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ALGEBRAIC DIVISIBILITY SEQUENCES OVER FUNCTION FIELDS

Part of: Sequences and sets

Published online by Cambridge University Press:  24 April 2012

PATRICK INGRAM ,
VALÉRY MAHÉ ,
JOSEPH H. SILVERMAN ,
KATHERINE E. STANGE  and
MARCO STRENG
PATRICK INGRAM
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, CO 80521, USA (email: pingram@math.colostate.edu)
VALÉRY MAHÉ
Affiliation:
EPF Lausanne, SB-IMB-CSAG, Station 8, CH-1015 Lausanne, Switzerland (email: valery.mahe@epfl.ch)
JOSEPH H. SILVERMAN*
Affiliation:
Mathematics Department, Brown University, Box 1917, Providence, RI 02912, USA (email: jhs@math.brown.edu)
KATHERINE E. STANGE
Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, CA 94305, USA (email: stange@math.stanford.edu)
MARCO STRENG
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: marco.streng@gmail.com)
*
✉For correspondence; e-mail: jhs@math.brown.edu
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Abstract

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In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.

Keywords

lucas sequenceelliptic divisibility sequenceprimitive divisorfunction field over number field

MSC classification

Secondary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 1 , February 2012 , pp. 99 - 126
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

Ingram’s research was supported by a grant from NSERC of Canada. Mahé’s research was supported by the Université de Franche-Comté. Silverman’s research was supported by DMS-0854755. Stange’s research was supported by NSERC PDF-373333 and NSF MSPRF 0802915. Streng’s research was supported by EPSRC grant no. EP/G004870/1.

