , Proceedings of the 36th international symposium on Symbolic and algebraic computation, ACM, pp. 163–170.