References

[1]Ayad, M., ‘Points S-entiers des courbes elliptiques’, Manuscripta Math. 76(3–4) (1992), 305–324.CrossRefGoogle Scholar
[2]Bang, A. S., ‘Taltheoretiske undersøgelser’, Tidskrift f. Math. 5 (1886), 70–80 and 130–137.Google Scholar
[3]Bilu, Yu., Hanrot, G. and Voutier, P. M., ‘Existence of primitive divisors of Lucas and Lehmer numbers’, J. reine angew. Math. 539 (2001), 75–122 With an appendix by M. Mignotte.Google Scholar
[4]Carmichael, R. D., ‘On the numerical factors of the arithmetic forms α n±β n’, Ann. of Math. (2) 15(1–4) (1913/14), 30–70.CrossRefGoogle Scholar
[5]Chudnovsky, D. V. and Chudnovsky, G. V., ‘Sequences of numbers generated by addition in formal groups and new primality and factorization tests’, Adv. Appl. Math. 7(4) (1986), 385–434.CrossRefGoogle Scholar
[6]Cornelissen, G. and Zahidi, K., ‘Elliptic divisibility sequences and undecidable problems about rational points’, J. reine angew. Math. 613 (2007), 1–33.CrossRefGoogle Scholar
[7]Corrales-Rodrigáñez, C. and Schoof, R., ‘The support problem and its elliptic analogue’, J. Number Theory 64(2) (1997), 276–290.CrossRefGoogle Scholar
[8]Dubner, H. and Keller, W., ‘New Fibonacci and Lucas primes’, Math. Comp. 68(225) (1999), 417–427, S1–S12.CrossRefGoogle Scholar
[9]Edixhoven, B., ‘Rational torsion points on elliptic curves over number fields (after Kamienny and Mazur)’, Astérisque (227) (1995), Exp. No. 782, 4, 209–227. Séminaire Bourbaki, Vol. 1993/94.Google Scholar
[10]Einsiedler, M., Everest, G. and Ward, T., ‘Primes in elliptic divisibility sequences’, LMS J. Comput. Math. 4 (2001), 1–13 (electronic).CrossRefGoogle Scholar
[11]Eisenträger, K. and Everest, G., ‘Descent on elliptic curves and Hilbert’s tenth problem’, Proc. Amer. Math. Soc. 137(6) (2009), 1951–1959.CrossRefGoogle Scholar
[12]Elkies, N. D., ‘Distribution of supersingular primes’, Astérisque (198–200) (1992), 127–132.Google Scholar
[13]Everest, G., Ingram, P., Mahé, V. and Stevens, S., ‘The uniform primality conjecture for elliptic curves’, Acta Arith. 134(2) (2008), 157–181.CrossRefGoogle Scholar
[14]Everest, G. and King, H., ‘Prime powers in elliptic divisibility sequences’, Math. Comp. 74(252) (2005), 2061–2071 (electronic).CrossRefGoogle Scholar
[15]Everest, G., Mclaren, G. and Ward, T., ‘Primitive divisors of elliptic divisibility sequences’, J. Number Theory 118(1) (2006), 71–89.CrossRefGoogle Scholar
[16]Everest, G., Miller, V. and Stephens, N., ‘Primes generated by elliptic curves’, Proc. Amer. Math. Soc. 132(4) (2004), 955–963 (electronic).CrossRefGoogle Scholar
[17]Everest, G. and Ward, T., ‘Primes in divisibility sequences’, Cubo Mat. Educ. 3(2) (2001), 245–259.Google Scholar
[18]Flatters, A. and Ward, T., ‘Polynomial Zsigmondy theorems’, J. Algebra. 343 (2011), 138–142.CrossRefGoogle Scholar
[19] Great internet Mersenne prime search, Mersenne Research Inc. http://mersenne.org.Google Scholar
[20]Hindry, M. and Silverman, J. H., ‘The canonical height and integral points on elliptic curves’, Invent. Math. 93(2) (1988), 419–450.CrossRefGoogle Scholar
[21]Ingram, P., ‘Elliptic divisibility sequences over certain curves’, J. Number Theory 123(2) (2007), 473–486.CrossRefGoogle Scholar
[22]Ingram, P., ‘A quantitative primitive divisor result for points on elliptic curves’, J. Théor. Nombres Bordeaux 21(3) (2009), 609–634.CrossRefGoogle Scholar
[23]Ingram, P. and Silverman, J. H., ‘Uniform estimates for primitive divisors in elliptic divisibility sequences’, in: Number Theory, Analysis and Geometry (In Memory of Serge Lang) (Springer, 2011), pp. 233–263.Google Scholar
[24]Lauter, K. and Stange, K. E., ‘The elliptic curve discrete logarithm problem and equivalent hard problems for elliptic divisibility sequences’, in: Selected Areas in Cryptography 2008, Lecture Notes in Computer Science, 5381 (Springer, Berlin, 2009), pp. 309–327.CrossRefGoogle Scholar
[25]Luca, F. and Stănică, P., ‘Prime divisors of Lucas sequences and a conjecture of Skałba’, Int. J. Number Theory 1(4) (2005), 583–591.CrossRefGoogle Scholar
[26]Mahé, V., ‘Prime power terms in elliptic divisibility sequences’, Preprint, January 2010.Google Scholar
[27]Neukirch, J., Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 322 (Springer, Berlin, 1999), translated from the 1992 German original and with a note by Norbert Schappacher, with a foreword by G. Harder.CrossRefGoogle Scholar
[28]Parent, P., ‘Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres’, J. reine angew. Math. 506 (1999), 85–116.CrossRefGoogle Scholar
[29]Poonen, B., ‘Hilbert’s tenth problem and Mazur’s conjecture for large subrings of ℚ’, J. Amer. Math. Soc. 16(4) (2003), 981–990 (electronic).CrossRefGoogle Scholar
[30]Schinzel, A., ‘Primitive divisors of the expression A n−B n in algebraic number fields’, J. reine angew. Math. 268/269 (1974), 27–33. Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II.Google Scholar
[31]Seres, I., ‘Über die Irreduzibilität gewisser Polynome’, Acta Arith. 8 (1962/63), 321–341.CrossRefGoogle Scholar
[32]Seres, I., ‘Irreducibility of polynomials’, J. Algebra 2 (1965), 283–286.CrossRefGoogle Scholar
[33]Serre, J.-P., ‘Propriétés galoisiennes des points d’ordre fini des courbes elliptiques’, Invent. Math. 15(4) (1972), 259–331.CrossRefGoogle Scholar
[34]Serre, J.-P., ‘Quelques applications du théorème de densité de Chebotarev’, Publ. Math. Inst. Hautes Études Sci. 54 (1981), 323–401.CrossRefGoogle Scholar
[35]Serre, J.-P., Abelian l-adic Representations and Elliptic Curves, Research Notes in Mathematics, 7 (A K Peters Ltd., Wellesley, MA, 1998).Google Scholar
[36]Shipsey, R., ‘Elliptic divisibility sequences’, PhD Thesis, Goldsmith’s College, University of London, 2000.Google Scholar
[37]Silverman, J. H., ‘Wieferich’s criterion and the abc-conjecture’, J. Number Theory 30(2) (1988), 226–237.CrossRefGoogle Scholar
[38]Silverman, J. H., ‘The difference between the Weil height and the canonical height on elliptic curves’, Math. Comp. 55(192) (1990), 723–743.CrossRefGoogle Scholar
[39]Silverman, J. H., Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 151 (Springer, New York, 1994).CrossRefGoogle Scholar
[40]Silverman, J. H., ‘Common divisors of elliptic divisibility sequences over function fields’, Manuscripta Math. 114(4) (2004), 431–446.CrossRefGoogle Scholar
[41]Silverman, J. H., ‘p-adic properties of division polynomials and elliptic divisibility sequences’, Math. Ann. 332(2) (2005), 443–471, addendum 473–474.CrossRefGoogle Scholar
[42]Silverman, J. H., The Arithmetic of Elliptic Curves, 2nd edn. Graduate Texts in Mathematics, 106 (Springer, Dordrecht, 2009).CrossRefGoogle Scholar
[43]Silverman, J. H. and Stange, K. E., ‘Terms in elliptic divisibility sequences divisible by their indices’, Acta Arith. 146(4) (2011), 355–378.CrossRefGoogle Scholar
[44]Silverman, J. H. and Stephens, N., ‘The sign of an elliptic divisibility sequence’, J. Ramanujan Math. Soc. 21(1) (2006), 1–17.Google Scholar
[45]Stange, K. E., ‘The Tate pairing via elliptic nets’, in: Pairing-based Cryptography—Pairing 2007, Lecture Notes in Computer Science, 4575 (Springer, Berlin, 2007), pp. 329–348.CrossRefGoogle Scholar
[46]Stange, K. E., ‘Elliptic nets and elliptic curves’, Algebra Number Theory 5(2) (2011), 197–229.CrossRefGoogle Scholar
[47]Stein, W. A.et al., Sage Mathematics Software (Version 4.6.2). The Sage Development Team. http://www.sagemath.org, 2011.Google Scholar
[48]Stewart, C. L., ‘Primitive divisors of Lucas and Lehmer numbers’, in: Transcendence Theory: Advances and Applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976) (Academic Press, London, 1977), pp. 79–92.Google Scholar
[49]Streng, M., ‘Divisibility sequences for elliptic curves with complex multiplication’, Algebra Number Theory 2(2) (2008), 183–208.CrossRefGoogle Scholar
[50]Voutier, P. M., ‘Primitive divisors of Lucas and Lehmer sequences’, Math. Comp. 64(210) (1995), 869–888.CrossRefGoogle Scholar
[51]Voutier, P. M. and Yabuta, M., ‘Primitive divisors of certain elliptic divisibility sequences’, arXiv:1009.0872, 2010.Google Scholar
[52]Wagstaff, S. S. Jr., ‘Divisors of Mersenne numbers’, Math. Comp. 40(161) (1983), 385–397.CrossRefGoogle Scholar
[53]Ward, M., ‘The law of repetition of primes in an elliptic divisibility sequence’, Duke Math. J. 15 (1948), 941–946.CrossRefGoogle Scholar
[54]Ward, M., ‘Memoir on elliptic divisibility sequences’, Amer. J. Math. 70 (1948), 31–74.CrossRefGoogle Scholar
[55]Watson, G. N., ‘The problem of the square pyramid’, Messenger of Math. 48 (1918), 1–22.Google Scholar
[56]Zimmer, H. G., ‘On the difference of the Weil height and the Néron-Tate height’, Math. Z. 147(1) (1976), 35–51.CrossRefGoogle Scholar
[57]Zsigmondy, K., ‘Zur Theorie der Potenzreste’, Monatsh. Math. Phys. 3(1) (1892), 265–284.CrossRefGoogle Scholar